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BERRY–ESSEEN BOUNDS FOR GENERALIZED UU STATISTICS

Zhuo-Song Zhang
(National University of Singapore)
Abstract

In this paper, we establish optimal Berry–Esseen bounds for the generalized UU-statistics. The proof is based on a new Berry–Esseen theorem for exchangeable pair approach by Stein’s method under a general linearity condition setting. As applications, an optimal convergence rate of the normal approximation for subgraph counts in Erdös–Rényi graphs and graphon-random graph is obtained.
MSC: Primnary 60F05; secondary 60K35.
Keywords: Generalized UU-statistics, Stein’s method, Exchangeable pair approach, Berry-Esseen bound, graphon-generated random graph, Erdös-Rényi model.

1 Introduction

Let X=(X1,,Xn)𝒳nX=(X_{1},\dots,X_{n})\in\mathcal{X}^{n} and Y=(Yi,j,1i<jn)𝒴n(n1)/2Y=(Y_{i,j},1\leqslant i<j\leqslant n)\in\mathcal{Y}^{n(n-1)/2} be two families of i.i.d.  random variables; moreover, XX and YY are also mutually independent and we set Yj,i=Yi,jY_{j,i}=Y_{i,j} for j>ij>i. For k1k\geqslant 1, let f:𝒳k×𝒴k(k1)/2f:\mathcal{X}^{k}\times\mathcal{Y}^{k(k-1)/2}\to\mathbb{R} be a function and we say ff is symmetric if the value of the function f(Xi1,,Xik;Yi1,i2,,Yik1,ik)f(X_{i_{1}},\dots,X_{i_{k}};Y_{i_{1},i_{2}},\dots,Y_{i_{k-1},i_{k}}) remains unchanged for any permutation of indices 1i1i2ikn1\leqslant i_{1}\neq i_{2}\neq\dots\neq i_{k}\leqslant n. In this paper, we consider the generalized UU-statistic defined by

Sn,k(f)=αn,kf(Xα(1),,Xα(k);Yα(1),α(2),,Yα(k1),α(k)),\displaystyle S_{n,k}(f)=\sum_{\alpha\in\mathcal{I}_{n,k}}f(X_{\alpha(1)},\dots,X_{\alpha(k)};Y_{\alpha(1),\alpha(2)},\dots,Y_{\alpha(k-1),\alpha(k)}), (1.1)

where for every 1\ell\geqslant 1 and nn\geqslant\ell,

n,={α=(α(1),,α()):1α(1)<<α()n}.\displaystyle\mathcal{I}_{n,\ell}=\{\alpha=(\alpha(1),\dots,\alpha(\ell)):1\leqslant\alpha(1)<\dots<\alpha(\ell)\leqslant n\}. (1.2)

We note that every αn,\alpha\in\mathcal{I}_{n,\ell} is an \ell-fold ordered index.

As a generalization of the classical UU-statistic, generalized UU-statistics have been widely applied in the random graph theory as a count random variable. Janson and Nowicki (1991) studied the limiting behavior of Sn,k(f)S_{n,k}(f) via a projection method. Specifically, the function ff can be represented as an orthogonal sum of terms indexed by subgraphs of the complete graph with kk vertices. Janson and Nowicki (1991) showed that the limiting behavior of Sn,k(f)S_{n,k}(f) depends on topology of the principle support graphs (see more details in Subsection 2.1) of ff. In particular, the random variable Sn,k(f)S_{n,k}(f) is asymptotically normally distributed if the principle support graphs are all connected. However, the convergence rate is still unknown.

The main purpose of this paper is to establish a Berry–Esseen bound for SnS_{n} by using Stein’s method. Stein’s method is a powerful tool to estimating convergence rates for distributional approximation. Since introduced by Stein (1972) in 1972, Stein’s method has shown to be a powerful tool to evaluate distributional distances for dependent random variables. One of the most important techniques in Stein’s method is the exchangeable pair approach, which is commonly taken in computing the Berry–Esseen bound for both normal and nonnormal approximations. We refer to Stein (1986); Rinott and Rotar (1997); Chatterjee and Shao (2011) and Shao and Zhang (2016) for more details on Berry–Esseen bound for bounded exchangeable pairs. It is worth mentioning that Shao and Zhang (2019) obtained a Berry–Esseen bound for unbounded exchangeable pairs.

Let WW be the random variable of interest, and we say (W,W)(W,W^{\prime}) is an exchangeable pair if (W,W)=d.(W,W)(W,W^{\prime})\stackrel{{\scriptstyle d.}}{{=}}(W^{\prime},W). For normal approximation, it is often to assume the following condition holds:

𝔼{WW|W}=λ(W+R),\displaystyle\mathbb{E}\{W-W^{\prime}|W\}=\lambda(W+R), (1.3)

where λ>0\lambda>0 and RR is a random variable with a small 𝔼|R|\mathop{{}\mathbb{E}}\mathopen{}|R|. The condition 1.3 can be understood as a linear regression condition. Although an exchangeable pair can be easily constructed, it may be not easy to verify the linearity condition 1.3 in some applications.

In this paper, we aim to establish an optimal Berry–Esseen bound for the generalized UU-statistics by developing a new Berry–Esseen theorem for exchangeable pair approach by assuming a more general condition than 1.3. More specifically, we replace WWW-W^{\prime} in 1.3 by a random variable DD that is an antisymmetric function of (X,X)(X,X^{\prime}). The new result is given in Section 4. There are several advantages of our result. Firstly, we propose a new condition more general than 1.3 that may be easy to verify. For instance, the condition can be verified by constructing an antisymmetric random variable by the Gibbs sampling method, embedding method, generalized perturbative approach and so on. Secondly, the Berry–Esseen bound often provides an optimal convergence rate for many practical applications.

The rest of this paper is organized as follows. In Section 2, we give the Berry–Esseen bounds for Sn,k(f)S_{n,k}(f). Applications to subgraph counts in κ\kappa-random graphs are given in Section 3. The new Berry–Esseen theorem for exchangeable pair approach under a new setting is established in Section 4. We give the proofs of our main results in Section 5. The proofs of other results are postponed to Section 6.

2 Main results

Let (X,Y)(X,Y), ff and Sn,k(f)S_{n,k}(f) be defined in Section 1. For any 1\ell\geqslant 1, []={1,,}[\ell]=\{1,\dots,\ell\} and []2={(i,j):1i,j}[\ell]_{2}=\{(i,j):1\leqslant i,j\leqslant\ell\}. Let A[]A\subset[\ell] and let B[]2B\subset[\ell]_{2}, and let XA=(Xi:iA)X_{A}=(X_{i}:i\in A) and YB=(Yi,j:(i,j)B)Y_{B}=(Y_{i,j}:(i,j)\in B). Specially, we can simply write f(X1,,Xk;Y1,2,,Yk1,k)f(X_{1},\dots,X_{k};Y_{1,2},\dots,Y_{k-1,k}) as f(X[k];Y[k]2)f(X_{[k]};Y_{[k]_{2}}). Let GA,BG_{A,B} be the graph with vertex set AA and edge set BB, and let vA,Bv_{A,B} be the number of nodes in GA,BG_{A,B}.

By the Hoeffding decomposition, we have

f(X[k];Y[k]2)=A[k],B[k]2fA,B(XA;YB),\displaystyle f(X_{[k]};Y_{[k]_{2}})=\sum_{A\subset[k],B\subset[k]_{2}}f_{A,B}(X_{A};Y_{B}),

where fA,B:𝒳|A|×𝒴|B|f_{A,B}:\mathcal{X}^{|A|}\times\mathcal{Y}^{|B|}\to\mathbb{R} is defined as

fA,B(xA;yB)=(A,B):AA,BB(1)|A|+|B||A||B|×𝔼{f(X1,,Xk;Y1,2,,Yk1,k)|XA=xA,YB=yB},f_{A,B}(x_{A};y_{B})=\sum_{(A^{\prime},B^{\prime}):A^{\prime}\subset A,B^{\prime}\subset B}(-1)^{|A|+|B|-|A^{\prime}|-|B^{\prime}|}\\ \times\mathop{{}\mathbb{E}}\mathopen{}\bigl{\{}f(X_{1},\dots,X_{k};Y_{1,2},\dots,Y_{k-1,k})\bigm{|}X_{A^{\prime}}=x_{A^{\prime}},Y_{B^{\prime}}=y_{B^{\prime}}\bigr{\}}, (2.1)

where xA={xi:iA}x_{A}=\{x_{i}:i\in A\} and yB={yi,j:(i,j)B}y_{B}=\{y_{i,j}:(i,j)\in B\} for A[k]A\subset[k] and B[k]2B\subset[k]_{2}. We remark that if A=A=\varnothing and B=B=\varnothing, then f,(X;Y)=𝔼{f(X[k];Y[k]2)}f_{\varnothing,\varnothing}(X_{\varnothing};Y_{\varnothing})=\mathop{{}\mathbb{E}}\mathopen{}\{f(X_{[k]};Y_{[k]_{2}})\}. For =0,1,,k\ell=0,1,\dots,k, let

f()(X[k];Y[k]2)={𝔼{f(X[k];Y[k]2)} if =0vA,B=f(XA;YB) if 1,\displaystyle f_{(\ell)}(X_{[k]};Y_{[k]_{2}})=\begin{dcases}\mathop{{}\mathbb{E}}\mathopen{}\{f(X_{[k]};Y_{[k]_{2}})\}&\text{ if $\ell=0$, }\\ \sum_{v_{A,B}=\ell}f(X_{A};Y_{B})&\text{ if $\ell\geqslant 1$},\end{dcases} (2.2)

where vA,Bv_{A,B} is the number of nodes in GA,BG_{A,B}. Let d=min{>0:f()0}d=\min\{\ell>0:f_{(\ell)}\neq 0\}, and we call dd the principal degree of ff. We say f(d)f_{(d)} is the principal part of ff. Moreover, we say the subgraphs GA,BG_{A,B} such that vA,B=dv_{A,B}=d and fA,B0f_{A,B}\neq 0 are the principal support graphs of ff.

The central limit theorems for Sn,k(f)S_{n,k}(f) is proved by Janson and Nowicki (1991). Let σA,B=fA,B(XA;YB)\sigma_{A,B}=\lVert f_{A,B}(X_{A};Y_{B})\rVert, and let 𝒢f,d={GA,B:σA,B0,vA,B=d}\mathcal{G}_{f,d}=\{G_{A,B}:\sigma_{A,B}\neq 0,v_{A,B}=d\} be the set of principal index graph. We remark that if ff has the principal degree dd, then Var(Sn,k(f))\mathop{\mathrm{Var}}(S_{n,k}(f)) is of order n2kdn^{2k-d}, see Lemmas 2 and 3 in Janson and Nowicki (1991). Janson and Nowicki (1991) proved that if all graphs in 𝒢f\mathcal{G}_{f} are connected, then

Sn,k(f)𝔼{Sn,k(f)}(Var(Sn,k(f)))1/2d.N(0,1).\displaystyle\frac{S_{n,k}(f)-\mathop{{}\mathbb{E}}\mathopen{}\{S_{n,k}(f)\}}{(\mathop{\mathrm{Var}}(S_{n,k}(f)))^{1/2}}\stackrel{{\scriptstyle d.}}{{\to}}N(0,1).

Note that if not all principal support graphs are connected, then the limiting distribution of the scaled version of Sn,kS_{n,k} is nonnormal (see Theorems 2 and 3 in Janson and Nowicki (1991)), and we will consider this case in another paper.

Now, assume that ff is a symmetric function having principal degree dd (1dk1\leqslant d\leqslant k). In this subsection, we give a Berry–Esseen bound for Sn,k(f)S_{n,k}(f). For x𝒳x\in\mathcal{X}, let

f1(x)f{1},(x)=𝔼{f(X[k];Y[k]2)|X1=x}𝔼{f(X[k];Y[k]2)}.\displaystyle f_{1}(x)\coloneqq f_{\{1\},\varnothing}(x)=\mathop{{}\mathbb{E}}\mathopen{}\{f(X_{[k]};Y_{[k]_{2}})\,|\,X_{1}=x\}-\mathop{{}\mathbb{E}}\mathopen{}\{f(X_{[k]};Y_{[k]_{2}})\}.

If f1(X1)2>0\lVert f_{1}(X_{1})\rVert_{2}>0, then it follows that d=1d=1. Here and in the sequel, we denote by Zp(𝔼|Z|p)1/p\|Z\|_{p}\coloneqq(\mathop{{}\mathbb{E}}\mathopen{}|Z|^{p})^{1/p} for p>0p>0 and we denote by Φ()\Phi(\cdot) the distribution function of N(0,1)N(0,1). The following theorem provides the Berry–Esseen bound for Sn,k(f)S_{n,k}(f) in the case where f1(X1)2>0\lVert f_{1}(X_{1})\rVert_{2}>0.

Theorem 2.1.

If σ1f1(X1)2>0\sigma_{1}\coloneqq\lVert f_{1}(X_{1})\rVert_{2}>0, then

supz|[Sn,k(f)𝔼{Sn,k(f)}Var{Sn,k(f)}z]Φ(z)|12kf(X[k];Y[k]2)42nσ12.\displaystyle\sup_{z\in\mathbb{R}}\biggl{\lvert}\mathop{{}\mathbb{P}}\mathopen{}\biggl{[}\frac{S_{n,k}(f)-\mathop{{}\mathbb{E}}\mathopen{}\{S_{n,k}(f)\}}{\sqrt{\mathop{\mathrm{Var}}\{S_{n,k}(f)\}}}\leqslant z\biggr{]}-\Phi(z)\biggr{\rvert}\leqslant\frac{12k\lVert f(X_{[k]};Y_{[k]_{2}})\rVert_{4}^{2}}{\sqrt{n}\sigma_{1}^{2}}. (2.3)
Remark 2.2.

We remark that Var(Sn,k(f))=O(n2k1)\mathop{\mathrm{Var}}(S_{n,k}(f))=O(n^{2k-1}) as nn\to\infty. Typically, the right hand side of 2.3 is of order n1/2n^{-1/2}. Specially, if f(X[k],Y[k]2)=h(X[k])f(X_{[k]},Y_{[k]_{2}})=h(X_{[k]}) for some symmetric function h:𝒳kh:\mathcal{X}^{k}\to\mathbb{R}, then Sn,kS_{n,k} is the classical UU-statistic. In this case, Chen and Shao (2007) obtained a Berry–Esseen bound of order n1/2n^{-1/2} under the assumption that h(X[k])3<\lVert h(X_{[k]})\rVert_{3}<\infty.

If σ1=0\sigma_{1}=0, then d2d\geqslant 2, that is, the principal degree of ff is at least 22. We have the following theorem.

Theorem 2.3.

Let τf(X[k];Y[k]2)4<\tau\coloneqq\lVert f(X_{[k]};Y_{[k]_{2}})\rVert_{4}<\infty and let σminmin(σA,B:GA,B𝒢f,d)\sigma_{\min}\coloneqq\min(\sigma_{A,B}:G_{A,B}\in\mathcal{G}_{f,d}). Assume that ff is a symmetric function having principal degree dd for some 2dk2\leqslant d\leqslant k, and assume further that for all graphs in 𝒢f,d\mathcal{G}_{f,d} are connected. Then, we have

supz|[(Sn,k(f)𝔼{Sn,k(f)})Var{Sn,k(f)}z]Φ(z)|Cn1/2,\displaystyle\sup_{z\in\mathbb{R}}\biggl{\lvert}\mathop{{}\mathbb{P}}\mathopen{}\biggl{[}\frac{(S_{n,k}(f)-\mathop{{}\mathbb{E}}\mathopen{}\{S_{n,k}(f)\})}{\sqrt{\mathop{\mathrm{Var}}\{S_{n,k}(f)\}}}\leqslant z\biggr{]}-\Phi(z)\biggr{\rvert}\leqslant Cn^{-1/2},

where C>0C>0 is a constant depending only on k,d,σmink,d,\sigma_{\min}, and τ\tau.

If we further assume that the function ff does not depend on XX, i.e., f(X;Y)=g(Y)f(X;Y)=g(Y) for some symmetric g:𝒴k(k1)/2g:\mathcal{Y}^{k(k-1)/2}\to\mathbb{R}, we obtain a sharper convergence rate. To give the theorem, we first introduce some more notation. Let G(r)G^{(r)} be the graph generated from GG by deleting the node rr and all the edges connecting to the node rr. We say GG is strongly connected if G(r)G^{(r)} is connected or empty for all rV(G)r\in V(G). We note that all strongly connected graphs are also connected. The following theorem provides a sharper Berry–Esseen bound than that in Theorem 2.3.

Theorem 2.4.

Assume that f(X[k];Y[k]2)=g(Y[k]2)f(X_{[k]};Y_{[k]_{2}})=g(Y_{[k]_{2}}) almost surely for some symmetric g:𝒴k(k1)/2g:\mathcal{Y}^{k(k-1)/2}\to\mathbb{R}. Let τ\tau and σA,B\sigma_{A,B} be defined in Theorem 2.3. Assume that the conditions in Theorem 2.3 are satisfied and assume further that all graphs in 𝒢f,d\mathcal{G}_{f,d} are strongly connected. Then,

supz|[(Sn,k(g)𝔼{Sn,k(g)})Var{Sn,k(g)}z]Φ(z)|Cn1,\displaystyle\sup_{z\in\mathbb{R}}\biggl{\lvert}\mathop{{}\mathbb{P}}\mathopen{}\biggl{[}\frac{(S_{n,k}(g)-\mathop{{}\mathbb{E}}\mathopen{}\{S_{n,k}(g)\})}{\sqrt{\mathop{\mathrm{Var}}\{S_{n,k}(g)\}}}\leqslant z\biggr{]}-\Phi(z)\biggr{\rvert}\leqslant Cn^{-1},

where C>0C>0 is a constant depending on k,d,σmink,d,\sigma_{\min}, and τ\tau.

3 Applications

3.1 Subgraphs counts in random graphs generated from graphons

A symmetric Lebesgue measurable function κ:[0,1]2[0,1]\kappa:[0,1]^{2}\to[0,1] is called a graphon, which was firstly introduced by Lovász and Szegedy (2006) to represent the graph limit. Given a graphon κ\kappa and n2n\geqslant 2, the κ\kappa-random graph 𝔾(n,κ)\mathbb{G}(n,\kappa) can be generated as follows: Let n1n\geqslant 1 and let X=(X1,,Xn)X=(X_{1},\dots,X_{n}) be a vector of independent uniformly distributed random variables on [0,1][0,1]. Given XX, we generate the graph 𝔾(n,κ)\mathbb{G}(n,\kappa) by connecting the node pair (i,j)(i,j) independently with probability κ(Xi,Xj)\kappa(X_{i},X_{j}). This construction was firstly introduced by Diaconis and Freedman (1981), which can be used to study large dense and sparse random graphs and random trees generated from graphons. We refer to Lovász and Szegedy (2006); Bollobás et al. (2007); Lovász (2012) for more details.

Subgraph counts are important statistics in estimating graphons. As a special case, when κp\kappa\equiv p for some p(0,1)p\in(0,1), the κ\kappa-random graph model becomes the classical Erdös–Rényi model ER(p)\mathrm{ER}(p). The study of asymptotic properties of subgraph counts in ER(p)\mathrm{ER}(p) dates back to Nowicki (1989); Barbour et al. (1989); Janson and Nowicki (1991) for more details. Recently, Krokowski et al. (2017), Röllin (2017) and Privault and Serafin (2018) applied Stein’s method to obtain an optimal Berry–Esseen bound for triangle counts in ER(p)\mathrm{ER}(p). For subgraph counts in κ\kappa-random graph, Kaur and Röllin (2020) proved an upper bound of the Kolmogorov distance for multivariate normal approximations for centered subgraph counts with order n1/(p+2)n^{-1/(p+2)} for some p>0p>0. However, the Berry–Esseen bounds for subgraph counts of κ\kappa-random graph is still unknown so far. In this subsection, we apply Theorems 2.3 and 2.4 to prove sharp Berry–Esseen bounds for subgraph counts statistics.

Let Ξ=(ξi,j)1i<jn\Xi=(\xi_{i,j})_{1\leqslant i<j\leqslant n} be the adjacency matrix of 𝔾(n,κ)\mathbb{G}(n,\kappa), where for each (i,j)(i,j), the binary random variable ξi,j\xi_{i,j} indicates the connection of the graph. Formally, let Y=(Y1,1,,Yn1,n)Y=(Y_{1,1},\dots,Y_{n-1,n}) be a vector of independent uniformly distributed random variables that is also independent of XX, and then we can write ξi,j=Yi,jκ(Xi,Xj)\xi_{i,j}=\mathbb{N}{Y_{i,j}\leqslant\kappa(X_{i},X_{j})}. For any nonrandom simple FF with v(F)=kv(F)=k, the (injective) subgraph counts and induced subgraph counts in 𝔾(n,κ)\mathbb{G}(n,\kappa) are defined by

TFinjTFinj(𝔾(n,κ))\displaystyle T_{F}^{\operatorname{inj}}\coloneqq T_{F}^{\operatorname{inj}}(\mathbb{G}(n,\kappa)) =αn,kφFinj(ξα(1),α(2),,ξα(k1),α(k)),\displaystyle=\sum_{\alpha\in\mathcal{I}_{n,k}}\varphi_{F}^{\operatorname{inj}}(\xi_{\alpha(1),\alpha(2)},\dots,\xi_{\alpha({k-1}),\alpha({k})}),
TFindTFind(𝔾(n,κ))\displaystyle T_{F}^{\operatorname{ind}}\coloneqq T_{F}^{\operatorname{ind}}(\mathbb{G}(n,\kappa)) =αn,kφFind(ξα(1),α(2),,ξα(k1),α(k)),\displaystyle=\sum_{\alpha\in\mathcal{I}_{n,k}}\varphi_{F}^{\operatorname{ind}}(\xi_{\alpha(1),\alpha(2)},\dots,\xi_{\alpha({k-1}),\alpha(k)}),

respectively, where for (x1,1,,xk1,k)k(k1)/2(x_{1,1},\dots,x_{k-1,k})\in\mathbb{R}^{k(k-1)/2},

φFinj(x1,1,,xk1,k)\displaystyle\varphi_{F}^{\operatorname{inj}}(x_{1,1},\dots,x_{k-1,k}) =H:HF(i,j)E(H)xi,j,\displaystyle=\sum_{H:H\cong F}\prod_{(i,j)\in E(H)}x_{i,j},
φFind(x1,2,,xk1,k)\displaystyle\varphi_{F}^{\operatorname{ind}}(x_{1,2},\dots,x_{{k-1},k}) =H:HF(i,j)E(H)xi,j(i,j)E(H)(1xi,j).\displaystyle=\sum_{H:H\cong F}\prod_{(i,j)\in E(H)}x_{i,j}\prod_{(i,j)\not\in E(H)}(1-x_{i,j}).

Here, the summation H:HF\sum_{H:H\cong F} ranges over the subgraphs with v(F)v(F) nodes that are isomorphic to FF and thus contains v(F)!/|Aut(F)|v(F)!/|\operatorname{Aut}(F)| terms, where |Aut(F)|\lvert\operatorname{Aut}(F)\rvert is the number of automorphisms of FF. Moreover, we note that both φFinj\varphi_{F}^{\operatorname{inj}} and φFind\varphi_{F}^{\operatorname{ind}} are symmetric. For example, if FF is the 22-star, then k=3k=3, |Aut(F)|=2|\operatorname{Aut}(F)|=2 and

φFinj(ξ1,2,ξ1,3,ξ2,3)\displaystyle\varphi_{F}^{\operatorname{inj}}(\xi_{1,2},\xi_{1,3},\xi_{2,3}) =ξ1,2ξ1,3+ξ1,2ξ2,3+ξ1,3ξ2,3,\displaystyle=\xi_{1,2}\xi_{1,3}+\xi_{1,2}\xi_{2,3}+\xi_{1,3}\xi_{2,3},
φFind(ξ1,2,ξ1,3,ξ2,3)\displaystyle\varphi_{F}^{\operatorname{ind}}(\xi_{1,2},\xi_{1,3},\xi_{2,3}) =ξ1,2ξ1,3(1ξ2,3)+ξ1,2ξ2,3(1ξ1,3)+ξ1,3ξ2,3(1ξ1,2).\displaystyle=\xi_{1,2}\xi_{1,3}(1-\xi_{2,3})+\xi_{1,2}\xi_{2,3}(1-\xi_{1,3})+\xi_{1,3}\xi_{2,3}(1-\xi_{1,2}).

If FF is a triangle, then |Aut(F)|=6|\operatorname{Aut}(F)|=6 and

φFinj(ξ1,2,ξ1,3,ξ2,3)=φFind(ξ1,2,ξ1,3,ξ2,3)=ξ1,2ξ1,3ξ2,3.\displaystyle\varphi_{F}^{\operatorname{inj}}(\xi_{1,2},\xi_{1,3},\xi_{2,3})=\varphi_{F}^{\operatorname{ind}}(\xi_{1,2},\xi_{1,3},\xi_{2,3})=\xi_{1,2}\xi_{1,3}\xi_{2,3}.

Let

tF(κ)\displaystyle t_{F}(\kappa) =[0,1]k(i,j)E(F)κ(xi,xj)iV(F)dxi,\displaystyle=\int_{[0,1]^{k}}\prod_{(i,j)\in E(F)}\kappa(x_{i},x_{j})\prod_{i\in V(F)}dx_{i},
tFind(κ)\displaystyle t_{F}^{\operatorname{ind}}(\kappa) =[0,1]k(i,j)E(F)κ(xi,xj)(i,j)E(F)(1κ(xi,xj))iV(F)dxi.\displaystyle=\int_{[0,1]^{k}}\prod_{(i,j)\in E(F)}\kappa(x_{i},x_{j})\prod_{(i,j)\not\in E(F)}(1-\kappa(x_{i},x_{j}))\prod_{i\in V(F)}dx_{i}.

Then, we have

𝔼{φFinj(ξ1,1,,ξk1,k)}\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\{\varphi_{F}^{\operatorname{inj}}(\xi_{1,1},\dots,\xi_{k-1,k})\} =k!|Aut(F)|tF(κ),\displaystyle=\frac{k!}{|\operatorname{Aut}(F)|}t_{F}(\kappa),
𝔼{φFind(ξ1,1,,ξk1,k)}\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\{\varphi_{F}^{\operatorname{ind}}(\xi_{1,1},\dots,\xi_{k-1,k})\} =k!|Aut(F)|tFind(κ).\displaystyle=\frac{k!}{|\operatorname{Aut}(F)|}t_{F}^{\operatorname{ind}}(\kappa).

As ξi,j=Yi,jκ(Xi,Xj)\xi_{i,j}=\mathbb{N}{Y_{i,j}\leqslant\kappa(X_{i},X_{j})}, let

fFinj(X[k];Y[k]2)\displaystyle f_{F}^{\operatorname{inj}}(X_{[k]};Y_{[k]_{2}}) =φFinj(ξ1,1,,ξk1,k),\displaystyle=\varphi_{F}^{\operatorname{inj}}(\xi_{1,1},\dots,\xi_{k-1,k}),

Now, as random variables (ξi,j)1i<jn(\xi_{i,j})_{1\leqslant i<j\leqslant n} are conditionally independent given XX, we have

𝔼{fFinj(X[k];Y[k]2)|X}\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\{f_{F}^{\operatorname{inj}}(X_{[k]};Y_{[k]_{2}})\,|\,X\} =HF(i,j)E(H)κ(Xi,Xj),\displaystyle=\sum_{H\cong F}\prod_{(i,j)\in E(H)}\kappa(X_{i},X_{j}),
𝔼{fFind(X[k];Y[k]2)|X}\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\{f_{F}^{\operatorname{ind}}(X_{[k]};Y_{[k]_{2}})\,|\,X\} =HF(i,j)E(H)κ(Xi,Xj)(i,j)E(H)(1κ(Xi,Xj)).\displaystyle=\sum_{H\cong F}\prod_{(i,j)\in E(H)}\kappa(X_{i},X_{j})\prod_{(i,j)\not\in E(H)}(1-\kappa(X_{i},X_{j})).

Let

f1inj(x)\displaystyle f_{1}^{\operatorname{inj}}(x) =𝔼{fFinj(X[k];Y[k]2)|X1=x}\displaystyle=\mathop{{}\mathbb{E}}\mathopen{}\{f_{F}^{\operatorname{inj}}(X_{[k]};Y_{[k]_{2}})\,|\,X_{1}=x\}
=HF𝔼{(i,j)E(H)κ(Xi,Xj)|X1=x},\displaystyle=\sum_{H\cong F}\mathop{{}\mathbb{E}}\mathopen{}\biggl{\{}\prod_{(i,j)\in E(H)}\kappa(X_{i},X_{j})\biggm{|}X_{1}=x\biggr{\}},

and similarly, let

f1ind(x)\displaystyle f_{1}^{\operatorname{ind}}(x) =𝔼{fFind(X[k];Y[k]2)|X1=x}\displaystyle=\mathop{{}\mathbb{E}}\mathopen{}\{f_{F}^{\operatorname{ind}}(X_{[k]};Y_{[k]_{2}})\,|\,X_{1}=x\}
=HF𝔼{(i,j)E(H)κ(Xi,Xj)(i,j)E(H)(1κ(Xi,Xj))|X1=x}.\displaystyle=\sum_{H\cong F}\mathop{{}\mathbb{E}}\mathopen{}\biggl{\{}\prod_{(i,j)\in E(H)}\kappa(X_{i},X_{j})\prod_{(i,j)\not\in E(H)}(1-\kappa(X_{i},X_{j}))\biggm{|}X_{1}=x\biggr{\}}.

We have the following theorem, which follows from Theorem 2.1 directly.

Theorem 3.1.

Let σ1inj=f1inj(X1)𝔼{f1inj(X1)}2\sigma_{1}^{\operatorname{inj}}=\lVert f_{1}^{\operatorname{inj}}(X_{1})-\mathop{{}\mathbb{E}}\mathopen{}\{f_{1}^{\operatorname{inj}}(X_{1})\}\rVert_{2} and σ1ind=f1ind(X1)𝔼{gFind(X1)}2\sigma_{1}^{\operatorname{ind}}=\lVert f_{1}^{\operatorname{ind}}(X_{1})-\mathop{{}\mathbb{E}}\mathopen{}\{g_{F}^{\operatorname{ind}}(X_{1})\}\rVert_{2}. Assume that σ1inj>0\sigma_{1}^{\operatorname{inj}}>0, then

supz|[nkσ1inj(n\@@atopk)1(TFinj𝔼{TFinj})z]Φ(z)\displaystyle\sup_{z\in\mathbb{R}}\biggl{\lvert}\mathop{{}\mathbb{P}}\mathopen{}\biggl{[}\frac{\sqrt{n}}{k\sigma_{1}^{\operatorname{inj}}}\binom{n}{k}^{-1}(T_{F}^{\operatorname{inj}}-\mathop{{}\mathbb{E}}\mathopen{}\{T_{F}^{\operatorname{inj}}\})\leqslant z\biggr{]}-\Phi(z) Cn1/2.\displaystyle\leqslant Cn^{-1/2}.

Moreover, assume that σ1ind>0\sigma_{1}^{\operatorname{ind}}>0, then

supz|[nkσ1ind(n\@@atopk)1(TFind𝔼{TFind})z]Φ(z)|\displaystyle\sup_{z\in\mathbb{R}}\biggl{\lvert}\mathop{{}\mathbb{P}}\mathopen{}\biggl{[}\frac{\sqrt{n}}{k\sigma_{1}^{\operatorname{ind}}}\binom{n}{k}^{-1}(T_{F}^{\operatorname{ind}}-\mathop{{}\mathbb{E}}\mathopen{}\{T_{F}^{\operatorname{ind}}\})\leqslant z\biggr{]}-\Phi(z)\biggr{\rvert} Cn1/2.\displaystyle\leqslant Cn^{-1/2}.

If κp\kappa\equiv p for a fixed number 0<p<10<p<1, then the random variables (ξi,j)1i<jn(\xi_{i,j})_{1\leqslant i<j\leqslant n} are i.i.d.  and the functions φFinj\varphi_{F}^{\operatorname{inj}} and φFind\varphi_{F}^{\operatorname{ind}} do not depend on XX. We have the following theorem:

Theorem 3.2.

Let κp\kappa\equiv p for 0<p<10<p<1. Then

supz|[TFinj𝔼{TFinj}(Var{TFinj})1/2z]Φ(z)|\displaystyle\sup_{z\in\mathbb{R}}\biggl{\lvert}\mathop{{}\mathbb{P}}\mathopen{}\biggl{[}\frac{T_{F}^{\operatorname{inj}}-\mathop{{}\mathbb{E}}\mathopen{}\{T_{F}^{\operatorname{inj}}\}}{(\mathop{\mathrm{Var}}\{T_{F}^{\operatorname{inj}}\})^{1/2}}\leqslant z\biggr{]}-\Phi(z)\biggr{\rvert} Cn1.\displaystyle\leqslant Cn^{-1}.
Remark 3.3.

For the L1L_{1} bound, Barbour et al. (1989) proved the same order of O(n1)O(n^{-1}) in the case that pp is a constant. For the Berry–Esseen bound, Privault and Serafin (2018) proved a general Berry–Esseen bound for subgraph counts for Erdös–Rényi random graph using a different method. Specially, if pp is a constant, then Theorem 3.2 provides the same result as in Privault and Serafin (2018).

For induced subgraph counts, we need to consider some separate cases. Let s(F)s(F) and t(F)t(F) denote the number of 2-stars and triangles in FF, respectively. If any of the following conditions holds, then it has been proven by Janson and Nowicki (1991) that (TFind𝔼{TFind})/(Var{TFind})1/2(T_{F}^{\operatorname{ind}}-\mathop{{}\mathbb{E}}\mathopen{}\{T_{F}^{\operatorname{ind}}\})/(\mathop{\mathrm{Var}}\{T_{F}^{\operatorname{ind}}\})^{1/2} converges to a standard normal distribution:

  1. (G1)

    If e(F)p(v(F)\@@atop2)e(F)\neq p\binom{v(F)}{2};

  2. (G2)

    if e(F)=p(v(F)\@@atop2)e(F)=p\binom{v(F)}{2}, s(F)3p2(v(F)\@@atop3)s(F)\neq 3p^{2}\binom{v(F)}{3};

  3. (G3)

    if e(F)=p(v(F)\@@atop2)e(F)=p\binom{v(F)}{2}, s(F)=3p2(v(F)\@@atop3)s(F)=3p^{2}\binom{v(F)}{3} and t(F)p3(v(F)\@@atop3)t(F)\neq p^{3}\binom{v(F)}{3}.

The following theorem gives the Berry–Esseen bounds for induced subgraph counts.

Theorem 3.4.

Let κp\kappa\equiv p for 0<p<10<p<1. If (G1) or(G3) holds, then

supz|[TFind𝔼{TFind}(Var{TFind})1/2z]Φ(z)|\displaystyle\sup_{z\in\mathbb{R}}\biggl{\lvert}\mathop{{}\mathbb{P}}\mathopen{}\biggl{[}\frac{T_{F}^{\operatorname{ind}}-\mathop{{}\mathbb{E}}\mathopen{}\{T_{F}^{\operatorname{ind}}\}}{(\mathop{\mathrm{Var}}\{T_{F}^{\operatorname{ind}}\})^{1/2}}\leqslant z\biggr{]}-\Phi(z)\biggr{\rvert} Cn1.\displaystyle\leqslant Cn^{-1}. (3.1)

If (G2) holds, then

supz|[TFind𝔼{TFind}(Var{TFind})1/2z]Φ(z)|\displaystyle\sup_{z\in\mathbb{R}}\biggl{\lvert}\mathop{{}\mathbb{P}}\mathopen{}\biggl{[}\frac{T_{F}^{\operatorname{ind}}-\mathop{{}\mathbb{E}}\mathopen{}\{T_{F}^{\operatorname{ind}}\}}{(\mathop{\mathrm{Var}}\{T_{F}^{\operatorname{ind}}\})^{1/2}}\leqslant z\biggr{]}-\Phi(z)\biggr{\rvert} Cn1/2.\displaystyle\leqslant Cn^{-1/2}. (3.2)

4 A new Berry–Esseen bound for exchangeable pair approach

4.1 Berry–Esseen bound

In this section, we establish a new Berry–Esseen theorem for exchangeable pair approach under a new setting. Let X𝒳X\in\mathcal{X} be a random variable valued on a measurable space and let W=φ(X)W=\varphi(X) be the random variable of interest where φ:𝒳\varphi:\mathcal{X}\to\mathbb{R}. Assume that 𝔼{W}=0\mathop{{}\mathbb{E}}\mathopen{}\{W\}=0 and 𝔼{W2}=1\mathop{{}\mathbb{E}}\mathopen{}\{W^{2}\}=1. We propose the following condition:

  1. (A)

    Let (X,X)(X,X^{\prime}) be an exchangeable pair and let F:𝒳×𝒳F:\mathcal{X}\times\mathcal{X}\to\mathbb{R} be an antisymmetric function. Assume that DF(X,X)D\coloneqq F(X,X^{\prime}) satisfies the following condition:

    𝔼{D|X}=λ(W+R),\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\{D|X\}=\lambda(W+R), (4.1)

    where λ>0\lambda>0 is a constant and RR is a random variable.

We remark that the operator of antisymmetric functions was firstly mentioned by Holmes and Reinert (2004), and the condition (A) was considered by Chatterjee (2007), who applied the exchangeable pair approach to prove concentration inequalities.

The following theorem provides a uniform Berry–Esseen bound for exchangeable pair approach under the assumption (A).

Theorem 4.1.

Let (X,X)(X,X^{\prime}) and DD satisfy the condition (A). Let W=φ(X)W^{\prime}=\varphi(X^{\prime}) and Δ=WW\Delta=W-W^{\prime}. Then,

supz|[Wz]Φ(z)|\displaystyle\sup_{z\in\mathbb{R}}\lvert\mathop{{}\mathbb{P}}\mathopen{}[W\leqslant z]-\Phi(z)\rvert 𝔼|112λ𝔼{DΔ|W}|+1λ𝔼|𝔼{DΔ|W}|+𝔼|R|,\displaystyle\leqslant\mathop{{}\mathbb{E}}\mathopen{}\biggl{\lvert}1-\frac{1}{2\lambda}\mathop{{}\mathbb{E}}\mathopen{}\{D\Delta\,|\,W\}\biggr{\rvert}+\frac{1}{\lambda}\mathop{{}\mathbb{E}}\mathopen{}\bigl{\lvert}\mathop{{}\mathbb{E}}\mathopen{}\{D^{*}\Delta\,|\,W\}\bigr{\rvert}+\mathop{{}\mathbb{E}}\mathopen{}\lvert R\rvert, (4.2)

provided that DF(X,X)|D|D^{*}\coloneqq F^{*}(X,X^{\prime})\geqslant|D|, where FF^{*} is a symmetric function.

Remark 4.2.

Assume that 1.3 is satisfied. Then, we can choose D=Δ=WWD=\Delta=W-W^{\prime}, and the right hand side of 4.2 reduces to

𝔼|112λ𝔼{Δ2|W}|+1λ𝔼|𝔼{ΔΔ}|+𝔼|R|,\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\biggl{\lvert}1-\frac{1}{2\lambda}\mathop{{}\mathbb{E}}\mathopen{}\{\Delta^{2}\,|\,W\}\biggr{\rvert}+\frac{1}{\lambda}\mathop{{}\mathbb{E}}\mathopen{}\bigl{\lvert}\mathop{{}\mathbb{E}}\mathopen{}\{\Delta^{*}\Delta\}\bigr{\rvert}+\mathop{{}\mathbb{E}}\mathopen{}|R|,

where ΔΔ(W,W)\Delta^{*}\coloneqq\Delta^{*}(W,W^{\prime}) is a symmetric function for WW and WW^{\prime} such that Δ|Δ|\Delta^{*}\geqslant|\Delta|. Thus, Theorem 4.1 recovers to Theorem 2.1 in Shao and Zhang (2019).

The following corollary is useful for random variables that can be decomposed as a sum of WW and a remainder term. Specifically, let TT(X)T\coloneqq T(X) be a random variable such that T=W+UT=W+U, where W=φ(X)W=\varphi(X) is as defined at the beginning of this section, and UU(X)U\coloneqq U(X) is a remainder term. The following corollary gives a Berry–Esseen bound for TT.

Corollary 4.3.

Let (X,X)𝒳×𝒳(X,X^{\prime})\in\mathcal{X}\times\mathcal{X} be an exchangeable pair and let DF(X,X)D\coloneqq F(X,X^{\prime}) where F:𝒳×𝒳F:\mathcal{X}\times\mathcal{X}\to\mathbb{R} is antisymmetric. Assume that

𝔼{D|X}=λ(W+R)\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\{D\,|\,X\}=\lambda(W+R) (4.3)

for some λ>0\lambda>0 and some random variable RR. Let UU(X)U^{\prime}\coloneqq U(X^{\prime}) and Δ=φ(X)φ(X)\Delta=\varphi(X)-\varphi(X^{\prime}). Then, we have

supz|[Tz]Φ(z)|𝔼|112λ𝔼{DΔ|X}|+1λ𝔼|𝔼{DΔ|X}|+32λ𝔼|D(UU)|+𝔼|R|+𝔼|U|,\sup_{z\in\mathbb{R}}\bigl{\lvert}\mathop{{}\mathbb{P}}\mathopen{}[T\leqslant z]-\Phi(z)\bigr{\rvert}\leqslant\mathop{{}\mathbb{E}}\mathopen{}\biggl{\lvert}1-\frac{1}{2\lambda}\mathop{{}\mathbb{E}}\mathopen{}\{D\Delta\,|\,X\}\biggr{\rvert}\\ +\frac{1}{\lambda}\mathop{{}\mathbb{E}}\mathopen{}\bigl{\lvert}\mathop{{}\mathbb{E}}\mathopen{}\{D^{*}\Delta\,|\,X\}\bigr{\rvert}+\frac{3}{2\lambda}\mathop{{}\mathbb{E}}\mathopen{}\lvert D(U-U^{\prime})\rvert+\mathop{{}\mathbb{E}}\mathopen{}|R|+\mathop{{}\mathbb{E}}\mathopen{}|U|,

provided that DD(X,X)D^{*}\coloneqq D^{*}(X,X^{\prime}) is any symmetric function of XX and XX^{\prime} such that D|D|D^{*}\geqslant|D|.

Remark 4.4.

Assume that X=(X1,,Xn)X=(X_{1},\dots,X_{n}) is a family of independent random variables. Let W=i=1nξiW=\sum_{i=1}^{n}\xi_{i} be a linear statistic, where ξi=hi(Xi)\xi_{i}=h_{i}(X_{i}) and hih_{i} is a nonrandom function, such that 𝔼{ξi}=0\mathop{{}\mathbb{E}}\mathopen{}\{\xi_{i}\}=0 and i=1n𝔼{ξi2}=1\sum_{i=1}^{n}\mathop{{}\mathbb{E}}\mathopen{}\{\xi_{i}^{2}\}=1, and let U=U(X1,,Xn)U=U(X_{1},\dots,X_{n})\in\mathbb{R} be a random variable. Let T=W+UT=W+U, β2=i=1n𝔼{|ξi|2|ξi|>1}\beta_{2}=\sum_{i=1}^{n}\mathop{{}\mathbb{E}}\mathopen{}\{\lvert\xi_{i}\rvert^{2}\mathbb{N}{\lvert\xi_{i}\rvert>1}\} and β3=i=1n𝔼{|ξi|3|ξi|1}\beta_{3}=\sum_{i=1}^{n}\mathop{{}\mathbb{E}}\mathopen{}\{\lvert\xi_{i}\rvert^{3}\mathbb{N}{\lvert\xi_{i}\rvert\leqslant 1}\}. Chen and Shao (2007) (see also Shao and Zhou (2016)) proved the following result:

supz|[Tz]Φ(z)|\displaystyle\sup_{z\in\mathbb{R}}\lvert\mathop{{}\mathbb{P}}\mathopen{}[T\leqslant z]-\Phi(z)\rvert 17(β2+β3)+5𝔼|U|+2i=1n𝔼|ξi(UU(i))|,\displaystyle\leqslant 17(\beta_{2}+\beta_{3})+5\mathop{{}\mathbb{E}}\mathopen{}|U|+2\sum_{i=1}^{n}\mathop{{}\mathbb{E}}\mathopen{}\lvert\xi_{i}(U-U^{(i)})\rvert, (4.4)

where U(i)U^{(i)} is any random variable independent of ξi\xi_{i}.

The Berry–Esseen bound in Corollary 4.3 improves Chen and Shao (2007)’s result in the sense that the random variable WW in our result is not necessarily a partial sum of independent random variables, and our result in Corollary 4.3 can be applied to a general class of random variables.

4.2 Proof of Theorem 4.1

In this subsection, we prove Theorem 4.1 by Stein’s method. The proof is similar to that of Theorem 2.1 in Shao and Zhang (2019). To begin with, we need to prove the following lemma, which is useful in the proof of Theorem 4.1.

Lemma 4.5.

Let ff be a nondecreasing function. Then,

12λ|𝔼{DΔ0(f(W+u)f(W))𝑑u}|12λ𝔼{DΔf(W)},\displaystyle\frac{1}{2\lambda}\biggl{\lvert}\mathop{{}\mathbb{E}}\mathopen{}\biggl{\{}D\int_{-\Delta}^{0}\bigl{(}f(W+u)-f(W)\bigr{)}\mathop{}\!{d}u\biggr{\}}\biggr{\rvert}\leqslant\frac{1}{2\lambda}\mathop{{}\mathbb{E}}\mathopen{}\bigl{\{}D^{*}\Delta f(W)\bigr{\}},

where DD^{*} is as defined in Theorem 4.1.

Proof of Lemma 4.5.

Since f()f(\cdot) is nondecreasing, it follows that

Δ(f(W)f(W))0\Delta\bigl{(}f(W)-f(W^{\prime})\bigr{)}\geqslant 0

and

0\displaystyle 0 Δ0(f(W+u)f(W))𝑑u\displaystyle\geqslant\int_{-\Delta}^{0}\bigl{(}f(W+u)-f(W)\bigr{)}\mathop{}\!{d}u
Δ(f(W)f(W)),\displaystyle\geqslant-\Delta\bigl{(}f(W)-f(W^{\prime})\bigr{)},

which yields

𝔼{DD>0Δ(f(W)f(W))}\displaystyle-\mathop{{}\mathbb{E}}\mathopen{}\bigl{\{}D\mathbb{N}{D>0}\Delta\bigl{(}f(W)-f(W^{\prime})\bigr{)}\bigr{\}} 𝔼{DΔ0(f(W+u)f(W))𝑑u}\displaystyle\leqslant\mathop{{}\mathbb{E}}\mathopen{}\biggl{\{}D\int_{-\Delta}^{0}\bigl{(}f(W+u)-f(W)\bigr{)}\mathop{}\!{d}u\biggr{\}}
𝔼{DD<0Δ(f(W)f(W))}.\displaystyle\leqslant-\mathop{{}\mathbb{E}}\mathopen{}\bigl{\{}D\mathbb{N}{D<0}\Delta\bigl{(}f(W)-f(W^{\prime})\bigr{)}\bigr{\}}.

Recalling that W=φ(X)W=\varphi({X}), D=F(X,X)D=F({X},{X}^{\prime}) is antisymmetric and D=F(X,X)D^{*}=F^{*}({X},{X}^{\prime}) is symmetric, as (X,X)({X},{X}^{\prime}) is exchangeable, we have

𝔼{DD>0Δ{f(W)f(W)}}=𝔼{DD<0Δ(f(W)f(W))},\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\bigl{\{}D\mathbb{N}{D>0}\Delta\bigl{\{}f(W)-f(W^{\prime})\bigr{\}}\bigr{\}}=-\mathop{{}\mathbb{E}}\mathopen{}\bigl{\{}D\mathbb{N}{D<0}\Delta\bigl{(}f(W)-f(W^{\prime})\bigr{)}\bigr{\}},
and
𝔼{DD>0Δf(W)}=𝔼{DD<0Δf(W)}.\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\bigl{\{}D^{*}\mathbb{N}{D>0}\Delta f(W)\bigr{\}}=-\mathop{{}\mathbb{E}}\mathopen{}\bigl{\{}D^{*}\mathbb{N}{D<0}\Delta f(W^{\prime})\bigr{\}}.

Moreover, as 𝔼{DΔD=0(f(W)f(W))}0\mathop{{}\mathbb{E}}\mathopen{}\bigl{\{}D^{*}\Delta\mathbb{N}{D=0}\bigl{(}f(W)-f(W^{\prime})\bigr{)}\bigr{\}}\geqslant 0 and 𝔼{DD=0Δf(W)}=𝔼{DD=0Δφ(W)}\mathop{{}\mathbb{E}}\mathopen{}\bigl{\{}D^{*}\mathbb{N}{D=0}\Delta f(W)\bigr{\}}=-\mathop{{}\mathbb{E}}\mathopen{}\bigl{\{}D^{*}\mathbb{N}{D=0}\Delta\varphi(W^{\prime})\bigr{\}}, it follows that

𝔼{DΔD=0f(W)}0.\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\{D^{*}\Delta\mathbb{N}{D=0}f(W)\}\geqslant 0.

Therefore,

12λ|𝔼{DΔ0{f(W+u)f(W)}𝑑u}|\displaystyle\frac{1}{2\lambda}\biggl{\lvert}\mathop{{}\mathbb{E}}\mathopen{}\biggl{\{}D\int_{-\Delta}^{0}\bigl{\{}f(W+u)-f(W)\bigr{\}}\mathop{}\!{d}u\biggr{\}}\biggr{\rvert}
12λ𝔼{DD<0Δ(f(W)f(W))}\displaystyle\leqslant-\frac{1}{2\lambda}\mathop{{}\mathbb{E}}\mathopen{}\bigl{\{}D\mathbb{N}{D<0}\Delta\bigl{(}f(W)-f(W^{\prime})\bigr{)}\bigr{\}}
12λ𝔼{DD<0Δ(f(W)f(W))}\displaystyle\leqslant\frac{1}{2\lambda}\mathop{{}\mathbb{E}}\mathopen{}\bigl{\{}D^{*}\mathbb{N}{D<0}\Delta\bigl{(}f(W)-f(W^{\prime})\bigr{)}\bigr{\}}
=12λ𝔼{DΔ(D>0+D<0)f(W)}\displaystyle=\frac{1}{2\lambda}\mathop{{}\mathbb{E}}\mathopen{}\bigl{\{}D^{*}\Delta\bigl{(}\mathbb{N}{D>0}+\mathbb{N}{D<0}\bigr{)}f(W)\bigr{\}}
12λ𝔼{DΔf(W)}.\displaystyle\leqslant\frac{1}{2\lambda}\mathop{{}\mathbb{E}}\mathopen{}\{D^{*}\Delta f(W)\}.\qed
Proof of Theorem 4.1.

We apply some ideas of Theorem 2.1 in Shao and Zhang (2019) to prove the desired result. Let z0z\geqslant 0 be a fixed real number, and fzf_{z} the solution to the Stein equation:

f(w)wf(w)=wzΦ(z),\displaystyle f^{\prime}(w)-wf(w)=\mathbb{N}{w\leqslant z}-\Phi(z), (4.5)

where Φ()\Phi(\cdot) is the distribution function of the standard normal distribution. It is well known that (see, e.g., Chen et al. (2011))

fz(w)={2πew2/2Φ(w){1Φ(z)}if wz,2πew2/2Φ(z){1Φ(w)}otherwise,\displaystyle f_{z}(w)=\begin{dcases}\sqrt{2\pi}e^{w^{2}/2}{\Phi(w)\bigl{\{}1-\Phi(z)\bigr{\}}}&\text{if }w\leqslant z,\\ \sqrt{2\pi}e^{w^{2}/2}{\Phi(z)\bigl{\{}1-\Phi(w)\bigr{\}}}&\text{otherwise},\end{dcases} (4.6)

Since 𝔼{D|W}=λ(W+R)\mathop{{}\mathbb{E}}\mathopen{}\{D|W\}=\lambda(W+R), and D=F(X,X)D=F(X,X^{\prime}) is antisymmetric, it follows that, for any absolutely continuous function ff,

0\displaystyle 0 =𝔼{D(f(W)+f(W))}\displaystyle=\mathop{{}\mathbb{E}}\mathopen{}\bigl{\{}D\bigl{(}f(W)+f(W^{\prime})\bigr{)}\bigr{\}}
=2𝔼{Df(W)}𝔼{D(f(W)f(W))}\displaystyle=2\mathop{{}\mathbb{E}}\mathopen{}\{Df(W)\}-\mathop{{}\mathbb{E}}\mathopen{}\bigl{\{}D\bigl{(}f(W)-f(W^{\prime})\bigr{)}\bigr{\}}
=2λ𝔼{(W+R)f(W)}𝔼{DΔ0f(W+u)𝑑u}.\displaystyle=2\lambda\mathop{{}\mathbb{E}}\mathopen{}\{(W+R)f(W)\}-\mathop{{}\mathbb{E}}\mathopen{}\biggl{\{}D\int_{-\Delta}^{0}f^{\prime}(W+u)\mathop{}\!{d}u\biggr{\}}.

Rearranging the foregoing equality, we have

𝔼{Wf(W)}=12λ𝔼{DΔ0f(W+u)𝑑u}𝔼{Rf(W)}.\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\{Wf(W)\}=\frac{1}{2\lambda}\mathop{{}\mathbb{E}}\mathopen{}\biggl{\{}D\int_{-\Delta}^{0}f^{\prime}(W+u)\mathop{}\!{d}u\biggr{\}}-\mathop{{}\mathbb{E}}\mathopen{}\{Rf(W)\}. (4.7)

By 4.7,

𝔼{Wfz(W)}=12λ𝔼{DΔ0fz(W+t)𝑑t}𝔼{Rfz(W)},\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\bigl{\{}Wf_{z}(W)\bigr{\}}=\frac{1}{2\lambda}\mathop{{}\mathbb{E}}\mathopen{}\biggl{\{}D\int_{-\Delta}^{0}f_{z}^{\prime}(W+t)\mathop{}\!{d}t\biggr{\}}-\mathop{{}\mathbb{E}}\mathopen{}\bigl{\{}Rf_{z}(W)\bigr{\}},

and thus,

(W>z){1Φ(z)}\displaystyle\mathop{{}\mathbb{P}}\mathopen{}(W>z)-\bigl{\{}1-\Phi(z)\bigr{\}} =𝔼{fz(W)Wfz(W)}\displaystyle=\mathop{{}\mathbb{E}}\mathopen{}\bigl{\{}f_{z}^{\prime}(W)-Wf_{z}(W)\bigr{\}} (4.8)
=J1J2+J3,\displaystyle={}J_{1}-J_{2}+J_{3},

where

J1\displaystyle J_{1} =𝔼{fz(W)(112λ𝔼{DΔ|W})},\displaystyle=\mathop{{}\mathbb{E}}\mathopen{}\biggl{\{}f_{z}^{\prime}(W)\mathopen{}\mathclose{{}\left(1-\frac{1}{2\lambda}\mathop{{}\mathbb{E}}\mathopen{}\mathopen{}\mathclose{{}\left\{D\Delta\,\middle|\,W}\right\}}\right)\biggr{\}},
J2\displaystyle J_{2} =12λ𝔼{DΔ0(fz(W+u)fz(W))𝑑u},\displaystyle=\frac{1}{2\lambda}\mathop{{}\mathbb{E}}\mathopen{}\biggl{\{}D\int_{-\Delta}^{0}\mathopen{}\mathclose{{}\left(f_{z}^{\prime}(W+u)-f_{z}^{\prime}(W)}\right)\mathop{}\!{d}u\biggr{\}},
J3\displaystyle J_{3} =𝔼{Rfz(W)}.\displaystyle=\mathop{{}\mathbb{E}}\mathopen{}\bigl{\{}Rf_{z}(W)\bigr{\}}.

We now bound J1J_{1}, J2J_{2} and J3J_{3}, separately. By Chen et al. (2011, Lemma 2.3), we have

fz1,fz1,supz|wf(w)|1.\displaystyle\|f_{z}\|\leqslant 1,\quad\|f_{z}^{\prime}\|\leqslant 1,\quad\sup_{z\in\mathbb{R}}\bigl{\lvert}wf(w)\bigr{\rvert}\leqslant 1. (4.9)

Therefore,

|J1|\displaystyle|J_{1}| 𝔼|112λ𝔼{DΔ|W}|,\displaystyle\leqslant\mathop{{}\mathbb{E}}\mathopen{}\biggl{\lvert}1-\frac{1}{2\lambda}\mathop{{}\mathbb{E}}\mathopen{}\mathopen{}\mathclose{{}\left\{D\Delta\,\middle|\,W}\right\}\biggr{\rvert}, (4.10)
|J3|\displaystyle|J_{3}| 𝔼|R|.\displaystyle\leqslant\mathop{{}\mathbb{E}}\mathopen{}|R|.

For J2,J_{2}, observe that fz(w)=wf(w)w>z+{1Φ(z)}f_{z}^{\prime}(w)=wf(w)-\mathbb{N}{w>z}+\bigl{\{}1-\Phi(z)\bigr{\}}, and both wfz(w)wf_{z}(w) and w>z\mathbb{N}{w>z} are increasing functions (see, e.g. Chen et al. (2011, Lemma 2.3)), by Lemma 4.5,

|J2|\displaystyle|J_{2}| 12λ|𝔼{DΔ0{(W+u)fz(W+u)Wfz(W)}𝑑u}|\displaystyle\leqslant\frac{1}{2\lambda}\biggl{\lvert}\mathop{{}\mathbb{E}}\mathopen{}\biggl{\{}D\int_{-\Delta}^{0}\mathopen{}\mathclose{{}\left\{(W+u)f_{z}(W+u)-Wf_{z}^{\prime}(W)}\right\}\mathop{}\!{d}u\biggr{\}}\biggr{\rvert} (4.11)
+12λ|𝔼{DΔ0{W+u>zW>z}𝑑u}|\displaystyle\quad+\frac{1}{2\lambda}\biggl{\lvert}\mathop{{}\mathbb{E}}\mathopen{}\biggl{\{}D\int_{-\Delta}^{0}\mathopen{}\mathclose{{}\left\{\mathbb{N}{W+u>z}-\mathbb{N}{W>z}}\right\}\mathop{}\!{d}u\biggr{\}}\biggr{\rvert}
12λ𝔼{|𝔼{DΔ|W}|(|Wfz(W)|+W>z)}\displaystyle\leqslant\frac{1}{2\lambda}\mathop{{}\mathbb{E}}\mathopen{}\bigl{\{}\bigl{\lvert}\mathop{{}\mathbb{E}}\mathopen{}\mathopen{}\mathclose{{}\left\{D^{*}\Delta\,\middle|\,W}\right\}\bigr{\rvert}\bigl{(}|Wf_{z}(W)|+\mathbb{N}{W>z}\bigr{)}\bigr{\}}
J21+J22,\displaystyle\leqslant J_{21}+J_{22},

where

J21\displaystyle J_{21} =12λ𝔼{|𝔼{DΔ|W}||Wfz(W)|},\displaystyle=\frac{1}{2\lambda}\mathop{{}\mathbb{E}}\mathopen{}\Bigl{\{}\bigl{\lvert}\mathop{{}\mathbb{E}}\mathopen{}\mathopen{}\mathclose{{}\left\{D^{*}\Delta\,\middle|\,W}\right\}\bigr{\rvert}\cdot\bigl{\lvert}Wf_{z}(W)\bigr{\rvert}\Bigr{\}},
J22\displaystyle J_{22} =12λ𝔼{|𝔼{DΔ|W}|W>z}.\displaystyle=\frac{1}{2\lambda}\mathop{{}\mathbb{E}}\mathopen{}\Bigl{\{}\bigl{\lvert}\mathop{{}\mathbb{E}}\mathopen{}\mathopen{}\mathclose{{}\left\{D^{*}\Delta\,\middle|\,W}\right\}\bigr{\rvert}\mathbb{N}{W>z}\Bigr{\}}.

Then, by 4.9, |J2|1λ𝔼|𝔼{DΔ|W}||J_{2}|\leqslant\frac{1}{\lambda}\mathop{{}\mathbb{E}}\mathopen{}\bigl{\lvert}\mathop{{}\mathbb{E}}\mathopen{}\mathopen{}\mathclose{{}\left\{D^{*}\Delta\,\middle|\,W}\right\}\bigr{\rvert}. This proves Theorem 4.1 together with 4.10. ∎

4.3 Proof of Corollary 4.3

In this subsection, we apply Theorem 4.1 to prove Corollary 4.3. By 4.3, we have

𝔼{D|X}=λ(T+UR).\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\{D\,|\,X\}=\lambda(T+U-R).

Let T=φ(X)+U(X)T^{\prime}=\varphi(X^{\prime})+U(X^{\prime}), then we have (T,T)(T,T^{\prime}) is exchangeable. Then, by Theorem 4.1, we have

supz|[Tz]Φ(z)|\displaystyle\sup_{z\in\mathbb{R}}\bigl{\lvert}\mathop{{}\mathbb{P}}\mathopen{}[T\leqslant z]-\Phi(z)\bigr{\rvert}
𝔼|112λ𝔼{D(TT)|X}|\displaystyle\leqslant\mathop{{}\mathbb{E}}\mathopen{}\bigl{\lvert}1-\frac{1}{2\lambda}\mathop{{}\mathbb{E}}\mathopen{}\{D(T-T^{\prime})\,|\,X\}\bigr{\rvert}
+1λ𝔼|𝔼{D(TT)|X}|+𝔼|U|+𝔼|R|\displaystyle\quad+\frac{1}{\lambda}\mathop{{}\mathbb{E}}\mathopen{}\bigl{\lvert}\mathop{{}\mathbb{E}}\mathopen{}\{D^{*}(T-T^{\prime})\,|\,X\}\bigr{\rvert}+\mathop{{}\mathbb{E}}\mathopen{}|U|+\mathop{{}\mathbb{E}}\mathopen{}|R|
𝔼|112λ𝔼{D(φ(X)φ(X))|X}|\displaystyle\leqslant\mathop{{}\mathbb{E}}\mathopen{}\bigl{\lvert}1-\frac{1}{2\lambda}\mathop{{}\mathbb{E}}\mathopen{}\{D(\varphi(X)-\varphi(X^{\prime}))\,|\,X\}\bigr{\rvert}
+1λ𝔼|𝔼{D(φ(X)φ(X))|X}|+𝔼|U|+𝔼|R|+32λ𝔼|D(RR)|.\displaystyle\quad+\frac{1}{\lambda}\mathop{{}\mathbb{E}}\mathopen{}\bigl{\lvert}\mathop{{}\mathbb{E}}\mathopen{}\{D^{*}(\varphi(X)-\varphi(X^{\prime}))\,|\,X\}\bigr{\rvert}+\mathop{{}\mathbb{E}}\mathopen{}|U|+\mathop{{}\mathbb{E}}\mathopen{}|R|+\frac{3}{2\lambda}\mathop{{}\mathbb{E}}\mathopen{}|D(R-R^{\prime})|.

This completes the proof by recalling that Δ=φ(X)Φ(X)\Delta=\varphi(X)-\Phi(X^{\prime}).

5 Proofs of Theorems 2.1, 2.3 and 2.4

In this section, we give the proofs of Theorems 2.1, 2.3 and 2.4.

5.1 Proof of Theorem 2.1

Without loss of generality, we assume that nmax(2,k2)n\geqslant\max(2,k^{2}), otherwise the inequality is trivial. We use Corollary 4.3 to prove this theorem. For each α=(α(1),,α(k))n,k\alpha=(\alpha(1),\dots,\alpha(k))\in\mathcal{I}_{n,k}, let

r(Xα(1),,Xα(k);Yα(1),α(2),,Yα(k1),α(k))\displaystyle r(X_{\alpha(1)},\dots,X_{\alpha(k)};Y_{\alpha(1),\alpha(2)},\dots,Y_{\alpha(k-1),\alpha(k)}) (5.1)
=f(Xα(1),,Xα(k);Yα(1),α(2),,Yα(k1),α(k))j=1kf1(Xα(j)).\displaystyle=f(X_{\alpha(1)},\dots,X_{\alpha(k)};Y_{\alpha(1),\alpha(2)},\dots,Y_{\alpha(k-1),\alpha(k)})-\sum_{j=1}^{k}f_{1}(X_{\alpha(j)}).

Let σn=Var{Sn,k(f)}\sigma_{n}=\sqrt{\mathop{\mathrm{Var}}\{S_{n,k}(f)\}}, and

T=1σn(Sn,k(f)𝔼{Sn,k(f)})=W+U,\displaystyle T=\frac{1}{\sigma_{n}}(S_{n,k}(f)-\mathop{{}\mathbb{E}}\mathopen{}\{S_{n,k}(f)\})=W+U,

where

W\displaystyle W =1σn(n1\@@atopnk)i=1nf1(Xi),\displaystyle=\frac{1}{\sigma_{n}}\binom{n-1}{n-k}\sum_{i=1}^{n}f_{1}(X_{i}),
U\displaystyle U =1σnαn,kr(Xα(1),,Xα(k);Yα(1),α(2),,Yα(k1),α(k)).\displaystyle=\frac{1}{\sigma_{n}}\sum_{\alpha\in\mathcal{I}_{n,k}}r(X_{\alpha(1)},\dots,X_{\alpha(k)};Y_{\alpha(1),\alpha(2)},\dots,Y_{\alpha(k-1),\alpha(k)}).

By orthogonality we have Cov(W,U)=0\mathop{\mathrm{Cov}}(W,U)=0, and thus

σn2Var(W)=(n1\@@atopnk)2Var(j=1nf1(Xj))=(n\@@atopk)2k2σ12n.\displaystyle\sigma_{n}^{2}\geqslant\mathop{\mathrm{Var}}(W)=\binom{n-1}{n-k}^{2}\mathop{\mathrm{Var}}\biggl{(}\sum_{j=1}^{n}f_{1}(X_{j})\biggr{)}=\binom{n}{k}^{2}\frac{k^{2}\sigma_{1}^{2}}{n}. (5.2)

Let (X1,,Xn)(X_{1}^{\prime},\dots,X_{n}^{\prime}) be an independent copy of (X1,,Xn)(X_{1},\dots,X_{n}). For each i=1,,ni=1,\dots,n, define X(i)=(X1(i),,Xn(i))X^{(i)}=(X_{1}^{(i)},\dots,X_{n}^{(i)}) where

Xj(i)={Xjif ji,Xiif j=i,\displaystyle X_{j}^{(i)}=\begin{cases}X_{j}&\text{if $j\neq i$,}\\ X_{i}^{\prime}&\text{if $j=i$,}\end{cases}

and let

U(i)=1σnαn,kr(Xα(1)(i),,Xα(k)(i);Yα(1),α(2),,Yα(k1),α(k)).\displaystyle U^{(i)}=\frac{1}{\sigma_{n}}\sum_{\alpha\in\mathcal{I}_{n,k}}r(X_{\alpha(1)}^{(i)},\dots,X_{\alpha(k)}^{(i)};Y_{\alpha(1),\alpha(2)},\dots,Y_{\alpha(k-1),\alpha(k)}).

The following lemma provides the upper bounds of 𝔼{R12}\mathop{{}\mathbb{E}}\mathopen{}\{R_{1}^{2}\} and 𝔼{(R1R1(i))2}\mathop{{}\mathbb{E}}\mathopen{}\{(R_{1}-R_{1}^{(i)})^{2}\}.

Lemma 5.1.

For n2n\geqslant 2 and k2k\geqslant 2,

𝔼{U2}\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\{U^{2}\} (k1)2τ22(n1)σ12\displaystyle\leqslant\frac{(k-1)^{2}\tau^{2}}{2(n-1)\sigma_{1}^{2}} (5.3)
𝔼{(UU(i))2}\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\{(U-U^{(i)})^{2}\} 2(k1)2τ2n(n1)σ12.\displaystyle\leqslant\frac{2(k-1)^{2}\tau^{2}}{n(n-1)\sigma_{1}^{2}}. (5.4)

The proof of Lemma 5.1 is put in the appendix.

Now, we apply Corollary 4.3 to prove the Berry–Esseen bound for TT. To this end, let ξi=σn1f1(Xi)\xi_{i}=\sigma_{n}^{-1}f_{1}(X_{i}) for each 1in1\leqslant i\leqslant n. Let II be a random index uniformly distributed over {1,,n}\{1,\dots,n\}, which is independent of all others. Let

D=Δ=1σn(n1\@@atopnk)(f1(XI)f1(XI)),\displaystyle D=\Delta=\frac{1}{\sigma_{n}}\binom{n-1}{n-k}\bigl{(}f_{1}(X_{I})-f_{1}(X_{I}^{\prime})\bigr{)},

then it follows that

𝔼{D|W}=1nW.\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\{D\,|\,W\}=\frac{1}{n}W.

Thus, 4.3 is satisfied with λ=1/n\lambda=1/n and R=0R=0. Moreover, we have

12λ𝔼{DΔ|X}\displaystyle\frac{1}{2\lambda}\mathop{{}\mathbb{E}}\mathopen{}\{D\Delta\,|\,X\} =12σn2(n1\@@atopnk)2i=1n(f1(Xi)f1(Xi))2,\displaystyle=\frac{1}{2\sigma_{n}^{2}}\binom{n-1}{n-k}^{2}\sum_{i=1}^{n}\bigl{(}f_{1}(X_{i})-f_{1}(X_{i}^{\prime})\bigr{)}^{2},
1λ𝔼{|D|Δ|X}\displaystyle\frac{1}{\lambda}\mathop{{}\mathbb{E}}\mathopen{}\{|D|\Delta\,|\,X\} =1σn2(n1\@@atopnk)2i=1n(f1(Xi)f1(Xi))|f1(Xi)f1(Xi)|.\displaystyle=\frac{1}{\sigma_{n}^{2}}\binom{n-1}{n-k}^{2}\sum_{i=1}^{n}\bigl{(}f_{1}(X_{i})-f_{1}(X_{i}^{\prime})\bigr{)}\bigl{\lvert}f_{1}(X_{i})-f_{1}(X_{i}^{\prime})\bigr{\rvert}.

Also,

12λ𝔼{DΔ}\displaystyle\frac{1}{2\lambda}\mathop{{}\mathbb{E}}\mathopen{}\{D\Delta\} =𝔼{W2}=1𝔼{U2},𝔼{|D|Δ}=0.\displaystyle=\mathop{{}\mathbb{E}}\mathopen{}\{W^{2}\}=1-\mathop{{}\mathbb{E}}\mathopen{}\{U^{2}\},\quad\mathop{{}\mathbb{E}}\mathopen{}\{|D|\Delta\}=0.

Therefore, by the Cauchy inequality and Lemma 5.1, we have for nmax(2,k2)n\geqslant\max(2,k^{2}),

𝔼|12λ𝔼{DΔ|X}1|\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\biggl{\lvert}\frac{1}{2\lambda}\mathop{{}\mathbb{E}}\mathopen{}\{D\Delta\,|\,X\}-1\biggr{\rvert}
𝔼|12λ𝔼{DΔ|X}12λ𝔼{DΔ}|+𝔼{U2}\displaystyle\leqslant\mathop{{}\mathbb{E}}\mathopen{}\biggl{\lvert}\frac{1}{2\lambda}\mathop{{}\mathbb{E}}\mathopen{}\{D\Delta\,|\,X\}-\frac{1}{2\lambda}\mathop{{}\mathbb{E}}\mathopen{}\{D\Delta\}\biggr{\rvert}+\mathop{{}\mathbb{E}}\mathopen{}\{U^{2}\}
12σn2(n1\@@atopnk)2(Var{i=1n(f1(Xi)f1(Xi))2})1/2+(k1)2τ22(n1)σ12\displaystyle\leqslant\frac{1}{2\sigma_{n}^{2}}\binom{n-1}{n-k}^{2}\biggl{(}\mathop{\mathrm{Var}}\biggl{\{}\sum_{i=1}^{n}\bigl{(}f_{1}(X_{i})-f_{1}(X_{i}^{\prime})\bigr{)}^{2}\biggr{\}}\biggr{)}^{1/2}+\frac{(k-1)^{2}\tau^{2}}{2(n-1)\sigma_{1}^{2}}
2τ2nσ12+(k1)τ2nσ12(k+1)τ2nσ12,\displaystyle\leqslant\frac{2\tau^{2}}{\sqrt{n}\sigma_{1}^{2}}+\frac{(k-1)\tau^{2}}{\sqrt{n}\sigma_{1}^{2}}\leqslant\frac{(k+1)\tau^{2}}{\sqrt{n}\sigma_{1}^{2}},

where we used 5.2 in the last line. Using the same argument, we have for nmax{2,k2}n\geqslant\max\{2,k^{2}\},

𝔼|1λ𝔼{|D|Δ|X}|\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\biggl{\lvert}\frac{1}{\lambda}\mathop{{}\mathbb{E}}\mathopen{}\{|D|\Delta\,|\,X\}\biggr{\rvert}
1σn2(n1\@@atopnk)2(Var{i=1n(f1(Xi)f1(Xi))})1/2\displaystyle\leqslant\frac{1}{\sigma_{n}^{2}}\binom{n-1}{n-k}^{2}\biggl{(}\mathop{\mathrm{Var}}\biggl{\{}\sum_{i=1}^{n}\bigl{(}f_{1}(X_{i})-f_{1}(X_{i}^{\prime})\bigr{)}\biggr{\}}\biggr{)}^{1/2}
4τ2nσ12.\displaystyle\leqslant\frac{4\tau^{2}}{\sqrt{n}\sigma_{1}^{2}}.

Now we give the bounds for UU and U(i)U^{(i)}. We have two cases. For the case where k=1k=1, then it follows that U=0U=0 and U(i)=0U^{(i)}=0. As for k2k\geqslant 2, noting that (n1)1/22n1/2(n-1)^{-1/2}\leqslant 2n^{-1/2} for n2n\geqslant 2, by Lemma 5.1 and the Cauchy inequality, we have

𝔼|U|\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\lvert U\rvert 0.71(k1)τ(n1)1/2σ12(k1)τnσ1,\displaystyle\leqslant\frac{0.71(k-1)\tau}{(n-1)^{1/2}\sigma_{1}}\leqslant\frac{2(k-1)\tau}{\sqrt{n}\sigma_{1}},

and

i=1n𝔼{|(ξiξi)(UU(i))|}\displaystyle\sum_{i=1}^{n}\mathop{{}\mathbb{E}}\mathopen{}\{\lvert(\xi_{i}-\xi_{i}^{\prime})(U-U^{(i)})\rvert\} 2.84(k1)τ(n1)1/2σ16(k1)τnσ1.\displaystyle\leqslant\frac{2.84(k-1)\tau}{(n-1)^{1/2}\sigma_{1}}\leqslant\frac{6(k-1)\tau}{\sqrt{n}\sigma_{1}}.

By Corollary 4.3 and noting that σ12𝔼{f(X{α};Y{α})2}τ1/2\sigma_{1}^{2}\leqslant\mathop{{}\mathbb{E}}\mathopen{}\{f(X_{\{\alpha\}};Y_{\{\alpha\}})^{2}\}\leqslant\tau^{1/2}, we have

supz|[Tz]Φ(z)|\displaystyle\sup_{z\in\mathbb{R}}\bigl{\lvert}\mathop{{}\mathbb{P}}\mathopen{}[T\leqslant z]-\Phi(z)\bigr{\rvert} (k+5)τ2nσ12+11(k1)τnσ1\displaystyle\leqslant\frac{(k+5)\tau^{2}}{\sqrt{n}\sigma_{1}^{2}}+\frac{11(k-1)\tau}{\sqrt{n}\sigma_{1}}
12kτ2nσ12.\displaystyle\leqslant\frac{12k\tau^{2}}{\sqrt{n}\sigma_{1}^{2}}.

This proves 2.3.

5.2 Proof of Theorem 2.2

We first prove a proposition for the Hoeffding decomposition.

Proposition 5.2.

For A[n],B[n]2A\subset[n],B\subset[n]_{2} such that (A,B)(,)(A,B)\neq(\varnothing,\varnothing), and for any A~,B~\widetilde{A},\widetilde{B} such that A~A\widetilde{A}\subset A and B~B\widetilde{B}\subset B but (A~,B~)(A,B)(\widetilde{A},\widetilde{B})\neq(A,B), we have

𝔼{fA,B(XA;YB)|XA~,YB~}=0.\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\bigl{\{}f_{A,B}(X_{A};Y_{B})\bigm{|}X_{\widetilde{A}},Y_{\widetilde{B}}\bigr{\}}=0. (5.5)
Proof.

If |A|+|B|=1|A|+|B|=1, then for (A~,B~)=(,)(\widetilde{A},\widetilde{B})=(\varnothing,\varnothing), by definition,

𝔼{fA,B(XA;YB)|XA~,YB~}\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\bigl{\{}f_{A,B}(X_{A};Y_{B})\bigm{|}X_{\widetilde{A}},Y_{\widetilde{B}}\bigr{\}} =𝔼{fA,B(XA;YB)}\displaystyle=\mathop{{}\mathbb{E}}\mathopen{}\bigl{\{}f_{A,B}(X_{A};Y_{B})\bigr{\}}
=𝔼{f(X[k];Y[k]2)}𝔼{f(X[k];Y[k]2)}=0.\displaystyle=\mathop{{}\mathbb{E}}\mathopen{}\{f(X_{[k]};Y_{[k]_{2}})\}-\mathop{{}\mathbb{E}}\mathopen{}\{f(X_{[k]};Y_{[k]_{2}})\}=0.

We prove the proposition by induction. Assume that 5.5 holds for 1|A|+|B|m1\leqslant|A|+|B|\leqslant m. Let 𝒜A~,B~={(A,B):AA~,BB~,}\mathcal{A}_{\widetilde{A},\widetilde{B}}=\{(A^{\prime},B^{\prime}):A^{\prime}\subset\widetilde{A},B^{\prime}\subset\widetilde{B},\} and let 𝒜A~,B~c={(A,B):AA,BB,(A,B)(A,B)}𝒜A~,B~c\mathcal{A}_{\widetilde{A},\widetilde{B}}^{c}=\{(A^{\prime},B^{\prime}):A^{\prime}\subset A,B^{\prime}\subset B,(A^{\prime},B^{\prime})\neq(A,B)\}\setminus\mathcal{A}_{\widetilde{A},\widetilde{B}}^{c}. Reordering 2.1 by the inclusive-exclusive formula we have

fA,B(XA;YB)\displaystyle f_{A,B}(X_{A};Y_{B}) =𝔼{f(X[k];Y[k]2)|XA,YB}|A|+|B|<|A|+|B|fA,B(XA;YB)\displaystyle=\mathop{{}\mathbb{E}}\mathopen{}\{f(X_{[k]};Y_{[k]_{2}})|X_{A},Y_{B}\}-\sum_{|A^{\prime}|+|B^{\prime}|<|A|+|B|}f_{A^{\prime},B^{\prime}}(X_{A^{\prime}};Y_{B^{\prime}})
=𝔼{f(X[k];Y[k]2)|XA,YB}(A,B)𝒜A~,B~fA,B(XA;YB)\displaystyle=\mathop{{}\mathbb{E}}\mathopen{}\{f(X_{[k]};Y_{[k]_{2}})|X_{A},Y_{B}\}-\sum_{(A^{\prime},B^{\prime})\in\mathcal{A}_{\widetilde{A},\widetilde{B}}}f_{A^{\prime},B^{\prime}}(X_{A^{\prime}};Y_{B^{\prime}})
(A,B)𝒜A~,B~cfA,B(XA;YB)\displaystyle\quad-\sum_{(A^{\prime},B^{\prime})\in\mathcal{A}_{\widetilde{A},\widetilde{B}}^{c}}f_{A^{\prime},B^{\prime}}(X_{A^{\prime}};Y_{B^{\prime}})
=𝔼{f(X[k];Y[k]2)|XA,YB}𝔼{f(X[k];Y[k]2)|XA~,YB~}\displaystyle=\mathop{{}\mathbb{E}}\mathopen{}\{f(X_{[k]};Y_{[k]_{2}})|X_{A},Y_{B}\}-\mathop{{}\mathbb{E}}\mathopen{}\{f(X_{[k]};Y_{[k]_{2}})|X_{\widetilde{A}},Y_{\widetilde{B}}\}
(A,B)𝒜A~,B~cfA,B(XA;YB).\displaystyle\quad-\sum_{(A^{\prime},B^{\prime})\in\mathcal{A}_{\widetilde{A},\widetilde{B}}^{c}}f_{A^{\prime},B^{\prime}}(X_{A^{\prime}};Y_{B^{\prime}}).

By the induction assumption, we have

(A,B)𝒜A~,B~c𝔼{fA,B(XA;YB)|XA~,YB~}=0.\displaystyle\sum_{(A^{\prime},B^{\prime})\in\mathcal{A}_{\widetilde{A},\widetilde{B}}^{c}}\mathop{{}\mathbb{E}}\mathopen{}\{f_{A^{\prime},B^{\prime}}(X_{A^{\prime}};Y_{B^{\prime}})|X_{\widetilde{A}},Y_{\widetilde{B}}\}=0.

Then, the desired result follows. ∎

Let

𝒜n,={α=(α(1),,α()):1α(1)α()n}.\displaystyle\mathcal{A}_{n,\ell}=\{\alpha=(\alpha(1),\dots,\alpha(\ell)):1\leqslant\alpha(1)\neq\dots\neq\alpha(\ell)\leqslant n\}.

Then, n,𝒜n,\mathcal{I}_{n,\ell}\subset\mathcal{A}_{n,\ell}. For A[]A\subset[\ell] and B[]2B\in[\ell]_{2} and α=(α(1),,α())𝒜n,\alpha=(\alpha(1),\dots,\alpha(\ell))\in\mathcal{A}_{n,\ell}, write

α(A)\displaystyle\alpha(A) =(α(i))iA,\displaystyle=(\alpha(i))_{i\in A}, α(B)\displaystyle\alpha(B) =((α(i),α(j)))(i,j)B,\displaystyle=\bigl{(}(\alpha(i),\alpha(j))\bigr{)}_{(i,j)\in B},
Xα(A)\displaystyle X_{\alpha(A)} =(Xi)iα(A),\displaystyle=(X_{i})_{i\in\alpha(A)}, Yα(B)\displaystyle Y_{\alpha(B)} =(Yi,j)(i,j)α(B).\displaystyle=(Y_{i,j})_{(i,j)\in\alpha(B)}.

Moreover, for any αn,\alpha\in\mathcal{I}_{n,\ell} and fA,B:𝒳|A|×𝒴|B|f_{A,B}:\mathcal{X}^{|A|}\times\mathcal{Y}^{|B|}\to\mathbb{R}, let

S~n,(fA,B)=α𝒜n,fA,B(Xα(A);Yα(B)),\displaystyle\widetilde{S}_{n,\ell}(f_{A,B})=\sum_{\alpha\in\mathcal{A}_{n,\ell}}f_{A,B}(X_{\alpha(A)};Y_{\alpha(B)}),

and similarly, Sn,(fA,B)S_{n,\ell}(f_{A,B}) can be represented as αn,fA,B(Xα(A);Yα(B)).\sum_{\alpha\in\mathcal{I}_{n,\ell}}f_{A,B}(X_{\alpha(A)};Y_{\alpha(B)}).

Let (Y1,1,,Yn1,n)(Y_{1,1}^{\prime},\dots,Y_{n-1,n}^{\prime}) be an independent copy of Y=(Y1,1,,Yn1,n)Y=(Y_{1,1},\dots,Y_{n-1,n}). For any (i,j)𝒜n,2(i,j)\in\mathcal{A}_{n,2}, let Y(i,j)=(Y1,1(i,j),,Yn1,n(i,j))Y^{(i,j)}=(Y_{1,1}^{(i,j)},\dots,Y_{n-1,n}^{(i,j)}) with

Yp,q(i,j)={Yp,q if {p,q}{i,j},Yp,q if {p,q}={i,j},for (p,q)n,2.\displaystyle Y_{p,q}^{(i,j)}=\begin{cases}Y_{p,q}&\text{ if $\{p,q\}\neq\{i,j\}$,}\\ Y_{p,q}^{\prime}&\text{ if $\{p,q\}=\{i,j\}$,}\end{cases}\quad\text{for }(p,q)\in\mathcal{I}_{n,2}.

Then, it follows that for each (i,j)𝒜n,2(i,j)\in\mathcal{A}_{n,2}, ((X,Y),(X,Y(i,j)))((X,Y),(X,Y^{(i,j)})) is an exchangeable pair. For any B[n]2B\subset[n]_{2}, let YB(i,j)=(Yp,q(i,j))(p,q)BY_{B}^{(i,j)}=(Y_{p,q}^{(i,j)})_{(p,q)\in B}. For any A[]A\subset[\ell], B[]2B\subset[\ell]_{2}, α=(α(1),,α())n,\alpha=(\alpha(1),\dots,\alpha(\ell))\in\mathcal{I}_{n,\ell} and fA,B:𝒳|A|×𝒴|B|f_{A,B}:\mathcal{X}^{|A|}\times\mathcal{Y}^{|B|}\to\mathbb{R}, define

Yα(B)(i,j)\displaystyle Y_{\alpha(B)}^{(i,j)} =(Yα(p),α(q)(i,j))(p,q)B,\displaystyle=(Y_{\alpha(p),\alpha(q)}^{(i,j)})_{(p,q)\in B},
Sn,(i,j)(fA,B)\displaystyle S_{n,\ell}^{(i,j)}(f_{A,B}) =αn,fA,B(Xα(A);Yα(B)(i,j)),\displaystyle=\sum_{\alpha\in\mathcal{I}_{n,\ell}}f_{A,B}(X_{\alpha(A)};Y_{\alpha(B)}^{(i,j)}),
S~n,(i,j)(fA,B)\displaystyle\widetilde{S}_{n,\ell}^{(i,j)}(f_{A,B}) =α𝒜n,fA,B(Xα(A),Yα(B)(i,j)).\displaystyle=\sum_{\alpha\in\mathcal{A}_{n,\ell}}f_{A,B}(X_{\alpha(A)},Y_{\alpha(B)}^{(i,j)}).

Let f()f_{(\ell)} be defined as in 2.2, and it follows that

f==0kf(),f(0)=𝔼{f(X[k];Y[k]2)},Sn,k(f(0))=𝔼{Sn,k(f)}.\displaystyle f=\sum_{\ell=0}^{k}f_{(\ell)},\quad f_{(0)}=\mathop{{}\mathbb{E}}\mathopen{}\{f(X_{[k]};Y_{[k]_{2}})\},\quad S_{n,k}(f_{(0)})=\mathop{{}\mathbb{E}}\mathopen{}\{S_{n,k}(f)\}.

Moreover, by assumption, as ff has principal degree dd, and it follows that f()0f_{(\ell)}\equiv 0 for =1,,d1\ell=1,\dots,d-1. Let σn=(Var{Sn,k(f)})1/2\sigma_{n}=(\mathop{\mathrm{Var}}\{S_{n,k}(f)\})^{1/2} and σn,=(Var{Sn,k(f())})1/2\sigma_{n,\ell}=(\mathop{\mathrm{Var}}\{S_{n,k}(f_{(\ell)})\})^{1/2}. The next lemma estimates the upper and lower bounds of σn2\sigma_{n}^{2} and σn,d2\sigma_{n,d}^{2}. The proof is similar to that of Lemma 4 of Janson and Nowicki (1991), and we omit the details.

Lemma 5.3.

We have for each (i,j)𝒜n,2(i,j)\in\mathcal{A}_{n,2} and dkd\leqslant\ell\leqslant k,

σn,2=(A,B):vA,B=n!(n)!σA,B2(nk)!2(k)!2|Aut(GA,B)|Cn2kτ2,\displaystyle\sigma_{n,\ell}^{2}=\sum_{(A,B):v_{A,B}=\ell}\frac{n!(n-\ell)!\sigma_{A,B}^{2}}{(n-k)!^{2}(k-\ell)!^{2}\lvert\operatorname{Aut}(G_{A,B})\rvert}\leqslant Cn^{2k-\ell}\tau^{2}, (5.6)
σn2==dk(A,B):vA,B=n!(n)!σA,B2(nk)!2(k)!2|Aut(GA,B)|Cn2kdτ2,\displaystyle\sigma_{n}^{2}=\sum_{\ell=d}^{k}\sum_{(A,B):v_{A,B}=\ell}\frac{n!(n-\ell)!\sigma_{A,B}^{2}}{(n-k)!^{2}(k-\ell)!^{2}\lvert\operatorname{Aut}(G_{A,B})\rvert}\leqslant Cn^{2k-d}\tau^{2}, (5.7)
𝔼{(Sn,k(f())Sn,k(i,j)(f()))2}Cn2k2τ2,\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\{(S_{n,k}(f_{(\ell)})-S_{n,k}^{(i,j)}(f_{(\ell)}))^{2}\}\leqslant Cn^{2k-\ell-2}\tau^{2}, (5.8)

and

σn2σn,d2cn2kdσmin2.\displaystyle\sigma_{n}^{2}\geqslant\sigma_{n,d}^{2}\geqslant cn^{2k-d}\sigma_{\min}^{2}. (5.9)

where |Aut(G)|\lvert\operatorname{Aut}(G)\rvert is the number of the automorphisms of GG, and c,C>0c,C>0 are some absolute constant.

For any A[k]A\subset[k] and B[k]2B\subset[k]_{2}, let

μA,B\displaystyle\mu_{A,B} 1|Aut(GA,B)||B|(nvA,B\@@atopnk),\displaystyle\coloneqq\frac{1}{\lvert\operatorname{Aut}(G_{A,B})\rvert|B|}\binom{n-v_{A,B}}{n-k},
νA,B\displaystyle\nu_{A,B} |B|×μA,B=1|Aut(GA,B)|(nvA,B\@@atopnk),\displaystyle\coloneqq|B|\times\mu_{A,B}=\frac{1}{\lvert\operatorname{Aut}(G_{A,B})\rvert}\binom{n-v_{A,B}}{n-k},

and for any α𝒜n,\alpha\in\mathcal{A}_{n,\ell} (=1,,k\ell=1,\dots,k), let

ξα(A,B)(i,j)=fA,B(Xα(A);Yα(B))fA,B(Xα(A);Yα(B)(i,j)).\displaystyle\xi_{\alpha(A,B)}^{(i,j)}=f_{A,B}(X_{\alpha(A)};Y_{\alpha(B)})-f_{A,B}(X_{\alpha(A)};Y_{\alpha(B)}^{(i,j)}).

Recall that GA,BG_{A,B} is the graph generated by (A,B)(A,B). For any (Aj,Bj)(A_{j},B_{j}) for j=1,2j=1,2, we simply write vj=vAj,Bjv_{j}=v_{A_{j},B_{j}} as the number of nodes of the graph GAj,BjG_{A_{j},B_{j}}. Recall that 𝒢f,d={(A,B):A[d],B[d]2,σA,B>0,vA,B=d}\mathcal{G}_{f,d}=\{(A,B):A\subset[d],B\subset[d]_{2},\sigma_{A,B}>0,v_{A,B}=d\} and we similarly define 𝒢f,d+1={(A,B):A[d+1],B[d+1]2,σA,B>0,vA,B=d+1}\mathcal{G}_{f,d+1}=\{(A,B):A\subset[d+1],B\subset[d+1]_{2},\sigma_{A,B}>0,v_{A,B}=d+1\}. We have the following lemmas.

Lemma 5.4.

For all (A1,B1)(A_{1},B_{1}), (A2,B2)𝒢f,d(A_{2},B_{2})\in\mathcal{G}_{f,d} such that GA1,B1G_{A_{1},B_{1}} and GA2,B2G_{A_{2},B_{2}} are connected, we have

Var{(i,j)𝒜n,2(α1𝒜n,d(i,j)ξα1(A1,B1)(i,j))(α2𝒜n,d(i,j)ξα2(A2,B2)(i,j))}Cn2d1τ4.\displaystyle\mathop{\mathrm{Var}}\Biggl{\{}\sum_{(i,j)\in\mathcal{A}_{n,2}}\biggl{(}\sum_{\alpha_{1}\in\mathcal{A}_{n,d}^{(i,j)}}\xi_{\alpha_{1}(A_{1},B_{1})}^{(i,j)}\biggr{)}\biggl{(}\sum_{\alpha_{2}\in\mathcal{A}_{n,d}^{(i,j)}}\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)}\biggr{)}\Biggr{\}}\leqslant Cn^{2d-1}\tau^{4}.
Lemma 5.5.

Assume that kd+1k\geqslant d+1. For all (A1,B1)(A_{1},B_{1}), (A2,B2)𝒢f,d𝒢f,d+1(A_{2},B_{2})\in\mathcal{G}_{f,d}\cup\mathcal{G}_{f,d+1}, we have

Var{(i,j)𝒜n,2(α1𝒜n,v1(i,j)ξα1(A1,B1)(i,j))|α2𝒜n,v2(i,j)ξα2(A2,B2)(i,j)|}Cn2max{v1,v2}2τ4.\displaystyle\mathop{\mathrm{Var}}\Biggl{\{}\sum_{(i,j)\in\mathcal{A}_{n,2}}\biggl{(}\sum_{\alpha_{1}\in\mathcal{A}_{n,v_{1}}^{(i,j)}}\xi_{\alpha_{1}(A_{1},B_{1})}^{(i,j)}\biggr{)}\biggl{\lvert}\sum_{\alpha_{2}\in\mathcal{A}_{n,v_{2}}^{(i,j)}}\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)}\biggr{\rvert}\Biggr{\}}\leqslant Cn^{2\max\{v_{1},v_{2}\}-2}\tau^{4}.

We are now ready to give the proof of Theorem 2.3.

Proof of Theorem 2.3.

We assume that nmax{k,2}n\geqslant\max\{k,2\} without loss of generality, otherwise the result is trivial. Recall that f(d)f_{(d)} is defined in 2.2. Write T=σn1(Sn,k(f)𝔼{Sn,k(f)})T=\sigma_{n}^{-1}(S_{n,k}(f)-\mathop{{}\mathbb{E}}\mathopen{}\{S_{n,k}(f)\}), and

W=σn1Sn,k(f(d)),U=TW=σn1=d+1kSn,k(f()).\displaystyle W=\sigma_{n}^{-1}S_{n,k}(f_{(d)}),\quad U=T-W=\sigma_{n}^{-1}\sum_{\ell=d+1}^{k}S_{n,k}(f_{(\ell)}). (5.10)

Here, if d+1>kd+1>k, then set =d+1kSn,k(f())=0\sum_{\ell=d+1}^{k}S_{n,k}(f_{(\ell)})=0. With a slight abuse of notation, we write (A,B)𝒢f,d(A,B)\in\mathcal{G}_{f,d} if GA,B𝒢f,dG_{A,B}\in\mathcal{G}_{f,d}. We have

W\displaystyle W =1σnα𝒜n,d(A,B)𝒢f,d(nd\@@atopkd)fA,B(Xα(A);Yα(B))|Aut(GA,B)|\displaystyle=\frac{1}{\sigma_{n}}\sum_{\alpha\in\mathcal{A}_{n,d}}\sum_{(A,B)\in\mathcal{G}_{f,d}}\binom{n-d}{k-d}\frac{f_{A,B}(X_{\alpha(A)};Y_{\alpha(B)})}{\lvert\operatorname{Aut}(G_{A,B})\rvert}
=1σnα𝒜n,d(A,B)𝒢f,d(nd\@@atopkd)fA,B(Xα(A);Yα(B))|Aut(GA,B)|,\displaystyle=\frac{1}{\sigma_{n}}\sum_{\alpha\in\mathcal{A}_{n,d}}\sum_{({A,B})\in\mathcal{G}_{f,d}}\binom{n-d}{k-d}\frac{f_{A,B}(X_{\alpha(A)};Y_{\alpha(B)})}{\lvert\operatorname{Aut}(G_{A,B})\rvert},

because by assumption, fA,B0f_{A,B}\equiv 0 for all (A,B)𝒢f,d(A,B)\in\mathcal{G}_{f,d}.

For each (i,j)𝒜n,2(i,j)\in\mathcal{A}_{n,2}, let

W(i,j)=1σnSn,k(i,j)(f(d)),U(i,j)=σn1=d+1kSn,k(i,j)(f()).\displaystyle W^{(i,j)}=\frac{1}{\sigma_{n}}S_{n,k}^{(i,j)}(f_{(d)}),\quad U^{(i,j)}=\sigma_{n}^{-1}\sum_{\ell=d+1}^{k}S_{n,k}^{(i,j)}(f_{(\ell)}).

Let (I,J)(I,J) be a random 2-fold index uniformly chosen in 𝒜n,2\mathcal{A}_{n,2}, which is independent of all others. Then, ((X,Y),(X,Y(I,J)))((X,Y),(X,Y^{(I,J)})) is an exchangeable pair. Let

Δ\displaystyle\Delta =WW(I,J)=1σnα𝒜n,d(A,B)𝒢f,dνA,Bξα(A,B)(I,J).\displaystyle=W-W^{(I,J)}=\frac{1}{\sigma_{n}}\sum_{\alpha\in\mathcal{A}_{n,d}}\sum_{(A,B)\in\mathcal{G}_{f,d}}\nu_{A,B}\xi_{\alpha(A,B)}^{(I,J)}.

Also, define

D=1σnα𝒜n,d(A,B)𝒢f,dμA,Bξα(A,B)(I,J).\displaystyle D=\frac{1}{\sigma_{n}}\sum_{\alpha\in\mathcal{A}_{n,d}}\sum_{(A,B)\in\mathcal{G}_{f,d}}\mu_{A,B}\xi_{\alpha(A,B)}^{(I,J)}.

Then, we have DD is antisymmetric with respect to (X,Y)(X,Y) and (X,Y(I,J))(X,Y^{(I,J)}).

Let 𝒜n,d(i,j)={α𝒜n,d:{i,j}{α}}\mathcal{A}_{n,d}^{(i,j)}=\{\alpha\in\mathcal{A}_{n,d}:\{i,j\}\subset\{\alpha\}\}. Then,

𝔼{D|X,Y}\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\{D\,|\,X,Y\}
=1n(n1)σn(i,j)𝒜n,2α𝒜n,d(i,j)(A,B)𝒢f,dμA,B𝔼{ξα(A,B)(i,j)|X,Y}.\displaystyle=\frac{1}{n(n-1)\sigma_{n}}\sum_{(i,j)\in\mathcal{A}_{n,2}}\sum_{\alpha\in\mathcal{A}_{n,d}^{(i,j)}}\sum_{(A,B)\in\mathcal{G}_{f,d}}\mu_{A,B}\mathop{{}\mathbb{E}}\mathopen{}\bigl{\{}\xi_{\alpha(A,B)}^{(i,j)}\,|\,X,Y\bigr{\}}.

By 5.5,

𝔼{fA,B(Xα(A);Yα(B)(i,j))|X,Y}\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\{f_{A,B}(X_{\alpha(A)};Y_{\alpha(B)}^{(i,j)})\,|\,X,Y\}
=𝔼{fA,B(Xα(A);Yα(B))|XA,YB{Yi,j}}\displaystyle=\mathop{{}\mathbb{E}}\mathopen{}\{f_{A,B}(X_{\alpha(A)};Y_{\alpha(B)})\,|\,X_{A},Y_{B}\setminus\{Y_{i,j}\}\}
={0if (i,j)BfA,B(Xα(A);Yα(B))otherwise.\displaystyle=\begin{cases}0&\text{if $(i,j)\in B$, }\\ f_{A,B}(X_{\alpha(A)};Y_{\alpha(B)})&\text{otherwise}.\end{cases}

Moreover, note that for α𝒜n,d\alpha\in\mathcal{A}_{n,d},

(i,j)𝒜n,2(i,j)α(B)=2|B|,\displaystyle\sum_{(i,j)\in\mathcal{A}_{n,2}}\mathbb{N}{(i,j)\in\alpha(B)}=2|B|,

and thus

𝔼{D|X,Y}\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\{D\,|\,X,Y\}
=1n(n1)σnα𝒜n,d(A,B)𝒢f,dμA,BfA,B(Xα(A);Yα(B))\displaystyle=\frac{1}{n(n-1)\sigma_{n}}\sum_{\alpha\in\mathcal{A}_{n,d}}\sum_{(A,B)\in\mathcal{G}_{f,d}}\mu_{A,B}f_{A,B}(X_{\alpha(A)};Y_{\alpha(B)}) (5.11)
×(i,j)𝒜n,2(i,j)α(B)\displaystyle\quad\times\sum_{(i,j)\in\mathcal{A}_{n,2}}\mathbb{N}{(i,j)\in\alpha(B)}
=2n(n1)σnα𝒜n,d(A,B)𝒢f,dνA,BfA,B(Xα(A);Yα(B))\displaystyle=\frac{2}{n(n-1)\sigma_{n}}\sum_{\alpha\in\mathcal{A}_{n,d}}\sum_{(A,B)\in\mathcal{G}_{f,d}}\nu_{A,B}f_{A,B}(X_{\alpha(A)};Y_{\alpha(B)})
=2n(n1)W.\displaystyle=\frac{2}{n(n-1)}W. (5.12)

Thus, 4.3 is satisfied with λ=2/(n(n1))\lambda=2/(n(n-1)) and R=0R=0. Moreover, by exchangeability,

𝔼{DΔ}=2𝔼{DW}=2λ𝔼{W2}=2λσn,d2/σn2.\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\{D\Delta\}=2\mathop{{}\mathbb{E}}\mathopen{}\{DW\}=2\lambda\mathop{{}\mathbb{E}}\mathopen{}\{W^{2}\}=2\lambda\sigma_{n,d}^{2}/\sigma_{n}^{2}. (5.13)

Then, we have

12λ𝔼{DΔ|X,Y,Y}\displaystyle\frac{1}{2\lambda}\mathop{{}\mathbb{E}}\mathopen{}\{D\Delta\,|\,X,Y,Y^{\prime}\}
=14σn2(A1,B1)𝒢f,d(A2,B2)𝒢f,dμA1,B1νA2,B2\displaystyle=\frac{1}{4\sigma_{n}^{2}}\sum_{(A_{1},B_{1})\in\mathcal{G}_{f,d}}\sum_{(A_{2},B_{2})\in\mathcal{G}_{f,d}}\mu_{A_{1},B_{1}}\nu_{A_{2},B_{2}}
×(i,j)𝒜n,2(α𝒜n,d(i,j)ξα(A1,B1)(i,j))(α𝒜n,d(i,j)ξα(A2,B2)(i,j)).\displaystyle\quad\times\sum_{(i,j)\in\mathcal{A}_{n,2}}\biggl{(}\sum_{\alpha\in\mathcal{A}_{n,d}^{(i,j)}}\xi_{\alpha(A_{1},B_{1})}^{(i,j)}\biggr{)}\biggl{(}\sum_{\alpha\in\mathcal{A}_{n,d}^{(i,j)}}\xi_{\alpha(A_{2},B_{2})}^{(i,j)}\biggr{)}.

Now, by the Cauchy inequality, 5.13 and Lemmas 5.3 and 5.4, we have

𝔼|12λ𝔼{DΔ|X,Y,Y}1|\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\biggl{\lvert}\frac{1}{2\lambda}\mathop{{}\mathbb{E}}\mathopen{}\{D\Delta\,|\,X,Y,Y^{\prime}\}-1\biggr{\rvert}
𝔼|12λ𝔼{DΔ|X,Y,Y}12λ𝔼{DΔ}|+σn2σn,d2σn2\displaystyle\leqslant\mathop{{}\mathbb{E}}\mathopen{}\biggl{\lvert}\frac{1}{2\lambda}\mathop{{}\mathbb{E}}\mathopen{}\{D\Delta\,|\,X,Y,Y^{\prime}\}-\frac{1}{2\lambda}\mathop{{}\mathbb{E}}\mathopen{}\{D\Delta\}\biggr{\rvert}+\frac{\sigma_{n}^{2}-\sigma_{n,d}^{2}}{\sigma_{n}^{2}}
14σn2(A1,B1)𝒢f,d(A2,B2)𝒢f,dμA1,B1νA2,B2\displaystyle\leqslant\frac{1}{4\sigma_{n}^{2}}\sum_{(A_{1},B_{1})\in\mathcal{G}_{f,d}}\sum_{(A_{2},B_{2})\in\mathcal{G}_{f,d}}\mu_{A_{1},B_{1}}\nu_{A_{2},B_{2}}
×(Var{(i,j)𝒜n,2(α𝒜n,d(i,j)ξα(A1,B1)(i,j))(α𝒜n(i,j)ξα(A2,B2)(i,j))})1/2\displaystyle\quad\times\Biggl{(}\mathop{\mathrm{Var}}\Biggl{\{}\sum_{(i,j)\in\mathcal{A}_{n,2}}\biggl{(}\sum_{\alpha\in\mathcal{A}_{n,d}^{(i,j)}}\xi_{\alpha(A_{1},B_{1})}^{(i,j)}\biggr{)}\biggl{(}\sum_{\alpha\in\mathcal{A}_{n}^{(i,j)}}\xi_{\alpha(A_{2},B_{2})}^{(i,j)}\biggr{)}\Biggr{\}}\Biggr{)}^{1/2}
+σn2σn,d2σn2\displaystyle\quad+\frac{\sigma_{n}^{2}-\sigma_{n,d}^{2}}{\sigma_{n}^{2}}
Cn1/2.\displaystyle\leqslant Cn^{-1/2}.

Taking D=|D|D^{*}=|D|, by Lemma 5.5,

1λ𝔼|𝔼{DΔ|X,Y,Y}|\displaystyle\frac{1}{\lambda}\mathop{{}\mathbb{E}}\mathopen{}\bigl{\lvert}\mathop{{}\mathbb{E}}\mathopen{}\{D^{*}\Delta\,|\,X,Y,Y^{\prime}\}\bigr{\rvert}
=1λ𝔼|𝔼{DΔ|X,Y,Y}|\displaystyle=\frac{1}{\lambda}\mathop{{}\mathbb{E}}\mathopen{}\bigl{\lvert}\mathop{{}\mathbb{E}}\mathopen{}\{D^{*}\Delta\,|\,X,Y,Y^{\prime}\}\bigr{\rvert}
14σn2(A1,B1)𝒢f,d(A2,B2)𝒢f,dμA1,B1νA2,B2\displaystyle\leqslant\frac{1}{4\sigma_{n}^{2}}\sum_{(A_{1},B_{1})\in\mathcal{G}_{f,d}}\sum_{(A_{2},B_{2})\in\mathcal{G}_{f,d}}\mu_{A_{1},B_{1}}\nu_{A_{2},B_{2}}
×(Var{(i,j)𝒜n,2|α𝒜n,d(i,j)ξα(A1,B1)(i,j)|(α𝒜n(i,j)ξα(A2,B2)(i,j))})1/2\displaystyle\quad\times\Biggl{(}\mathop{\mathrm{Var}}\Biggl{\{}\sum_{(i,j)\in\mathcal{A}_{n,2}}\biggl{\lvert}\sum_{\alpha\in\mathcal{A}_{n,d}^{(i,j)}}\xi_{\alpha(A_{1},B_{1})}^{(i,j)}\biggr{\rvert}\biggl{(}\sum_{\alpha\in\mathcal{A}_{n}^{(i,j)}}\xi_{\alpha(A_{2},B_{2})}^{(i,j)}\biggr{)}\Biggr{\}}\Biggr{)}^{1/2}
Cn1.\displaystyle\leqslant Cn^{-1}.

Now, by 5.10 and Lemma 5.3, we have

𝔼|U|2\displaystyle\mathop{{}\mathbb{E}}\mathopen{}|U|^{2} Cσn,d2(kd)=d+1k𝔼(Sn,k2(f()))Cn1,\displaystyle\leqslant C\sigma_{n,d}^{-2}(k-d)\sum_{\ell=d+1}^{k}\mathop{{}\mathbb{E}}\mathopen{}(S_{n,k}^{2}(f_{(\ell)}))\leqslant Cn^{-1},
𝔼(UU(i,j))2\displaystyle\mathop{{}\mathbb{E}}\mathopen{}(U-U^{(i,j)})^{2} Cσn,d2(kd)=d+1k𝔼{(Sn,k(f())Sn,k(i,j)(f()))2}Cn3,\displaystyle\leqslant C\sigma_{n,d}^{-2}(k-d)\sum_{\ell=d+1}^{k}\mathop{{}\mathbb{E}}\mathopen{}\{(S_{n,k}(f_{(\ell)})-S_{n,k}^{(i,j)}(f_{(\ell)}))^{2}\}\leqslant Cn^{-3},
𝔼(WW(i,j))2\displaystyle\mathop{{}\mathbb{E}}\mathopen{}(W-W^{(i,j)})^{2} Cσn,d2𝔼{(Sn,k(f(d))Sn,k(i,j)(f(d)))2}Cn2.\displaystyle\leqslant C\sigma_{n,d}^{-2}\mathop{{}\mathbb{E}}\mathopen{}\{(S_{n,k}(f_{(d)})-S_{n,k}^{(i,j)}(f_{(d)}))^{2}\}\leqslant Cn^{-2}.

Thus,

𝔼|U|\displaystyle\mathop{{}\mathbb{E}}\mathopen{}|U| Cn1/2,\displaystyle\leqslant Cn^{-1/2},
1λ𝔼|Δ(UU(I,J))|\displaystyle\frac{1}{\lambda}\mathop{{}\mathbb{E}}\mathopen{}\bigl{\lvert}\Delta(U-U^{(I,J)})\bigr{\rvert} =in,2𝔼{|(WW(i,j))(UU(i,j))|}\displaystyle=\sum_{i\in\mathcal{I}_{n,2}}\mathop{{}\mathbb{E}}\mathopen{}\{\lvert(W-W^{(i,j)})(U-U^{(i,j)})\rvert\}
Cn1/2.\displaystyle\leqslant Cn^{-1/2}.

Applying Corollary 4.3, we obtain the desired result. ∎

5.3 Proof of Theorem 2.4

The proof of Theorem 2.4 is similar to that of Theorem 2.3. Without loss of generality, we assume that kd+1k\geqslant d+1, otherwise the proof is even simpler.

For any A[k]A\subset[k] and B[k]2B\subset[k]_{2}, recall that

μA,B\displaystyle\mu_{A,B} 1|Aut(GA,B)||B|(nvA,B\@@atopnk),\displaystyle\coloneqq\frac{1}{\lvert\operatorname{Aut}(G_{A,B})\rvert|B|}\binom{n-v_{A,B}}{n-k},
νA,B\displaystyle\nu_{A,B} |B|μA,B=1|Aut(GA,B)|(nvA,B\@@atopnk).\displaystyle\coloneqq|B|\mu_{A,B}=\frac{1}{\lvert\operatorname{Aut}(G_{A,B})\rvert}\binom{n-v_{A,B}}{n-k}.

By Proposition 5.2, we have there exists a Hoeffding decomposition of gg as follows:

g(y)=B[k]2gB(yB),\displaystyle g(y)=\sum_{B\subset[k]_{2}}g_{B}(y_{B}),

where y=(y1,2,,yk1,k)y=(y_{1,2},\dots,y_{k-1,k}) and yB=(yi,j:(i,j)B).y_{B}=(y_{i,j}:(i,j)\in B). Also, for any B[k]2B\subset[k]_{2} and α𝒜n,\alpha\in\mathcal{A}_{n,\ell} (=1,,k\ell=1,\dots,k), let

ηα(B)(i,j)=gB(Yα(B))gB(Yα(B)(i,j)).\displaystyle\eta_{\alpha(B)}^{(i,j)}=g_{B}(Y_{\alpha(B)})-g_{B}(Y_{\alpha(B)}^{(i,j)}).

For any B[k]2B\in[k]_{2}, let VBV_{B} be the node set of the graph with edge set BB. For any rVBr\in V_{B}, let B(r)={(i,j):(i,j)B,ir,jr}B^{(r)}=\{(i,j):(i,j)\in B,i\neq r,j\neq r\}. Recall that 𝒢f,d+1={(A,B):A[k],B[k]2,vA,B=d+1,σA,B>0}\mathcal{G}_{f,d+1}=\{(A,B):A\subset[k],B\subset[k]_{2},v_{A,B}=d+1,\sigma_{A,B}>0\} and 𝒢~f,d={(A,B)𝒢f,d:GA,B is strongly connected.}\widetilde{\mathcal{G}}_{f,d}=\{(A,B)\in\mathcal{G}_{f,d}:G_{A,B}\text{ is strongly connected.}\}.

We need to apply the following lemma in the proof of Theorem 2.4.

Lemma 5.6.

Assume that kd+1k\geqslant d+1. For all (Aj,Bj)𝒢~f,d𝒢f,d+1(A_{j},B_{j})\in\widetilde{\mathcal{G}}_{f,d}\cup\mathcal{G}_{f,d+1} and let vj=vAj,Bjv_{j}=v_{A_{j},B_{j}} for j=1,2j=1,2, we have

Var{(i,j)𝒜n,2(α1𝒜n,v1(i,j)ηα1(B1)(i,j))(α2𝒜n,v2(i,j)ηα2(B2)(i,j))}Cn2d2τ4.\displaystyle\mathop{\mathrm{Var}}\biggl{\{}\sum_{(i,j)\in\mathcal{A}_{n,2}}\biggl{(}\sum_{\alpha_{1}\in\mathcal{A}_{n,v_{1}}^{(i,j)}}\eta_{\alpha_{1}(B_{1})}^{(i,j)}\biggr{)}\biggl{(}\sum_{\alpha_{2}\in\mathcal{A}_{n,v_{2}}^{(i,j)}}\eta_{\alpha_{2}(B_{2})}^{(i,j)}\biggr{)}\biggr{\}}\leqslant Cn^{2d-2}\tau^{4}.
Proof of Theorem 2.4.

Again, write T=σn1(Sn,k(g)𝔼{Sn,k(g)})T=\sigma_{n}^{-1}(S_{n,k}(g)-\mathop{{}\mathbb{E}}\mathopen{}\{S_{n,k}(g)\}), and let

W=σn1(Sn,k(g(d))+Sn,k(g(d+1))),U=σn1=d+2kSn,k(g()).\displaystyle W=\sigma_{n}^{-1}(S_{n,k}(g_{(d)})+S_{n,k}(g_{(d+1)})),\quad U=\sigma_{n}^{-1}\sum_{\ell=d+2}^{k}S_{n,k}(g_{(\ell)}). (5.14)

Here, if d+1>kd+1>k, then set =d+1kSn,k(g())=0\sum_{\ell=d+1}^{k}S_{n,k}(g_{(\ell)})=0. Then, T=W+UT=W+U. Now we apply Corollary 4.3 again to prove the desired result. To this end, we need to construct an exchangeable pair. For each (i,j)𝒜n,2(i,j)\in\mathcal{A}_{n,2}, let

W(i,j)=1σn(Sn,k(i,j)(g(d))+Sn,k(i,j)(g(d+1))),U(i,j)=σn1=d+2kSn,k(i,j)(g()).\displaystyle W^{(i,j)}=\frac{1}{\sigma_{n}}(S_{n,k}^{(i,j)}(g_{(d)})+S_{n,k}^{(i,j)}(g_{(d+1)})),\quad U^{(i,j)}=\sigma_{n}^{-1}\sum_{\ell=d+2}^{k}S_{n,k}^{(i,j)}(g_{(\ell)}).

By assumption, we have

W\displaystyle W =1σn(A,B)𝒢~f,d𝒢f,d+1α𝒜n,v(G)νA,BgA,B(Xα(A);Yα(B)).\displaystyle=\frac{1}{\sigma_{n}}\sum_{(A,B)\in\widetilde{\mathcal{G}}_{f,d}\cup\mathcal{G}_{f,d+1}}\sum_{\alpha\in\mathcal{A}_{n,v(G)}}\nu_{A,B}{g_{A,B}(X_{\alpha(A)};Y_{\alpha(B)})}.

Let (I,J)(I,J) be a random 2-fold index uniformly chosen in 𝒜n,2\mathcal{A}_{n,2}, which is independent of all others. Then, ((X,Y),(X,Y(I,J)))((X,Y),(X,Y^{(I,J)})) is an exchangeable pair. Let

Δ\displaystyle\Delta =WW(I,J)=1σn((A,B)𝒢~f,dα𝒜n,dνA,Bηα(B)(I,J)).\displaystyle=W-W^{(I,J)}=\frac{1}{\sigma_{n}}\biggl{(}\sum_{(A,B)\in\widetilde{\mathcal{G}}_{f,d}}\sum_{\alpha\in\mathcal{A}_{n,d}}\nu_{A,B}{\eta_{\alpha(B)}^{(I,J)}}\biggr{)}.

Also, define

D\displaystyle D =1σn((A,B)𝒢~f,dα𝒜n,dμA,Bηα(B)(I,J)).\displaystyle=\frac{1}{\sigma_{n}}\biggl{(}\sum_{(A,B)\in\widetilde{\mathcal{G}}_{f,d}}\sum_{\alpha\in\mathcal{A}_{n,d}}\mu_{A,B}{\eta_{\alpha(B)}^{(I,J)}}\biggr{)}.

Then, DD is antisymmetric with respect to (X,Y)(X,Y) and (X,Y(I,J))(X,Y^{(I,J)}).

Following a similar argument leading to 5.12,

𝔼{D|X,Y}=2n(n1)W.\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\{D\,|\,X,Y\}=\frac{2}{n(n-1)}W. (5.15)

Thus, 4.3 is satisfied with λ=2/(n(n1))\lambda=2/(n(n-1)) and R=0R=0. Moreover, by exchangeability,

𝔼{DΔ}=2𝔼{DW}=2λ𝔼{W2}=2λ(σn,d2+σn,d+12)/σn2.\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\{D\Delta\}=2\mathop{{}\mathbb{E}}\mathopen{}\{DW\}=2\lambda\mathop{{}\mathbb{E}}\mathopen{}\{W^{2}\}=2\lambda(\sigma_{n,d}^{2}+\sigma_{n,d+1}^{2})/\sigma_{n}^{2}. (5.16)

Now, by the Cauchy inequality, 5.16 and Lemmas 5.3 and 5.6, we have

𝔼|12λ𝔼{DΔ|X,Y,Y}1|\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\biggl{\lvert}\frac{1}{2\lambda}\mathop{{}\mathbb{E}}\mathopen{}\{D\Delta\,|\,X,Y,Y^{\prime}\}-1\biggr{\rvert}
𝔼|12λ𝔼{DΔ|X,Y,Y}12λ𝔼{DΔ}|+σn2σn,d2σn,d+12σn2\displaystyle\leqslant\mathop{{}\mathbb{E}}\mathopen{}\biggl{\lvert}\frac{1}{2\lambda}\mathop{{}\mathbb{E}}\mathopen{}\{D\Delta\,|\,X,Y,Y^{\prime}\}-\frac{1}{2\lambda}\mathop{{}\mathbb{E}}\mathopen{}\{D\Delta\}\biggr{\rvert}+\frac{\sigma_{n}^{2}-\sigma_{n,d}^{2}-\sigma_{n,d+1}^{2}}{\sigma_{n}^{2}}
Cn1.\displaystyle\leqslant Cn^{-1}.

With D=|D|D^{*}=|D|, and by Lemma 5.5 again,

1λ𝔼|𝔼{DΔ|X,Y,Y}|Cn1.\displaystyle\frac{1}{\lambda}\mathop{{}\mathbb{E}}\mathopen{}\bigl{\lvert}\mathop{{}\mathbb{E}}\mathopen{}\{D^{*}\Delta\,|\,X,Y,Y^{\prime}\}\bigr{\rvert}\leqslant Cn^{-1}.

Now, by 5.14 and Lemma 5.3, we have

𝔼|U|2\displaystyle\mathop{{}\mathbb{E}}\mathopen{}|U|^{2} Cσn2(kd)=d+2k𝔼(Sn,k2(g()))Cn2,\displaystyle\leqslant C\sigma_{n}^{-2}(k-d)\sum_{\ell=d+2}^{k}\mathop{{}\mathbb{E}}\mathopen{}(S_{n,k}^{2}(g_{(\ell)}))\leqslant Cn^{-2},
𝔼(UU(i,j))2\displaystyle\mathop{{}\mathbb{E}}\mathopen{}(U-U^{(i,j)})^{2} Cσn2(kd)=d+2k𝔼{(Sn,k(g())Sn,k(i,j)(g()))2}Cn4,\displaystyle\leqslant C\sigma_{n}^{-2}(k-d)\sum_{\ell=d+2}^{k}\mathop{{}\mathbb{E}}\mathopen{}\{(S_{n,k}(g_{(\ell)})-S_{n,k}^{(i,j)}(g_{(\ell)}))^{2}\}\leqslant Cn^{-4},
𝔼(WW(i,j))2\displaystyle\mathop{{}\mathbb{E}}\mathopen{}(W-W^{(i,j)})^{2} Cσn2{Sn,k(g(d))Sn,k(i,j)(g(d))22\displaystyle\leqslant C\sigma_{n}^{-2}\Bigl{\{}\|S_{n,k}(g_{(d)})-S_{n,k}^{(i,j)}(g_{(d)})\|_{2}^{2}
+Sn,k(g(d+1))Sn,k(i,j)(g(d+1))22}Cn2.\displaystyle\hskip 71.13188pt+\|S_{n,k}(g_{(d+1)})-S_{n,k}^{(i,j)}(g_{(d+1)})\|_{2}^{2}\Bigr{\}}\leqslant Cn^{-2}.

Thus,

𝔼|U|\displaystyle\mathop{{}\mathbb{E}}\mathopen{}|U| Cn1,\displaystyle\leqslant Cn^{-1},
1λ𝔼|Δ(UU(I,J))|\displaystyle\frac{1}{\lambda}\mathop{{}\mathbb{E}}\mathopen{}\bigl{\lvert}\Delta(U-U^{(I,J)})\bigr{\rvert} C(i,j)𝒜n,2𝔼{|(WW(i,j))(UU(i,j))|}\displaystyle\leqslant C\sum_{(i,j)\in\mathcal{A}_{n,2}}\mathop{{}\mathbb{E}}\mathopen{}\{\lvert(W-W^{(i,j)})(U-U^{(i,j)})\rvert\}
Cn1.\displaystyle\leqslant Cn^{-1}.

Applying Corollary 4.3, we obtain the desired result. ∎

6 Proof of other results

6.1 Proof of Theorem 3.2

As fFinjf_{F}^{\operatorname{inj}} does not dependent on XX if κp\kappa\equiv p for some 0<p<10<p<1. Fix FF. Define

ginj(Y)=fFinj(X;Y)\displaystyle g^{\operatorname{inj}}(Y)=f_{F}^{\operatorname{inj}}(X;Y)

and by Proposition 5.2, we have ginjg^{\operatorname{inj}} has the following decomposition:

ginj(Y)=B[k]2gBinj(YB).\displaystyle g^{\operatorname{inj}}(Y)=\sum_{B\subset[k]_{2}}g_{B}^{\operatorname{inj}}(Y_{B}). (6.1)

By (Janson and Nowicki, 1991, p. 361), we have

g{(1,2)}inj(y1,2)=2e(F)(v(F)2)!|Aut(G)|pe(F)1(y1,2p)0.\displaystyle g^{\operatorname{inj}}_{\{(1,2)\}}(y_{1,2})=\frac{2e(F)(v(F)-2)!}{\lvert\operatorname{Aut}(G)\rvert}p^{e(F)-1}(y_{1,2}-p)\neq 0.

Therefore, by Theorem 2.4 with d=2d=2, we complete the proof.

6.2 Proof of Theorem 3.3

Again, let

gind(Y)=fFind(X;Y),\displaystyle g^{\operatorname{ind}}(Y)=f^{\operatorname{ind}}_{F}(X;Y),

and similar to 6.1, we have

gind(Y)=B[k]2gBind(YB).\displaystyle g^{\operatorname{ind}}(Y)=\sum_{B\subset[k]_{2}}g_{B}^{\operatorname{ind}}(Y_{B}).

Recall that e(F)e(F) is the number of 2-stars in FF and t(F)t(F) is the number of triangles in FF. Let

e¯(F)=(v(F)\@@atop2)1e(F),s¯(F)=(v(F)\@@atop3)1s(F)3,t¯(F)=(v(F)\@@atop3)1t(F).\displaystyle\bar{e}(F)=\binom{v(F)}{2}^{-1}e(F),\quad\bar{s}(F)=\binom{v(F)}{3}^{-1}\frac{s(F)}{3},\quad\bar{t}(F)=\binom{v(F)}{3}^{-1}t(F).

Let

N(F)=v(F)!|Aut(F)|pe(F)(1p)(v(F)\@@atop2)e(F).\displaystyle N(F)=\frac{v(F)!}{\lvert\operatorname{Aut}(F)\rvert}p^{e(F)}(1-p)^{\binom{v(F)}{2}-e(F)}.

By Janson and Nowicki (1991), letting B1={(1,2)}B_{1}=\{(1,2)\}, B2={(1,2),(1,3)}B_{2}=\{(1,2),(1,3)\} and B3={(1,2),(1,3),(2,3)}B_{3}=\{(1,2),(1,3),(2,3)\}, we have

gB1ind(y)\displaystyle g^{\operatorname{ind}}_{B_{1}}(y) =N(F)p(1p)(e¯(F)p)(yp),\displaystyle=\frac{N(F)}{p(1-p)}(\bar{e}(F)-p)(y-p),
gB2ind(y12,y13)\displaystyle g^{\operatorname{ind}}_{B_{2}}(y_{12},y_{13}) =N(F)p2(1p)2(s¯(F)2pe¯(F)+p2)\displaystyle=\frac{N(F)}{p^{2}(1-p)^{2}}(\bar{s}(F)-2p\bar{e}(F)+p^{2})
×((y12p)(y13p),\displaystyle\quad\times((y_{12}-p)(y_{13}-p),
gB3ind(y12,y13,y23)\displaystyle g^{\operatorname{ind}}_{B_{3}}(y_{12},y_{13},y_{23}) =N(F)p3(1p)3(t¯(F)3ps¯(F)+3p2e¯(F)p3)\displaystyle=\frac{N(F)}{p^{3}(1-p)^{3}}(\bar{t}(F)-3p\bar{s}(F)+3p^{2}\bar{e}(F)-p^{3})
×(y12p)(y13p)(y23p).\displaystyle\quad\times(y_{12}-p)(y_{13}-p)(y_{23}-p).

We now consider the following three cases.

Case 1. If e(F)p(v(F)\@@atop2)e(F)\neq p\binom{v(F)}{2}. In this case, we have gB1ind0g^{\operatorname{ind}}_{B_{1}}\not\equiv 0. Then, by Theorem 2.4, we have 3.1 holds.

Case 2. If e¯(F)=p\bar{e}(F)=p and s¯(F)p2\bar{s}(F)\neq p^{2}. In this case, we have

gB1ind0,gB2ind0.\displaystyle g^{\operatorname{ind}}_{B_{1}}\equiv 0,\quad g^{\operatorname{ind}}_{B_{2}}\not\equiv 0.

However, the graph generated by B2B_{2} is a 2-star, which is not strongly connected. Then, by Theorem 2.3, we have 3.2 holds.

Case 3. If e¯(F)=p\bar{e}(F)=p, s¯(F)=p2\bar{s}(F)=p^{2} and t¯(F)p3\bar{t}(F)\neq p^{3}. In this case, we have

gB1ind0,gB2ind0,gB3ind0.\displaystyle g^{\operatorname{ind}}_{B_{1}}\equiv 0,\quad g^{\operatorname{ind}}_{B_{2}}\equiv 0,\quad g^{\operatorname{ind}}_{B_{3}}\not\equiv 0.

Because the graph generated by B3B_{3} is a triangle, which is strongly connected. Then, by Theorem 2.4, we have 3.1 holds.

Appendix A Proofs of some lemmas

A.1 Proof of Lemma 5.1

Proof of Lemma 5.1.

We write {α}={α(1),,α(k)}\{\alpha\}=\{\alpha(1),\dots,\alpha(k)\} for any α=(α(1),,α(k))𝒜n,k\alpha=(\alpha(1),\dots,\alpha(k))\in\mathcal{A}_{n,k}. Also, write rα=r(Xα(1),,Xα(k);Yα(1),α(2),,Yα(k1),α(k))r_{\alpha}=r(X_{\alpha(1)},\dots,X_{\alpha(k)};Y_{\alpha(1),\alpha(2)},\dots,Y_{\alpha(k-1),\alpha(k)}). Now, observe that

Var{αn,krα}=1σn2αn,kαn,kCov(rα,rα).\displaystyle\mathop{\mathrm{Var}}\biggl{\{}\sum_{\alpha\in\mathcal{I}_{n,k}}r_{\alpha}\biggr{\}}=\frac{1}{\sigma_{n}^{2}}\sum_{\alpha\in\mathcal{I}_{n,k}}\sum_{\alpha^{\prime}\in\mathcal{I}_{n,k}}\mathop{\mathrm{Cov}}\bigl{(}r_{\alpha},r_{\alpha^{\prime}}\bigr{)}. (A.1)

Note that if {α}{α}=\{\alpha\}\cap\{\alpha^{\prime}\}=\varnothing, then rαr_{\alpha} and rαr_{\alpha^{\prime}} are independent, then clearly it follows that

Cov(rα,rα)=0\mathop{\mathrm{Cov}}\bigl{(}r_{\alpha},r_{\alpha^{\prime}}\bigr{)}=0 (A.2)

if {α}{α}=\{\alpha\}\cap\{\alpha^{\prime}\}=\varnothing. If there exists i{1,,n}i\in\{1,\dots,n\} such that {α}{α}={i}\{\alpha\}\cap\{\alpha^{\prime}\}=\{i\}, then

Cov(rα,rα))\displaystyle\mathop{\mathrm{Cov}}\bigl{(}r_{\alpha},r_{\alpha^{\prime}})\bigr{)} =𝔼{Cov(rα,rα|Xi)}+Cov(𝔼{rα|Xi},𝔼{rα|Xi}).\displaystyle=\mathop{{}\mathbb{E}}\mathopen{}\Bigl{\{}\mathop{\mathrm{Cov}}\bigl{(}r_{\alpha},r_{\alpha^{\prime}}\bigm{|}X_{i}\bigr{)}\Bigr{\}}+\mathop{\mathrm{Cov}}\bigl{(}\mathop{{}\mathbb{E}}\mathopen{}\{r_{\alpha}|X_{i}\},\mathop{{}\mathbb{E}}\mathopen{}\{r_{\alpha^{\prime}}|X_{i}\}\bigr{)}. (A.3)

By independence, we have the first term of A.3 is 0. For the second term, note that for any i{α}i\in\{\alpha\}, then 𝔼{rα|Xi}=0\mathop{{}\mathbb{E}}\mathopen{}\{r_{\alpha}|X_{i}\}=0, and thus the second term of A.3 is also 0. Therefore,

Cov(rα,rα)=0, if |{α}{α}|=1.\displaystyle\mathop{\mathrm{Cov}}\bigl{(}r_{\alpha},r_{\alpha^{\prime}}\bigr{)}=0,\text{ if $\lvert\{\alpha\}\cap\{\alpha^{\prime}\}\rvert=1$. } (A.4)

For any α\alpha and α\alpha^{\prime} such that |{α}{α}|2\lvert\{\alpha\}\cap\{\alpha^{\prime}\}\rvert\geqslant 2, by the Cauchy inequality, we have

Cov(rα,rα)Var(rα).\displaystyle\mathop{\mathrm{Cov}}\bigl{(}r_{\alpha},r_{\alpha^{\prime}}\bigr{)}\leqslant\mathop{\mathrm{Var}}\bigl{(}r_{\alpha}\bigr{)}.

Recall that rαr_{\alpha} and g(Xj)g(X_{j}) are orthogonal for every j{α}j\in\{\alpha\}. By 5.1, we have

Var(rα)\displaystyle\mathop{\mathrm{Var}}\bigl{(}r_{\alpha}\bigr{)} =Var(f(Xα(1),,Xα(k);Yα(1),α(2),,Yα(k1),α(k)))\displaystyle=\mathop{\mathrm{Var}}\bigl{(}f(X_{\alpha(1)},\dots,X_{\alpha(k)};Y_{\alpha(1),\alpha(2)},\dots,Y_{\alpha(k-1),\alpha(k)})\bigr{)}
j{α}𝔼{f1(Xj)2}\displaystyle\quad-\sum_{j\in\{\alpha\}}\mathop{{}\mathbb{E}}\mathopen{}\{f_{1}(X_{j})^{2}\}
τ2.\displaystyle\leqslant\tau^{2}.

Thus, it follows that

|Cov(rα,rα)|\displaystyle\bigl{\lvert}\mathop{\mathrm{Cov}}\bigl{(}r_{\alpha},r_{\alpha^{\prime}}\bigr{)}\bigr{\rvert} τ2, if |{α}{α}|2.\displaystyle\leqslant\tau^{2},\text{ if $\lvert\{\alpha\}\cap\{\alpha^{\prime}\}\rvert\geqslant 2$. } (A.5)

Combining 5.2, A.1, A.2, A.4 and A.5, we have

𝔼{U2}\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\{U^{2}\} τ2σn2αn,kαn,k|{α}{α}|2\displaystyle\leqslant\frac{\tau^{2}}{\sigma_{n}^{2}}\sum_{\alpha\in\mathcal{I}_{n,k}}\sum_{\alpha^{\prime}\in\mathcal{I}_{n,k}}\mathbb{N}{\lvert\{\alpha\}\cap\{\alpha^{\prime}\}\rvert\geqslant 2} (A.6)
nτ2k2σ12(n\@@atopk)1(k\@@atop2)(nk\@@atopk2)\displaystyle\leqslant\frac{n\tau^{2}}{k^{2}\sigma_{1}^{2}}\binom{n}{k}^{-1}\binom{k}{2}\binom{n-k}{k-2}
(k1)2τ22(n1)σ12.\displaystyle\leqslant\frac{(k-1)^{2}\tau^{2}}{2(n-1)\sigma_{1}^{2}}.

This proves 5.3.

Now we prove 5.4. Let n,k(i)={α={α(1),,α(k)}:α(1)<<α(k),i{α}}\mathcal{I}_{n,k}^{(i)}=\{\alpha=\{\alpha(1),\dots,\alpha(k)\}:\alpha(1)<\dots<\alpha(k),i\in\{\alpha\}\}. Note that

UU(i)\displaystyle U-U^{(i)} =1σnαn,k(i)r{α}(i).\displaystyle=\frac{1}{\sigma_{n}}\sum_{\alpha\in\mathcal{I}_{n,k}^{(i)}}r_{\{\alpha\}}^{(i)}.

where

rα(i)=rαr(Xα(1)(i),,Xα(k)(i);Yα(1),α(2),,Yα(k1),α(k)).\displaystyle r_{\alpha}^{(i)}=r_{\alpha}-r(X_{\alpha(1)}^{(i)},\dots,X_{\alpha(k)}^{(i)};Y_{\alpha(1),\alpha(2)},\dots,Y_{\alpha(k-1),\alpha(k)}).

For each α\alpha, by independence, we have

Var(rα(i))\displaystyle\mathop{\mathrm{Var}}\bigl{(}r_{\alpha}^{(i)}\bigr{)} =2𝔼{Var(rα|Xj,j{α}{i},Yα(1),α(2),,Yα(k1),α(k))}\displaystyle=2\mathop{{}\mathbb{E}}\mathopen{}\{\mathop{\mathrm{Var}}\bigl{(}r_{\alpha}\bigm{|}X_{j},j\in\{\alpha\}\setminus\{i\},Y_{\alpha(1),\alpha(2)},\dots,Y_{\alpha(k-1),\alpha(k)}\bigr{)}\}
2Var(rα)2τ2.\displaystyle\leqslant 2\mathop{\mathrm{Var}}\bigl{(}r_{\alpha}\bigr{)}\leqslant 2\tau^{2}.

Similar to A.6, we have

𝔼{(UU(i))2}\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\{(U-U^{(i)})^{2}\} =1σn2αn,k(i)αn,k(i)Cov(rα(i),rα(i))\displaystyle=\frac{1}{\sigma_{n}^{2}}\sum_{\alpha\in\mathcal{I}_{n,k}^{(i)}}\sum_{\alpha^{\prime}\in\mathcal{I}_{n,k}^{(i)}}\mathop{\mathrm{Cov}}\bigl{(}r_{\alpha}^{(i)},r_{\alpha^{\prime}}^{(i)}\bigr{)}
2nτ2k2σ12(n\@@atopk)2αn,k(i)αn,k(i)|{α}{α}|2\displaystyle\leqslant\frac{2n\tau^{2}}{k^{2}\sigma_{1}^{2}}\binom{n}{k}^{-2}\sum_{\alpha\in\mathcal{I}_{n,k}^{(i)}}\sum_{\alpha^{\prime}\in\mathcal{I}_{n,k}^{(i)}}\mathbb{N}{\lvert\{\alpha\}\cap\{\alpha^{\prime}\}\rvert\geqslant 2}
2n(k1)τ2k2σ12(n\@@atopk)2(n1\@@atopk1)(nk\@@atopk2)\displaystyle\leqslant\frac{2n(k-1)\tau^{2}}{k^{2}\sigma_{1}^{2}}\binom{n}{k}^{-2}\binom{n-1}{k-1}\binom{n-k}{k-2}
2(k1)2τ2n(n1)σ12.\displaystyle\leqslant\frac{2(k-1)^{2}\tau^{2}}{n(n-1)\sigma_{1}^{2}}.

This completes the proof. ∎

A.2 Proof of Lemma 5.3

Recall that {α}={α(1),,α()}\{\alpha\}=\{\alpha(1),\dots,\alpha(\ell)\} for α𝒜n,\alpha\in\mathcal{A}_{n,\ell}. To prove Lemma 5.4, we need the following lemma.

Lemma A.1.

Let (A1,B1),(A2,B2)𝒢f,d(A_{1},B_{1}),(A_{2},B_{2})\in\mathcal{G}_{f,d}, (i,j),(i,j)𝒜n,2(i,j),(i^{\prime},j^{\prime})\in\mathcal{A}_{n,2}, α1,α2𝒜n,d(i,j)\alpha_{1},\alpha_{2}\in\mathcal{A}_{n,d}^{(i,j)} and α1,α2𝒜n,d(i,j)\alpha_{1}^{\prime},\alpha_{2}^{\prime}\in\mathcal{A}_{n,d}^{(i^{\prime},j^{\prime})}. Let

s=|{α1}{α2}|,t=|{α1}{α2}|.\displaystyle s=|\{\alpha_{1}\}\cap\{\alpha_{2}\}|,\quad t=|\{\alpha_{1}^{\prime}\}\cap\{\alpha_{2}^{\prime}\}|.

If |({α1}{α2})({α1}{α2})|2d(s+t)|(\{\alpha_{1}\}\cup\{\alpha_{2}\})\cap(\{\alpha_{1}^{\prime}\}\cap\{\alpha_{2}^{\prime}\})|\leqslant 2d-(s+t), then

Cov{ξα1(A1,B1)(i,j)ξα2(A2,B2)(i,j),ξα1(A1,B1)(i,j)ξα2(A2,B2)(i,j)}=0.\displaystyle\mathop{\mathrm{Cov}}\Bigl{\{}\xi_{\alpha_{1}(A_{1},B_{1})}^{(i,j)}\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)},\xi_{\alpha_{1}^{\prime}(A_{1},B_{1})}^{(i^{\prime},j^{\prime})}\xi_{\alpha_{2}^{\prime}(A_{2},B_{2})}^{(i^{\prime},j^{\prime})}\Bigr{\}}=0. (A.7)
Proof of Lemma A.1.

Let

V0\displaystyle V_{0} ={α1}{α2},\displaystyle=\{\alpha_{1}\}\cap\{\alpha_{2}\}, V1\displaystyle V_{1} ={α1}V0,\displaystyle=\{\alpha_{1}\}\setminus V_{0}, V2\displaystyle V_{2} ={α2}V0,\displaystyle=\{\alpha_{2}\}\setminus V_{0}, s\displaystyle s =|V0|,\displaystyle=\lvert V_{0}\rvert, (A.8)
V0\displaystyle V_{0}^{\prime} ={α1}{α2},\displaystyle=\{\alpha_{1}^{\prime}\}\cap\{\alpha_{2}^{\prime}\}, V1\displaystyle V_{1}^{\prime} ={α1}V0,\displaystyle=\{\alpha_{1}^{\prime}\}\setminus V_{0}^{\prime}, V2\displaystyle V_{2}^{\prime} ={α2}V0,\displaystyle=\{\alpha_{2}^{\prime}\}\setminus V_{0}^{\prime}, t\displaystyle t =|V0|.\displaystyle=\lvert V_{0}^{\prime}\rvert.

Then, we have V1V2=V_{1}\cap V_{2}=\varnothing, V1V2=V_{1}^{\prime}\cap V_{2}^{\prime}=\varnothing, 2s,td2\leqslant s,t\leqslant d. Without loss of generality, assume that sts\leqslant t.

If 2d(s+t)=02d-(s+t)=0, which is equivalent to s=d,t=ds=d,t=d, then {α1}={α2}\{\alpha_{1}\}=\{\alpha_{2}\} and {α1}={α2}\{\alpha_{1}^{\prime}\}=\{\alpha_{2}^{\prime}\}. If {a1}{a1}=\{a_{1}\}\cap\{a_{1}^{\prime}\}=\varnothing, then (ξα1(A1,B1)(i,j),ξα2(A2,B2)(i,j))(\xi_{\alpha_{1}(A_{1},B_{1})}^{(i,j)},\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)}) and (ξα1(A1,B1)(i,j),ξα2(A2,B2)(i,j))(\xi_{\alpha_{1}^{\prime}(A_{1},B_{1})}^{(i^{\prime},j^{\prime})},\xi_{\alpha_{2}^{\prime}(A_{2},B_{2})}^{(i^{\prime},j^{\prime})}) are independent, which implies that A.7 holds.

If 2d(s+t)>02d-(s+t)>0 and |({α1}{α2})({α1}{α2})|<2d(s+t)\lvert(\{\alpha_{1}\}\cup\{\alpha_{2}\})\cap(\{\alpha_{1}^{\prime}\}\cup\{\alpha_{2}^{\prime}\})\rvert<2d-(s+t), then there exists r[n]r\in[n] such that r(V1V2)({α1,α2})r\in(V_{1}^{\prime}\cup V_{2}^{\prime})\setminus(\{\alpha_{1},\alpha_{2}\}). Now, assume that rV2({α1,α2})r\in V_{2}^{\prime}\setminus(\{\alpha_{1},\alpha_{2}\}) without loss of generality. Let

r=σ(Xp,Yp,q,p,q[n]{r})σ(Yi,j).\displaystyle\mathcal{F}_{r}=\sigma(X_{p},Y_{p,q},p,q\in[n]\setminus\{r\})\vee\sigma(Y_{i^{\prime},j^{\prime}}^{\prime}). (A.9)

Therefore, we have ξα1(A1,B1)(i,j),ξα2(A2,B2)(i,j),ξα1(A1,B1)(i,j)r\xi_{\alpha_{1}(A_{1},B_{1})}^{(i,j)},\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)},\xi_{\alpha_{1}^{\prime}(A_{1},B_{1})}^{(i^{\prime},j^{\prime})}\in\mathcal{F}_{r}. Then, by 5.5,

𝔼{ξα2(A2,B2)(i,j)|r}\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\{\xi_{\alpha_{2}^{\prime}(A_{2},B_{2})}^{(i^{\prime},j^{\prime})}\,|\,\mathcal{F}_{r}\}
=𝔼{fG1(Xα2(A1,B1);Yα2(A1,B1))fG1(Xα2(A1,B1);Yα2(A1,B1)(i,j))|r}\displaystyle=\mathop{{}\mathbb{E}}\mathopen{}\Bigl{\{}f_{G_{1}}\bigl{(}X_{\alpha_{2}^{\prime}(A_{1},B_{1})};Y_{\alpha_{2}^{\prime}(A_{1},B_{1})}\bigr{)}-f_{G_{1}}\bigl{(}X_{\alpha_{2}^{\prime}(A_{1},B_{1})};Y_{\alpha_{2}^{\prime}(A_{1},B_{1})}^{(i^{\prime},j^{\prime})}\bigr{)}\Bigm{|}\mathcal{F}_{r}\Bigr{\}}
=0.\displaystyle=0. (A.10)

Hence,

𝔼{ξα1(A1,B1)(i,j)ξα2(A2,B2)(i,j)|r}=0,\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\Bigl{\{}\xi_{\alpha_{1}^{\prime}(A_{1},B_{1})}^{(i^{\prime},j^{\prime})}\xi_{\alpha_{2}^{\prime}(A_{2},B_{2})}^{(i^{\prime},j^{\prime})}\,|\,\mathcal{F}_{r}\Bigr{\}}=0,
which further implies that
𝔼{ξα1(A1,B1)(i,j)ξα2(A2,B2)(i,j)}=0,\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\Bigl{\{}\xi_{\alpha_{1}^{\prime}(A_{1},B_{1})}^{(i^{\prime},j^{\prime})}\xi_{\alpha_{2}^{\prime}(A_{2},B_{2})}^{(i^{\prime},j^{\prime})}\Bigr{\}}=0,

and

Cov{ξα1(A1,B1)(i,j)ξα2(A2,B2)(i,j),ξα1(A1,B1)(i,j)ξα2(A2,B2)(i,j)}\displaystyle\mathop{\mathrm{Cov}}\bigl{\{}\xi_{\alpha_{1}(A_{1},B_{1})}^{(i,j)}\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)},\xi_{\alpha_{1}^{\prime}(A_{1},B_{1})}^{(i^{\prime},j^{\prime})}\xi_{\alpha_{2}^{\prime}(A_{2},B_{2})}^{(i^{\prime},j^{\prime})}\bigr{\}} (A.11)
=𝔼{ξα1(A1,B1)(i,j)ξα2(A2,B2)(i,j)ξα1(A1,B1)(i,j)ξα2(A2,B2)(i,j)}\displaystyle=\mathop{{}\mathbb{E}}\mathopen{}\{\xi_{\alpha_{1}(A_{1},B_{1})}^{(i,j)}\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)}\xi_{\alpha_{1}^{\prime}(A_{1},B_{1})}^{(i^{\prime},j^{\prime})}\xi_{\alpha_{2}^{\prime}(A_{2},B_{2})}^{(i^{\prime},j^{\prime})}\}
=𝔼{𝔼{ξα1(A1,B1)(i,j)ξα2(A2,B2)(i,j)ξα1(A1,B1)(i,j)ξα2(A2,B2)(i,j)|r}}\displaystyle=\mathop{{}\mathbb{E}}\mathopen{}\Bigl{\{}\mathop{{}\mathbb{E}}\mathopen{}\bigl{\{}\xi_{\alpha_{1}(A_{1},B_{1})}^{(i,j)}\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)}\xi_{\alpha_{1}^{\prime}(A_{1},B_{1})}^{(i^{\prime},j^{\prime})}\xi_{\alpha_{2}^{\prime}(A_{2},B_{2})}^{(i^{\prime},j^{\prime})}\,|\,\mathcal{F}_{r}\bigr{\}}\Bigr{\}}
=𝔼{ξα1(A1,B1)(i,j)ξα2(A2,B2)(i,j)ξα1(A1,B1)(i,j)𝔼{ξα2(A2,B2)(i,j)|r}}\displaystyle=\mathop{{}\mathbb{E}}\mathopen{}\Bigl{\{}\xi_{\alpha_{1}(A_{1},B_{1})}^{(i,j)}\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)}\xi_{\alpha_{1}^{\prime}(A_{1},B_{1})}^{(i^{\prime},j^{\prime})}\mathop{{}\mathbb{E}}\mathopen{}\bigl{\{}\xi_{\alpha_{2}^{\prime}(A_{2},B_{2})}^{(i^{\prime},j^{\prime})}\,|\,\mathcal{F}_{r}\bigr{\}}\Bigr{\}}
=0.\displaystyle=0.

If 2d(s+t)>02d-(s+t)>0 and |({α1}{α2})({α1}{α2})|=2d(s+t)\lvert(\{\alpha_{1}\}\cup\{\alpha_{2}\})\cap(\{\alpha_{1}^{\prime}\}\cap\{\alpha_{2}^{\prime}\})\rvert=2d-(s+t), then either the following two conditions holds: (a) there exists rV1V2({α1}{α2})r\in V_{1}^{\prime}\cup V_{2}^{\prime}\setminus(\{\alpha_{1}\}\cup\{\alpha_{2}\}) or (b) V0V0=V_{0}\cap V_{0}^{\prime}=\varnothing. If (a) holds, then following a similar argument that leading to A.11, we have A.7 holds.

If (b) is true, letting =σ(X,{Yp,q:p,qV1V2V1V2})\mathcal{F}=\sigma(X,\{Y_{p,q}:p,q\in V_{1}\cup V_{2}\cup V_{1}^{\prime}\cup V_{2}^{\prime}\}), we have conditional on \mathcal{F}, (ξα1(A1,B1)(i,j),ξα2(A2,B2)(i,j))(\xi_{\alpha_{1}(A_{1},B_{1})}^{(i,j)},\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)}) is conditionally independent of (ξα1(A1,B1)(i,j),ξα2(A2,B2)(i,j))(\xi_{\alpha_{1}^{\prime}(A_{1},B_{1})}^{(i^{\prime},j^{\prime})},\xi_{\alpha_{2}^{\prime}(A_{2},B_{2})}^{(i^{\prime},j^{\prime})}), and thus,

Cov{ξα1(A1,B1)(i,j)ξα2(A2,B2)(i,j),ξα1(A1,B1)(i,j)ξα2(A2,B2)(i,j)}=Cov{𝔼{ξα1(A1,B1)(i,j)ξα2(A2,B2)(i,j)|},𝔼{ξα1(A1,B1)(i,j)ξα2(A2,B2)(i,j)|}}.\mathop{\mathrm{Cov}}\Bigl{\{}\xi_{\alpha_{1}(A_{1},B_{1})}^{(i,j)}\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)},\xi_{\alpha_{1}^{\prime}(A_{1},B_{1})}^{(i^{\prime},j^{\prime})}\xi_{\alpha_{2}^{\prime}(A_{2},B_{2})}^{(i^{\prime},j^{\prime})}\Bigr{\}}\\ =\mathop{\mathrm{Cov}}\biggl{\{}\mathop{{}\mathbb{E}}\mathopen{}\Bigl{\{}\xi_{\alpha_{1}(A_{1},B_{1})}^{(i,j)}\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)}\,|\,\mathcal{F}\Bigr{\}},\mathop{{}\mathbb{E}}\mathopen{}\Bigl{\{}\xi_{\alpha_{1}^{\prime}(A_{1},B_{1})}^{(i^{\prime},j^{\prime})}\xi_{\alpha_{2}^{\prime}(A_{2},B_{2})}^{(i^{\prime},j^{\prime})}\,|\,\mathcal{F}\Bigr{\}}\biggr{\}}.

Without loss of generality, we assume that V1V2V1V2V_{1}\cup V_{2}\cup V_{1}^{\prime}\cup V_{2}^{\prime}\neq\varnothing, otherwise the argument is even simpler. Moreover, we may assume that V1V_{1}\neq\varnothing. Let 0=σ(Yi,j,Yp,q:p,qV0)\mathcal{F}_{0}=\sigma(Y_{i,j}^{\prime},Y_{p,q}:p,q\in V_{0}), and we have ξα1(A1,B1)(i,j)\xi_{\alpha_{1}(A_{1},B_{1})}^{(i,j)} and ξα2(A2,B2)(i,j)\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)} are conditionally independent given 0\mathcal{F}\vee\mathcal{F}_{0}. Moreover, by 5.5, 𝔼{ξα1(A1,B1)(i,j)|0}=𝔼{ξα2(A2,B2)(i,j)|0}=0,\mathop{{}\mathbb{E}}\mathopen{}\{\xi_{\alpha_{1}(A_{1},B_{1})}^{(i,j)}\,|\,\mathcal{F}\vee\mathcal{F}_{0}\}=\mathop{{}\mathbb{E}}\mathopen{}\{\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)}\,|\,\mathcal{F}\vee\mathcal{F}_{0}\}=0, and thus 𝔼{ξα1(A1,B1)(i,j)ξα2(A2,B2)(i,j)|}=0\mathop{{}\mathbb{E}}\mathopen{}\{\xi_{\alpha_{1}(A_{1},B_{1})}^{(i,j)}\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)}\,|\,\mathcal{F}\}=0. Therefore, we have under the condition (b),

Cov{ξα1(A1,B1)(i,j)ξα2(A2,B2)(i,j),ξα1(A1,B1)(i,j)ξα2(A2,B2)(i,j)}=0\mathop{\mathrm{Cov}}\{\xi_{\alpha_{1}(A_{1},B_{1})}^{(i,j)}\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)},\xi_{\alpha_{1}^{\prime}(A_{1},B_{1})}^{(i^{\prime},j^{\prime})}\xi_{\alpha_{2}^{\prime}(A_{2},B_{2})}^{(i^{\prime},j^{\prime})}\}=0. (A.12)

Combining A.11 and A.12 we prove that A.7 holds for |{α1,α2}{α1,α2}|=2d(s+t)\lvert\{\alpha_{1},\alpha_{2}\}\cap\{\alpha_{1}^{\prime},\alpha_{2}^{\prime}\}\rvert=2d-(s+t). This completes the proof. ∎

Proof of Lemma 5.4.

In this proof, we denote by CC a constant depending on kk and dd, which may take different values in different places. Note that 2s,td2\leqslant s,t\leqslant d, and

Var{(i,j)𝒜n,2(α1𝒜n,d(i,j)ξα1(A1,B1)(i,j))(α2𝒜n,d(i,j)ξα2(A2,B2)(i,j))}\displaystyle\mathop{\mathrm{Var}}\Biggl{\{}\sum_{(i,j)\in\mathcal{A}_{n,2}}\biggl{(}\sum_{\alpha_{1}\in\mathcal{A}_{n,d}^{(i,j)}}\xi_{\alpha_{1}(A_{1},B_{1})}^{(i,j)}\biggr{)}\biggl{(}\sum_{\alpha_{2}\in\mathcal{A}_{n,d}^{(i,j)}}\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)}\biggr{)}\Biggr{\}}
=(i,j)𝒜n,2\@@atop(i,j)𝒜n,2α1𝒜n,d(i,j)\@@atopα2𝒜n,d(i,j)α1𝒜n,d(i,j)\@@atopα2𝒜n,d(i,j)Cov{ξα1(A1,B1)(i,j)ξα2(A2,B2)(i,j),ξα1(A1,B1)(i,j)ξα2(A2,B2)(i,j)}\displaystyle=\sum_{(i,j)\in\mathcal{A}_{n,2}\@@atop(i^{\prime},j^{\prime})\in\mathcal{A}_{n,2}}\sum_{\alpha_{1}\in\mathcal{A}_{n,d}^{(i,j)}\@@atop\alpha_{2}\in\mathcal{A}_{n,d}^{(i,j)}}\sum_{\alpha_{1}^{\prime}\in\mathcal{A}_{n,d}^{(i^{\prime},j^{\prime})}\@@atop\alpha_{2}^{\prime}\in\mathcal{A}_{n,d}^{(i^{\prime},j^{\prime})}}\mathop{\mathrm{Cov}}\bigl{\{}\xi_{\alpha_{1}(A_{1},B_{1})}^{(i,j)}\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)},\xi_{\alpha_{1}^{\prime}(A_{1},B_{1})}^{(i^{\prime},j^{\prime})}\xi_{\alpha_{2}^{\prime}(A_{2},B_{2})}^{(i^{\prime},j^{\prime})}\bigr{\}}
=s,t=2d(i,j)𝒜n,2\@@atop(i,j)𝒜n,2α1𝒜n,d(i,j)\@@atopα2𝒜n,d(i,j)α1𝒜n,d(i,j)\@@atopα2𝒜n,d(i,j)\displaystyle=\sum_{s,t=2}^{d}\sum_{(i,j)\in\mathcal{A}_{n,2}\@@atop(i^{\prime},j^{\prime})\in\mathcal{A}_{n,2}}\sum_{\alpha_{1}\in\mathcal{A}_{n,d}^{(i,j)}\@@atop\alpha_{2}\in\mathcal{A}_{n,d}^{(i,j)}}\sum_{\alpha_{1}^{\prime}\in\mathcal{A}_{n,d}^{(i^{\prime},j^{\prime})}\@@atop\alpha_{2}^{\prime}\in\mathcal{A}_{n,d}^{(i^{\prime},j^{\prime})}} (A.13)
×Cov{ξα1(A1,B1)(i,j)ξα2(A2,B2)(i,j),ξα1(A1,B1)(i,j)ξα2(A2,B2)(i,j)}Os,t,\displaystyle\quad\times\mathop{\mathrm{Cov}}\bigl{\{}\xi_{\alpha_{1}(A_{1},B_{1})}^{(i,j)}\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)},\xi_{\alpha_{1}^{\prime}(A_{1},B_{1})}^{(i^{\prime},j^{\prime})}\xi_{\alpha_{2}^{\prime}(A_{2},B_{2})}^{(i^{\prime},j^{\prime})}\bigr{\}}\mathbb{N}{O_{s,t}}, (A.14)

where Os,t={|{α1}{α2}|=s}{|{α1}{α2}|=t}O_{s,t}=\{\lvert\{\alpha_{1}\}\cap\{\alpha_{2}\}\rvert=s\}\cap\{\lvert\{\alpha_{1}^{\prime}\}\cap\{\alpha_{2}^{\prime}\}\rvert=t\}. If |({α1}{α2})({α1}{α2})|2d(s+t)\lvert(\{\alpha_{1}\}\cup\{\alpha_{2}\})\cap(\{\alpha_{1}^{\prime}\}\cap\{\alpha_{2}^{\prime}\})\rvert\leqslant 2d-(s+t), by A.7 in Lemma A.1, we have

Cov{ξα1(A1,B1)(i,j)ξα2(A2,B2)(i,j),ξα1(A1,B1)(i,j)ξα2(A2,B2)(i,j)}=0.\displaystyle\mathop{\mathrm{Cov}}\bigl{\{}\xi_{\alpha_{1}(A_{1},B_{1})}^{(i,j)}\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)},\xi_{\alpha_{1}^{\prime}(A_{1},B_{1})}^{(i^{\prime},j^{\prime})}\xi_{\alpha_{2}^{\prime}(A_{2},B_{2})}^{(i^{\prime},j^{\prime})}\bigr{\}}=0.

If |({α1}{α2})({α1}{α2})|>2d(s+t)\lvert(\{\alpha_{1}\}\cup\{\alpha_{2}\})\cap(\{\alpha_{1}^{\prime}\}\cap\{\alpha_{2}^{\prime}\})\rvert>2d-(s+t), then, recalling that (ξα1(A1,B1)(i,j),ξα2(A2,B2)(i,j))=d.(ξα1(A1,B1)(i,j),ξα2(A2,B2)(i,j))(\xi_{\alpha_{1}(A_{1},B_{1})}^{(i,j)},\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)})\stackrel{{\scriptstyle d.}}{{=}}(\xi_{\alpha_{1}^{\prime}(A_{1},B_{1})}^{(i^{\prime},j^{\prime})},\xi_{\alpha_{2}^{\prime}(A_{2},B_{2})}^{(i^{\prime},j^{\prime})}), we have

|Cov{ξα1(A1,B1)(i,j)ξα2(A2,B2)(i,j),ξα1(A1,B1)(i,j)ξα2(A2,B2)(i,j)}|\displaystyle\bigl{\lvert}\mathop{\mathrm{Cov}}\{\xi_{\alpha_{1}(A_{1},B_{1})}^{(i,j)}\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)},\xi_{\alpha_{1}^{\prime}(A_{1},B_{1})}^{(i^{\prime},j^{\prime})}\xi_{\alpha_{2}^{\prime}(A_{2},B_{2})}^{(i^{\prime},j^{\prime})}\}\bigr{\rvert}
𝔼{(ξα1(A1,B1)(i,j))2(ξα2(A2,B2)(i,j))2}\displaystyle\leqslant\mathop{{}\mathbb{E}}\mathopen{}\{(\xi_{\alpha_{1}(A_{1},B_{1})}^{(i,j)})^{2}(\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)})^{2}\}
C(𝔼{fA1,B14(Xα1(A1,B1);Yα1(A1,B1))}+𝔼{fA1,B14(Xα2(A2,B2);Yα2(A2,B2))})\displaystyle\leqslant C\bigl{(}\mathop{{}\mathbb{E}}\mathopen{}\{f_{A_{1},B_{1}}^{4}(X_{\alpha_{1}(A_{1},B_{1})};Y_{\alpha_{1}(A_{1},B_{1})})\}+\mathop{{}\mathbb{E}}\mathopen{}\{f_{A_{1},B_{1}}^{4}(X_{\alpha_{2}(A_{2},B_{2})};Y_{\alpha_{2}(A_{2},B_{2})})\}\bigr{)}
Cτ4.\displaystyle\leqslant C\tau^{4}. (A.15)

Therefore, with

O1={|({α1}{α2})({α1}{α2})|>2d(s+t)},\displaystyle O_{1}=\{\lvert(\{\alpha_{1}\}\cup\{\alpha_{2}\})\cap(\{\alpha_{1}^{\prime}\}\cup\{\alpha_{2}^{\prime}\})\rvert>2d-(s+t)\},

we have

Var{(i,j)𝒜n,2(α𝒜n,d(i,j)ξα(A1,B1)(i,j))(α𝒜n,d(i,j)ξα(A1,B1)(i,j))}\displaystyle\mathop{\mathrm{Var}}\Biggl{\{}\sum_{(i,j)\in\mathcal{A}_{n,2}}\biggl{(}\sum_{\alpha\in\mathcal{A}_{n,d}^{(i,j)}}\xi_{\alpha(A_{1},B_{1})}^{(i,j)}\biggr{)}\biggl{(}\sum_{\alpha\in\mathcal{A}_{n,d}^{(i,j)}}\xi_{\alpha(A_{1},B_{1})}^{(i,j)}\biggr{)}\Biggr{\}}
Cτ4s,t=0d(i,j)𝒜n,2\@@atop(i,j)𝒜n,2α1𝒜n,d(i,j)\@@atopα2𝒜n,d(i,j)α1𝒜n,d(i,j)\@@atopα2𝒜n,d(i,j)O1Os,t\displaystyle\leqslant C\tau^{4}\sum_{s,t=0}^{d}\sum_{(i,j)\in\mathcal{A}_{n,2}\@@atop(i^{\prime},j^{\prime})\in\mathcal{A}_{n,2}}\sum_{\alpha_{1}\in\mathcal{A}_{n,d}^{(i,j)}\@@atop\alpha_{2}\in\mathcal{A}_{n,d}^{(i,j)}}\sum_{\alpha_{1}^{\prime}\in\mathcal{A}_{n,d}^{(i^{\prime},j^{\prime})}\@@atop\alpha_{2}^{\prime}\in\mathcal{A}_{n,d}^{(i^{\prime},j^{\prime})}}\mathbb{N}{O_{1}\cap O_{s,t}}
Cτ4s,t=0dn(2ds)+(2dt)(2dst+1)\displaystyle\leqslant C\tau^{4}\sum_{s,t=0}^{d}n^{(2d-s)+(2d-t)-(2d-s-t+1)}
Cn2d1τ4.\displaystyle\leqslant Cn^{2d-1}\tau^{4}.\qed
Proof of Lemma 5.5.

If k<d+1k<d+1, then it follows that ξα(G)=0\xi_{\alpha(G)}=0 for all GΓd+1G\in\Gamma_{d+1} and α𝒜n,d+1\alpha\in\mathcal{A}_{n,d+1}. Therefore, we assume kd+1k\geqslant d+1 without loss of generality.

Observe that

Var{(i,j)𝒜n,2(α1𝒜n,v1(i,j)ξα1(A1,B1)(i,j))|α2𝒜n,v2(i,j)ξα2(A2,B2)(i,j)|}\displaystyle\mathop{\mathrm{Var}}\Biggl{\{}\sum_{(i,j)\in\mathcal{A}_{n,2}}\biggl{(}\sum_{\alpha_{1}\in\mathcal{A}_{n,v_{1}}^{(i,j)}}\xi_{\alpha_{1}(A_{1},B_{1})}^{(i,j)}\biggr{)}\biggl{\lvert}\sum_{\alpha_{2}\in\mathcal{A}_{n,v_{2}}^{(i,j)}}\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)}\biggr{\rvert}\Biggr{\}} (A.16)
=(i,j)𝒜n,2(i,j)𝒜n,2Cov{(α1𝒜n,v1(i,j)ξα1(A1,B1)(i,j))|α2𝒜n,v2(i,j)ξα2(A2,B2)(i,j)|,\displaystyle=\sum_{(i,j)\in\mathcal{A}_{n,2}}\sum_{(i^{\prime},j^{\prime})\in\mathcal{A}_{n,2}}\mathop{\mathrm{Cov}}\Biggl{\{}\biggl{(}\sum_{\alpha_{1}\in\mathcal{A}_{n,v_{1}}^{(i,j)}}\xi_{\alpha_{1}(A_{1},B_{1})}^{(i,j)}\biggr{)}\biggl{\lvert}\sum_{\alpha_{2}\in\mathcal{A}_{n,v_{2}}^{(i,j)}}\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)}\biggr{\rvert},
(α1𝒜n,v1(i,j)ξα1(A1,B1)(i,j))|α2𝒜n,v2(i,j)ξα2(A2,B2)(i,j)|}.\displaystyle\hskip 142.26378pt\biggl{(}\sum_{\alpha_{1}^{\prime}\in\mathcal{A}_{n,v_{1}}^{(i^{\prime},j^{\prime})}}\xi_{\alpha_{1}^{\prime}(A_{1},B_{1})}^{(i^{\prime},j^{\prime})}\biggr{)}\biggl{\lvert}\sum_{\alpha_{2}^{\prime}\in\mathcal{A}_{n,v_{2}}^{(i^{\prime},j^{\prime})}}\xi_{\alpha_{2}^{\prime}(A_{2},B_{2})}^{(i^{\prime},j^{\prime})}\biggr{\rvert}\Biggr{\}}.

Letting

1=σ(X)σ(Yp,q,Yp,q:{p,q}{i,j}),\displaystyle\mathcal{F}_{1}=\sigma(X)\vee\sigma(Y_{p,q},Y_{p,q}^{\prime}:\{p,q\}\neq\{i,j\}),

and noting that

(α1𝒜n,v1(i,j)ξα1(A1,B1)(i,j))|α2𝒜n,v2(i,j)ξα2(A2,B2)(i,j)|\displaystyle\biggl{(}\sum_{\alpha_{1}\in\mathcal{A}_{n,v_{1}}^{(i,j)}}\xi_{\alpha_{1}(A_{1},B_{1})}^{(i,j)}\biggr{)}\biggl{\lvert}\sum_{\alpha_{2}\in\mathcal{A}_{n,v_{2}}^{(i,j)}}\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)}\biggr{\rvert}

is anti-symmetric with respect to (Yij,Yij)(Y_{ij},Y_{ij}^{\prime}), we have

𝔼{(α1𝒜n,v1(i,j)ξα1(A1,B1)(i,j))|α2𝒜n,v2(i,j)ξα2(A2,B2)(i,j)|}=0.\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\Biggl{\{}\biggl{(}\sum_{\alpha_{1}\in\mathcal{A}_{n,v_{1}}^{(i,j)}}\xi_{\alpha_{1}(A_{1},B_{1})}^{(i,j)}\biggr{)}\biggl{\lvert}\sum_{\alpha_{2}\in\mathcal{A}_{n,v_{2}}^{(i,j)}}\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)}\biggr{\rvert}\Biggr{\}}=0.

Now, we consider the following two cases. First, if {i,j}{i,j}\{i,j\}\neq\{i^{\prime},j^{\prime}\}, we have

(α1𝒜n,v1(i,j)ξα1(A1,B1)(i,j))|α2𝒜n,v2(i,j)ξα2(A2,B2)(i,j)|is 1 measurable\displaystyle\biggl{(}\sum_{\alpha_{1}^{\prime}\in\mathcal{A}_{n,v_{1}}^{(i^{\prime},j^{\prime})}}\xi_{\alpha_{1}^{\prime}(A_{1},B_{1})}^{(i^{\prime},j^{\prime})}\biggr{)}\biggl{\lvert}\sum_{\alpha_{2}^{\prime}\in\mathcal{A}_{n,v_{2}}^{(i^{\prime},j^{\prime})}}\xi_{\alpha_{2}^{\prime}(A_{2},B_{2})}^{(i^{\prime},j^{\prime})}\biggr{\rvert}\quad\text{is $\mathcal{F}_{1}$ measurable}

and by anti-symmetry again,

𝔼{(α1𝒜n,v1(i,j)ξα1(A1,B1)(i,j))|α2𝒜n,v2(i,j)ξα2(A2,B2)(i,j)||1}=0.\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\Biggl{\{}\biggl{(}\sum_{\alpha_{1}\in\mathcal{A}_{n,v_{1}}^{(i,j)}}\xi_{\alpha_{1}(A_{1},B_{1})}^{(i,j)}\biggr{)}\biggl{\lvert}\sum_{\alpha_{2}\in\mathcal{A}_{n,v_{2}}^{(i,j)}}\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)}\biggr{\rvert}\Biggm{|}\mathcal{F}_{1}\Biggr{\}}=0.

Therefore,

Cov{(α1𝒜n,v1(i,j)ξα1(A1,B1)(i,j))|α2𝒜n,v2(i,j)ξα2(A2,B2)(i,j)|,(α1𝒜n,v1(i,j)ξα1(A1,B1)(i,j))|α2𝒜n,v2(i,j)ξα2(A2,B2)(i,j)|}=0\mathop{\mathrm{Cov}}\Biggl{\{}\biggl{(}\sum_{\alpha_{1}\in\mathcal{A}_{n,v_{1}}^{(i,j)}}\xi_{\alpha_{1}(A_{1},B_{1})}^{(i,j)}\biggr{)}\biggl{\lvert}\sum_{\alpha_{2}\in\mathcal{A}_{n,v_{2}}^{(i,j)}}\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)}\biggr{\rvert},\\ \quad\biggl{(}\sum_{\alpha_{1}^{\prime}\in\mathcal{A}_{n,v_{1}}^{(i^{\prime},j^{\prime})}}\xi_{\alpha_{1}^{\prime}(A_{1},B_{1})}^{(i^{\prime},j^{\prime})}\biggr{)}\biggl{\lvert}\sum_{\alpha_{2}^{\prime}\in\mathcal{A}_{n,v_{2}}^{(i^{\prime},j^{\prime})}}\xi_{\alpha_{2}^{\prime}(A_{2},B_{2})}^{(i^{\prime},j^{\prime})}\biggr{\rvert}\Biggr{\}}=0 (A.17)

for {i,j}{i,j}\{i,j\}\neq\{i^{\prime},j^{\prime}\}.

It suffices to consider the case where {i,j}={i,j}\{i,j\}=\{i^{\prime},j^{\prime}\}. Observe that

Cov{(α1𝒜n,v1(i,j)ξα1(A1,B1)(i,j))|α2𝒜n,v2(i,j)ξα2(A2,B2)(i,j)|,(α1𝒜n,v1(i,j)ξα1(A1,B1)(i,j))|α2𝒜n,v2(i,j)ξα2(A2,B2)(i,j)|}\displaystyle\mathop{\mathrm{Cov}}\Biggl{\{}\biggl{(}\sum_{\alpha_{1}\in\mathcal{A}_{n,v_{1}}^{(i,j)}}\xi_{\alpha_{1}(A_{1},B_{1})}^{(i,j)}\biggr{)}\biggl{\lvert}\sum_{\alpha_{2}\in\mathcal{A}_{n,v_{2}}^{(i,j)}}\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)}\biggr{\rvert},\biggl{(}\sum_{\alpha_{1}^{\prime}\in\mathcal{A}_{n,v_{1}}^{(i,j)}}\xi_{\alpha_{1}^{\prime}(A_{1},B_{1})}^{(i,j)}\biggr{)}\biggl{\lvert}\sum_{\alpha_{2}^{\prime}\in\mathcal{A}_{n,v_{2}}^{(i,j)}}\xi_{\alpha_{2}^{\prime}(A_{2},B_{2})}^{(i,j)}\biggr{\rvert}\Biggr{\}} (A.18)
=𝔼{(α1𝒜n,v1(i,j)ξα1(A1,B1)(i,j))(α1𝒜n,v1(i,j)ξα1(A1,B1)(i,j))|(α2𝒜n,v2(i,j)ξα2(A2,B2)(i,j))(α2𝒜n,v2(i,j)ξα2(A2,B2)(i,j))|}\displaystyle=\mathop{{}\mathbb{E}}\mathopen{}\Biggl{\{}\biggl{(}\sum_{\alpha_{1}\in\mathcal{A}_{n,v_{1}}^{(i,j)}}\xi_{\alpha_{1}(A_{1},B_{1})}^{(i,j)}\biggr{)}\biggl{(}\sum_{\alpha_{1}^{\prime}\in\mathcal{A}_{n,v_{1}}^{(i,j)}}\xi_{\alpha_{1}^{\prime}(A_{1},B_{1})}^{(i,j)}\biggr{)}\biggl{\lvert}\biggl{(}\sum_{\alpha_{2}\in\mathcal{A}_{n,v_{2}}^{(i,j)}}\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)}\biggr{)}\biggl{(}\sum_{\alpha_{2}^{\prime}\in\mathcal{A}_{n,v_{2}}^{(i,j)}}\xi_{\alpha_{2}^{\prime}(A_{2},B_{2})}^{(i,j)}\biggr{)}\biggr{\rvert}\Biggr{\}}
=α1𝒜n,v1(i,j)α1𝒜n,v1(i,j)𝔼{ξα1(A1,B1)(i,j)ξα1(A1,B1)(i,j)|α2𝒜n,v2(i,j)α2𝒜n,v2(i,j)ξα2(A2,B2)(i,j)ξα2(A2,B2)(i,j)|}.\displaystyle=\sum_{\alpha_{1}\in\mathcal{A}_{n,v_{1}}^{(i,j)}}\sum_{\alpha_{1}^{\prime}\in\mathcal{A}_{n,v_{1}}^{(i,j)}}\mathop{{}\mathbb{E}}\mathopen{}\Biggl{\{}\xi_{\alpha_{1}(A_{1},B_{1})}^{(i,j)}\xi_{\alpha_{1}^{\prime}(A_{1},B_{1})}^{(i,j)}\biggl{\lvert}\sum_{\alpha_{2}\in\mathcal{A}_{n,v_{2}}^{(i,j)}}\sum_{\alpha_{2}^{\prime}\in\mathcal{A}_{n,v_{2}}^{(i,j)}}\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)}\xi_{\alpha_{2}^{\prime}(A_{2},B_{2})}^{(i,j)}\biggr{\rvert}\Biggr{\}}.

Let H1={α1}{α1}H_{1}=\{\alpha_{1}\}\setminus\{\alpha_{1}^{\prime}\} and H1={α1}{α1}H_{1}^{\prime}=\{\alpha_{1}^{\prime}\}\setminus\{\alpha_{1}\}. Let t=|α1α1|t=\lvert\alpha_{1}\cap\alpha_{1}^{\prime}\rvert, and then we have 2tv12\leqslant t\leqslant v_{1}. Now, as

α2𝒜n,v2(i,j)α2𝒜n,v2(i,j)ξα2(A2,B2)(i,j)ξα2(A2,B2)(i,j)\displaystyle\sum_{\alpha_{2}\in\mathcal{A}_{n,v_{2}}^{(i,j)}}\sum_{\alpha_{2}^{\prime}\in\mathcal{A}_{n,v_{2}}^{(i,j)}}\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)}\xi_{\alpha_{2}^{\prime}(A_{2},B_{2})}^{(i,j)} =α2,α2𝒜1ξα2(A2,B2)(i,j)ξα2(A2,B2)(i,j)\displaystyle=\sum_{\alpha_{2},\alpha_{2}^{\prime}\in\mathcal{A}_{1}}\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)}\xi_{\alpha_{2}^{\prime}(A_{2},B_{2})}^{(i,j)}
+α2,α2𝒜2ξα2(A2,B2)(i,j)ξα2(A2,B2)(i,j),\displaystyle\quad+\sum_{\alpha_{2},\alpha_{2}^{\prime}\in\mathcal{A}_{2}}\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)}\xi_{\alpha_{2}^{\prime}(A_{2},B_{2})}^{(i,j)},

where 𝒜1={α2,α2𝒜n,v2(i,j):(H1H1){α2,α2}}\mathcal{A}_{1}=\{\alpha_{2},\alpha_{2}^{\prime}\in\mathcal{A}_{n,v_{2}}^{(i,j)}:(H_{1}\cup H_{1}^{\prime})\setminus\{\alpha_{2},\alpha_{2}^{\prime}\}\neq\varnothing\} and 𝒜2={α2,α2𝒜n,v2(i,j):(H1H1){α2,α2}=}\mathcal{A}_{2}=\{\alpha_{2},\alpha_{2}^{\prime}\in\mathcal{A}_{n,v_{2}}^{(i,j)}:(H_{1}\cup H_{1}^{\prime})\setminus\{\alpha_{2},\alpha_{2}^{\prime}\}=\varnothing\}. If there exists r(H1H1){α2,α2}r\in(H_{1}\cup H_{1}^{\prime})\setminus\{\alpha_{2},\alpha_{2}^{\prime}\}, letting r=σ(Xp,Yp,q,Yp,q:p,q[n]{r})\mathcal{F}_{r}=\sigma(X_{p},Y_{p,q},Y_{p,q}^{\prime}:p,q\in[n]\setminus\{r\}), then we have

α2,α2𝒜1ξα2(A2,B2)(i,j)ξα2(A2,B2)(i,j)r,\displaystyle\sum_{\alpha_{2},\alpha_{2}^{\prime}\in\mathcal{A}_{1}}\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)}\xi_{\alpha_{2}^{\prime}(A_{2},B_{2})}^{(i,j)}\in\mathcal{F}_{r},

and by orthogonality, we have

𝔼{ξα1(A1,B1)(i,j)ξα1(A1,B1)(i,j)|r}=0.\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\bigl{\{}\xi_{\alpha_{1}(A_{1},B_{1})}^{(i,j)}\xi_{\alpha_{1}^{\prime}(A_{1},B_{1})}^{(i,j)}|\mathcal{F}_{r}\bigr{\}}=0.

Therefore, we have

𝔼{ξα1(A1,B1)(i,j)ξα1(A1,B1)(i,j)|α2,α2𝒜1ξα2(A2,B2)(i,j)ξα2(A2,B2)(i,j)|}=0.\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\Biggl{\{}\xi_{\alpha_{1}(A_{1},B_{1})}^{(i,j)}\xi_{\alpha_{1}^{\prime}(A_{1},B_{1})}^{(i,j)}\biggl{\lvert}\sum_{\alpha_{2},\alpha_{2}^{\prime}\in\mathcal{A}_{1}}\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)}\xi_{\alpha_{2}^{\prime}(A_{2},B_{2})}^{(i,j)}\biggr{\rvert}\Biggr{\}}=0.

Hence, by Cauchy’s inequality, we have

|𝔼{ξα1(A1,B1)(i,j)ξα1(A1,B1)(i,j)|α2,α2𝒜n,v2(i,j)ξα2(A2,B2)(i,j)ξα2(A2,B2)(i,j)|}|\displaystyle\Biggl{\lvert}\mathop{{}\mathbb{E}}\mathopen{}\Biggl{\{}\xi_{\alpha_{1}(A_{1},B_{1})}^{(i,j)}\xi_{\alpha_{1}^{\prime}(A_{1},B_{1})}^{(i,j)}\biggl{\lvert}\sum_{\alpha_{2},\alpha_{2}^{\prime}\in\mathcal{A}_{n,v_{2}}^{(i,j)}}\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)}\xi_{\alpha_{2}^{\prime}(A_{2},B_{2})}^{(i,j)}\biggr{\rvert}\Biggr{\}}\Biggr{\rvert}
𝔼{|ξα1(A1,B1)(i,j)ξα1(A1,B1)(i,j)||α2,α2𝒜2ξα2(A2,B2)(i,j)ξα2(A2,B2)(i,j)|}\displaystyle\leqslant\mathop{{}\mathbb{E}}\mathopen{}\Biggl{\{}\bigl{\lvert}\xi_{\alpha_{1}(A_{1},B_{1})}^{(i,j)}\xi_{\alpha_{1}^{\prime}(A_{1},B_{1})}^{(i,j)}\bigr{\rvert}\biggl{\lvert}\sum_{\alpha_{2},\alpha_{2}^{\prime}\in\mathcal{A}_{2}}\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)}\xi_{\alpha_{2}^{\prime}(A_{2},B_{2})}^{(i,j)}\biggr{\rvert}\Biggr{\}}
Cτ2𝔼{|α2,α2𝒜2ξα2(A2,B2)(i,j)ξα2(A2,B2)(i,j)|2}.\displaystyle\leqslant C\tau^{2}\sqrt{\mathop{{}\mathbb{E}}\mathopen{}\Biggl{\{}\biggl{\lvert}\sum_{\alpha_{2},\alpha_{2}^{\prime}\in\mathcal{A}_{2}}\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)}\xi_{\alpha_{2}^{\prime}(A_{2},B_{2})}^{(i,j)}\biggr{\rvert}^{2}\Biggr{\}}}.

Following the similar argument in the proof of Lemma 5.4, and recalling that {α1α1}=t\{\alpha_{1}\cap\alpha_{1}^{\prime}\}=t and |𝒜2|Cn2(t2)(nv2v11)\lvert\mathcal{A}_{2}\rvert\leqslant Cn^{2(t-2)}(n^{v_{2}-v_{1}}\vee 1), we have

𝔼{|α2,α2𝒜2ξα2(A2,B2)(i,j)ξα2(A2,B2)(i,j)|2}\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\Biggl{\{}\biggl{\lvert}\sum_{\alpha_{2},\alpha_{2}^{\prime}\in\mathcal{A}_{2}}\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)}\xi_{\alpha_{2}^{\prime}(A_{2},B_{2})}^{(i,j)}\biggr{\rvert}^{2}\Biggr{\}} Cn2(t2)(nv2v11)τ4.\displaystyle\leqslant Cn^{2(t-2)}(n^{v_{2}-v_{1}}\vee 1)\tau^{4}.

Therefore, we have

|𝔼{ξα1(A1,B1)(i,j)ξα1(A1,B1)(i,j)|α2,α2𝒜n,v2(i,j)ξα2(A2,B2)(i,j)ξα2(A2,B2)(i,j)|}|Cn2(t2)(nv2v11)τ4.\displaystyle\Biggl{\lvert}\mathop{{}\mathbb{E}}\mathopen{}\Biggl{\{}\xi_{\alpha_{1}(A_{1},B_{1})}^{(i,j)}\xi_{\alpha_{1}^{\prime}(A_{1},B_{1})}^{(i,j)}\biggl{\lvert}\sum_{\alpha_{2},\alpha_{2}^{\prime}\in\mathcal{A}_{n,v_{2}}^{(i,j)}}\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)}\xi_{\alpha_{2}^{\prime}(A_{2},B_{2})}^{(i,j)}\biggr{\rvert}\Biggr{\}}\Biggr{\rvert}\leqslant Cn^{2(t-2)}(n^{v_{2}-v_{1}}\vee 1)\tau^{4}.

Substituting the foregoing inequality to A.18, we have

(i,j)𝒜n,2Cov{(α1𝒜n,v1(i,j)ξα1(A1,B1)(i,j))|α2𝒜n,v2(i,j)ξα2(A2,B2)(i,j)|,(α1𝒜n,v1(i,j)ξα1(A1,B1)(i,j))|α2𝒜n,v2(i,j)ξα2(A2,B2)(i,j)|}Cn2max{v1,v2}2τ4.\sum_{(i,j)\in\mathcal{A}_{n,2}}\mathop{\mathrm{Cov}}\Biggl{\{}\biggl{(}\sum_{\alpha_{1}\in\mathcal{A}_{n,v_{1}}^{(i,j)}}\xi_{\alpha_{1}(A_{1},B_{1})}^{(i,j)}\biggr{)}\biggl{\lvert}\sum_{\alpha_{2}\in\mathcal{A}_{n,v_{2}}^{(i,j)}}\xi_{\alpha_{2}(A_{2},B_{2})}^{(i,j)}\biggr{\rvert},\\ \biggl{(}\sum_{\alpha_{1}^{\prime}\in\mathcal{A}_{n,v_{1}}^{(i,j)}}\xi_{\alpha_{1}^{\prime}(A_{1},B_{1})}^{(i,j)}\biggr{)}\biggl{\lvert}\sum_{\alpha_{2}^{\prime}\in\mathcal{A}_{n,v_{2}}^{(i,j)}}\xi_{\alpha_{2}^{\prime}(A_{2},B_{2})}^{(i,j)}\biggr{\rvert}\Biggr{\}}\\ \leqslant Cn^{2\max\{v_{1},v_{2}\}-2}\tau^{4}. (A.19)

By A.16, A.17 and A.19, we complete the proof.

A.3 Proof of Lemma 5.6

Lemma 5.6 follows from a similar argument as that in the proof of Lemma 5.4 and the following lemma. Let 𝒢~f,={(A,B)𝒢f,:GA,B is strongly connected.}\widetilde{\mathcal{G}}_{f,\ell}=\{(A,B)\in\mathcal{G}_{f,\ell}:G_{A,B}\text{ is strongly connected.}\} Now, as the function gg does not depend on XX, we set Am=A_{m}=\varnothing in the following lemma. With a slight abuse of notation, For j=1,2j=1,2 and for Bm[k]2B_{m}\subset[k]_{2}, let GmG_{m} be the graph generated by BmB_{m} and let vmv_{m} be the number of nodes of GmG_{m}, and we write Bm𝒢B_{m}\in\mathcal{G} if Gm𝒢G_{m}\in\mathcal{G}.

Lemma A.2.

Let Bm𝒢~f,d𝒢f,d+1B_{m}\in\widetilde{\mathcal{G}}_{f,d}\cup\mathcal{G}_{f,d+1} for m=1,2m=1,2. Let (i,j),(i,j)𝒜n,2(i,j),(i^{\prime},j^{\prime})\in\mathcal{A}_{n,2}, and let αm𝒜n,vm(i,j)\alpha_{m}\in\mathcal{A}_{n,v_{m}}^{(i,j)}, αm𝒜n,vm(i,j)\alpha_{m}^{\prime}\in\mathcal{A}_{n,v_{m}}^{(i^{\prime},j^{\prime})} for m=1,2m=1,2. Let s=|{α1}{α2}|s=\lvert\{\alpha_{1}\}\cap\{\alpha_{2}\}\rvert and t=|{α1}{α2}|t=\lvert\{\alpha_{1}^{\prime}\}\cap\{\alpha_{2}^{\prime}\}\rvert. For m=1,2m=1,2, let γm\gamma_{m} indicate that Bm𝒢~f,d𝒢~f,d+1B_{m}\in\widetilde{\mathcal{G}}_{f,d}\cup\widetilde{\mathcal{G}}_{f,d+1}. Then

Cov{ηα1(B1)(i,j)ηα2(B2)(i,j),ηα1(B1)(i,j)ηα2(B2)(i,j)}=0\displaystyle\mathop{\mathrm{Cov}}\Bigl{\{}\eta_{\alpha_{1}(B_{1})}^{(i,j)}\eta_{\alpha_{2}(B_{2})}^{(i,j)},\eta_{\alpha_{1}^{\prime}(B_{1})}^{(i^{\prime},j^{\prime})}\eta_{\alpha_{2}^{\prime}(B_{2})}^{(i^{\prime},j^{\prime})}\Bigr{\}}=0 (A.20)

for |{α1,α2}{α1,α2}|<v1+v2+γ1+γ2(s+t)\lvert\{\alpha_{1},\alpha_{2}\}\cap\{\alpha_{1}^{\prime},\alpha_{2}^{\prime}\}\rvert<v_{1}+v_{2}+\gamma_{1}+\gamma_{2}-(s+t).

Proof.

The proof is similar to that of Lemma A.1.

Let V0,V0,V1,V1,V2,V2V_{0},V_{0}^{\prime},V_{1},V_{1}^{\prime},V_{2},V_{2}^{\prime} be defined as in A.8. Note that if GBG_{B} has isolated nodes, then ηα(B)=0\eta_{\alpha(B)}=0 for all α𝒜n,vB\alpha\in\mathcal{A}_{n,v_{B}}, where vBv_{B} is the number of nodes of the graph generated by the index set BB. If v1+v2=s+tv_{1}+v_{2}=s+t, then it follows that {α1}={α2}\{\alpha_{1}\}=\{\alpha_{2}\} and {α1}={α2}\{\alpha_{1}^{\prime}\}=\{\alpha_{2}^{\prime}\}. If |{α1}{α1}|<2\lvert\{\alpha_{1}\}\cap\{\alpha_{1}^{\prime}\}\rvert<2, then ηα1(B1)(i,j)ηα2(B2)(i,j)\eta_{\alpha_{1}(B_{1})}^{(i,j)}\eta_{\alpha_{2}(B_{2})}^{(i,j)} and ηα1(B1)(i,j)ηα2(B2)(i,j)\eta_{\alpha_{1}^{\prime}(B_{1})}^{(i^{\prime},j^{\prime})}\eta_{\alpha_{2}^{\prime}(B_{2})}^{(i^{\prime},j^{\prime})} are independent, which further implies that A.20 holds.

Now we consider the case where v1+v2>s+tv_{1}+v_{2}>s+t. If |{α1,α2}{α1,α2}|<v1+v2(s+t)\lvert\{\alpha_{1},\alpha_{2}\}\cap\{\alpha_{1}^{\prime},\alpha_{2}^{\prime}\}\rvert<v_{1}+v_{2}-(s+t), then following the same argument as that leading to A.11, we have A.20 holds.

If G1G_{1} is connected and |{α1,α2}{α1,α2}|=v1+v2(s+t)\lvert\{\alpha_{1},\alpha_{2}\}\cap\{\alpha_{1}^{\prime},\alpha_{2}^{\prime}\}\rvert=v_{1}+v_{2}-(s+t), then either the following two conditions holds: (a) there exists rV2({α1}{α2}V0V1)r\in V_{2}^{\prime}\setminus(\{\alpha_{1}\}\cup\{\alpha_{2}\}\cup V_{0}^{\prime}\cup V_{1}^{\prime}) or (b) V0V0=V_{0}\cap V_{0}^{\prime}=\varnothing. If (a) holds, then following a similar argument as before, we have A.20 holds. Now we consider that the case where (b) holds. Let H1={(p,q):pV0,qV1}H_{1}=\{(p,q):p\in V_{0},q\in V_{1}\} and

1=σ(Yp,q,Yp,q,:𝒜n,2H1).\displaystyle\mathcal{F}_{1}=\sigma(Y_{p,q},Y_{p,q}^{\prime},:\mathcal{A}_{n,2}\setminus H_{1}).

By orthogonality, we have 𝔼{ηα1(B1)(i,j)|1}=0\mathop{{}\mathbb{E}}\mathopen{}\{\eta_{\alpha_{1}(B_{1})}^{(i,j)}|\mathcal{F}_{1}\}=0.

Note that ηα2(B2),ηα1(B1),ηα2(B2)1\eta_{\alpha_{2}(B_{2})},\eta_{\alpha_{1}^{\prime}(B_{1})},\eta_{\alpha_{2}^{\prime}(B_{2})}\in\mathcal{F}_{1}, we have

𝔼{ηα1(B1)(i,j)ηα2(B2)(i,j)}\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\Bigl{\{}\eta_{\alpha_{1}(B_{1})}^{(i,j)}\eta_{\alpha_{2}(B_{2})}^{(i,j)}\Bigr{\}}
=𝔼{ηα2(B2)(i,j)𝔼{ηα1(B1)(i,j)|1}}=0,\displaystyle=\mathop{{}\mathbb{E}}\mathopen{}\Bigl{\{}\eta_{\alpha_{2}(B_{2})}^{(i,j)}\mathop{{}\mathbb{E}}\mathopen{}\Bigl{\{}\eta_{\alpha_{1}(B_{1})}^{(i,j)}\Bigm{|}\mathcal{F}_{1}\Bigr{\}}\Bigr{\}}=0,
Cov{ηα1(B1)(i,j)ηα2(B2)(i,j),ηα1(B1)(i,j)ηα2(B2)(i,j)}\displaystyle\mathop{\mathrm{Cov}}\Bigl{\{}\eta_{\alpha_{1}(B_{1})}^{(i,j)}\eta_{\alpha_{2}(B_{2})}^{(i,j)},\eta_{\alpha_{1}^{\prime}(B_{1})}^{(i^{\prime},j^{\prime})}\eta_{\alpha_{2}^{\prime}(B_{2})}^{(i^{\prime},j^{\prime})}\Bigr{\}}
=𝔼{𝔼{ηα1(B1)(i,j)ηα2(B2)(i,j)ηα1(B1)(i,j)ηα2(B2)(i,j)|1}\displaystyle=\mathop{{}\mathbb{E}}\mathopen{}\Bigl{\{}\mathop{{}\mathbb{E}}\mathopen{}\Bigl{\{}\eta_{\alpha_{1}(B_{1})}^{(i,j)}\eta_{\alpha_{2}(B_{2})}^{(i,j)}\eta_{\alpha_{1}^{\prime}(B_{1})}^{(i^{\prime},j^{\prime})}\eta_{\alpha_{2}^{\prime}(B_{2})}^{(i^{\prime},j^{\prime})}\Bigm{|}\mathcal{F}_{1}\Bigr{\}}
=𝔼{ηα2(B2)(i,j)ηα1(B1)(i,j)ηα2(B2)(i,j)𝔼{ηα1(B1)(i,j)|1}}=0.\displaystyle=\mathop{{}\mathbb{E}}\mathopen{}\Bigl{\{}\eta_{\alpha_{2}(B_{2})}^{(i,j)}\eta_{\alpha_{1}^{\prime}(B_{1})}^{(i^{\prime},j^{\prime})}\eta_{\alpha_{2}^{\prime}(B_{2})}^{(i^{\prime},j^{\prime})}\mathop{{}\mathbb{E}}\mathopen{}\Bigl{\{}\eta_{\alpha_{1}(B_{1})}^{(i,j)}\Bigm{|}\mathcal{F}_{1}\Bigr{\}}\Bigr{\}}=0.

This proves A.20 for the case where |{α1,α2}{α1,α2}|=v1+v2(s+t)\lvert\{\alpha_{1},\alpha_{2}\}\cap\{\alpha_{1}^{\prime},\alpha_{2}^{\prime}\}\rvert=v_{1}+v_{2}-(s+t).

Now, we further assume that γ1=γ2=1\gamma_{1}=\gamma_{2}=1. If G1G_{1} or G2G_{2} is a graph containing one single edge, then the proof is even simpler. Without loss of generality, we now assume that Gm(r)G_{m}^{(r)} is connected for every r[n]r\in[n] for m=1,2m=1,2. We then prove that A.20 holds when |{α1,α2}{α1,α2}|=v1+v2(s+t)+1\lvert\{\alpha_{1},\alpha_{2}\}\cap\{\alpha_{1}^{\prime},\alpha_{2}^{\prime}\}\rvert=v_{1}+v_{2}-(s+t)+1. Under this condition, additional to (a) and (b), there is still another event that may happen: (c) there exists r[n]r\in[n] such that {r}=V0V0\{r\}=V_{0}\cap V_{0}^{\prime}. As the cases (a) and (b) have been discussed, we only need to prove that A.20 holds under (c).

As {i,j}V0\{i,j\}\subset V_{0}, we have s2s\geqslant 2, and V0{r}V_{0}\setminus\{r\} is not empty. Let

2=σ{Yp,q,Yp,q:pV1V2V1V2,qV1V2V1V2{r}}.\displaystyle\mathcal{F}_{2}=\sigma\{Y_{p,q},Y_{p,q}^{\prime}:p\in V_{1}\cup V_{2}\cup V_{1}^{\prime}\cup V_{2}^{\prime},q\in V_{1}\cup V_{2}\cup V_{1}^{\prime}\cup V_{2}^{\prime}\cup\{r\}\}.

Then, conditional on 2\mathcal{F}_{2}, we have ηα1(B1)(i,j)ηα2(B2)(i,j)\eta_{\alpha_{1}(B_{1})}^{(i,j)}\eta_{\alpha_{2}(B_{2})}^{(i,j)} and ηα1(B1)(i,j)ηα2(B2)(i,j)\eta_{\alpha_{1}^{\prime}(B_{1})}^{(i^{\prime},j^{\prime})}\eta_{\alpha_{2}^{\prime}(B_{2})}^{(i^{\prime},j^{\prime})} are conditionally independent. Hence,

Cov{ηα1(B1)(i,j)ηα2(B2)(i,j),ηα1(B1)(i,j)ηα2(B2)(i,j)}\displaystyle\mathop{\mathrm{Cov}}\Bigl{\{}\eta_{\alpha_{1}(B_{1})}^{(i,j)}\eta_{\alpha_{2}(B_{2})}^{(i,j)},\eta_{\alpha_{1}^{\prime}(B_{1})}^{(i^{\prime},j^{\prime})}\eta_{\alpha_{2}^{\prime}(B_{2})}^{(i^{\prime},j^{\prime})}\Bigr{\}}
=Cov{𝔼{ηα1(B1)(i,j)ηα2(B2)(i,j)|2},𝔼{ηα1(B1)(i,j)ηα2(B2)(i,j)|2}}.\displaystyle=\mathop{\mathrm{Cov}}\Bigl{\{}\mathop{{}\mathbb{E}}\mathopen{}\{\eta_{\alpha_{1}(B_{1})}^{(i,j)}\eta_{\alpha_{2}(B_{2})}^{(i,j)}|\mathcal{F}_{2}\},\mathop{{}\mathbb{E}}\mathopen{}\{\eta_{\alpha_{1}^{\prime}(B_{1})}^{(i^{\prime},j^{\prime})}\eta_{\alpha_{2}^{\prime}(B_{2})}^{(i^{\prime},j^{\prime})}|\mathcal{F}_{2}\}\Bigr{\}}.

Letting

3=σ{Yp,q,Yp,q:pV0{r},qV2{r}}.\displaystyle\mathcal{F}_{3}=\sigma\{Y_{p,q},Y_{p,q}:p\in V_{0}\setminus\{r\},q\in V_{2}\cup\{r\}\}.

Now, if G1(r)G_{1}^{(r)} is connected for every r[n]r\in[n], there is at least one edge in G1G_{1} connecting V0{r}V_{0}\setminus\{r\} and V1V_{1}, and thus

𝔼{ηα1(B1)(i,j)ηα2(B2)(i,j)|23}\displaystyle\mathop{{}\mathbb{E}}\mathopen{}\{\eta_{\alpha_{1}(B_{1})}^{(i,j)}\eta_{\alpha_{2}(B_{2})}^{(i,j)}|\mathcal{F}_{2}\vee\mathcal{F}_{3}\} =ηα2(B2)(i,j)𝔼{ηα1(B1)(i,j)|23}=0,\displaystyle=\eta_{\alpha_{2}(B_{2})}^{(i,j)}\mathop{{}\mathbb{E}}\mathopen{}\{\eta_{\alpha_{1}(B_{1})}^{(i,j)}|\mathcal{F}_{2}\vee\mathcal{F}_{3}\}=0,

where the last equality follows from orthogonality. Noting that 23\mathcal{F}_{2}\subset\mathcal{F}_{3}, then 𝔼{ηα1(B1)(i,j)ηα2(B2)(i,j)|2}=0\mathop{{}\mathbb{E}}\mathopen{}\{\eta_{\alpha_{1}(B_{1})}^{(i,j)}\eta_{\alpha_{2}(B_{2})}^{(i,j)}|\mathcal{F}_{2}\}=0 and thus A.20 holds.

Acknowledgements

The research is supported by Singapore Ministry of Education Academic Research Fund MOE 2018-T2-076.

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