Asymptotic Behaviour of Linear eigenvalue statistics of Hankel matrices

Kiran Kumar A.S Department of Mathematics
Indian Institute of Technology Bombay
Powai, Mumbai, Maharashtra 400076, India kiran [at] math.iitb.ac.in and Shambhu Nath Maurya Department of Mathematics
Indian Institute of Technology Bombay
Powai, Mumbai, Maharashtra 400076, India snmaurya [at] math.iitb.ac.in

Abstract.

We study linear eigenvalue statistics of band Hankel matrices with Brownian motion entries. We prove that, the centred, normalized linear eigenvalue statistics of band Hankel matrices obey a central limit theorem (CLT) type result. We also discuss the convergence of linear eigenvalue statistics of band Hankel matrices with independent entries for odd power monomials. Our method is based on trace formula, moment method and some results of process convergence.

The research of Kiran Kumar A.S. is supported by IIT Bombay and the work of Shambhu Nath Maurya is partially supported by UGC Doctoral Fellowship, India.

Keywords : Linear eigenvalue statistics, band Hankel matrix, Central limit theorem, Brownian motion, time dependent fluctuations.

1. Introduction

For an $n\times n$ matrix $A_{n}$ , the linear eigenvalue statistics is defined as

\mathcal{A}_{n}(\phi)=\sum_{i=1}^{n}\phi(\lambda_{i}),

(1)

where $\lambda_{1},\lambda_{2},\ldots,\lambda_{n}$ are the eigenvalues of $A_{n}$ and $\phi$ is a ‘nice’ test function. The study of linear eigenvalue statistics is a major area of interest in random matrix theory. For an overview of the fluctuations of linear eigenvalue statistics for various types of random matrices, an interested reader can look into the following papers, [4], [10], [11], [1], [15], [16], [6].

Hankel and the closely related Toeplitz are two important classes of random matrices. In particular, Hankel matrices show up prominently in studies of Padé approximation, moment problems and orthogonal polynomials (see[17], [2]). In [3], Bai proposed the study of limiting spectral distributions (LSD) of Hankel and the closely related Toeplitz matrix. The existence of LSD for Hankel matrices was shown independently by Hammond and Miller [8] and Bryc et al. [7]. Later with certain band conditions, Basak, Bose [5] and Liu, Wang [12] independently found the limit spectral distribution of band Hankel matrices. In 2012, it was proved by Liu et al. that the linear eigenvalue statistics obey a central limit theorem type of result for $\phi(x)=x^{2k}$ for $k\in\mathbb{N}$ .

The fluctuation problem we are interested to consider in this article, is inspired by the results on band Toeplitz matrices by Li and Sun [11]. The authors studied time dependent fluctuations of linear eigenvalue statistics for band Toeplitz matrices with standard Brownian motion entries. We follow the definition of Hankel matrix adopted in [13].

Consider an input sequence $\{a_{n}(t);t\geq 0\}_{n\geq 1}$ of independent standard Brownian motions along with $a_{0}(t)\equiv 0$ . For a sequence $\{b_{n}\}$ , with $b_{n}\rightarrow\infty$ and $b_{n}/n\rightarrow b$ , we define a band Hankel matrix $H_{n}(t)$ as

H_{n}=P_{n}T_{n}(t)\mbox{ and }A_{n}(t):=\frac{1}{\sqrt{b_{n}}}H_{n}(t),

(2)

where $T_{n}=(a_{i-j}(t)\delta_{|i-j|\leq b_{n}})$ is the band Toeplitz matrix and $P_{n}=(\delta_{i-1,n-j})_{i,j=1}^{n}$ is the backward identity permutation.

Now for each $p\in\mathbb{N}$ , define

w_{p}(t):=\frac{\sqrt{b_{n}}}{n}\big{(}{\mbox{Tr}}(A_{n}(t))^{p}-\mbox{E}{\mbox{Tr}}(A_{n}(t))^{p}\big{)}.

(3)

Observe that ${\mbox{Tr}}(A_{n}(t))^{p}$ is a linear eigenvalue statistics of $A_{n}(t)$ as defined in (1) with test function $\phi(x)=x^{p}$ . Here, $n$ is suppressed in the notation of $w_{p}(t)$ to keep the notation simple.

The following theorem describes the process convergence of $\{w_{p}(t);t\geq 0\}$ for even integer $p\geq 2$ , inspired from the results in [16], [6].

Theorem 1.

Suppose $p\geq 2$ is an even positive integer and $b_{n}=o(n)$ . Then as $n\to\infty$

\{w_{p}(t);t\geq 0\}\stackrel{{\scriptstyle\mathcal{D}}}{{\rightarrow}}\{W_{p}(t);t\geq 0\},

where $\{W_{p}(t);t\geq 0,p\geq 2\}$ are Gaussian with mean zero and the following covariance structure:

\mbox{\rm Cov}[W_{p}(t_{1}),W_{q}(t_{2})]=2^{\frac{p+q}{2}}\sum_{r=2,4,\ldots,q}{q\choose r}t_{1}^{\frac{p+r}{2}}\left(t_{2}-t_{1}\right)^{\frac{q-r}{2}}R(p,r)\Gamma\left(\frac{q-r}{2}+1\right),

(4)

with $R(p,r)$ as in (9), $\Gamma(\cdot)$ denotes the Gamma function and $\stackrel{{\scriptstyle\mathcal{D}}}{{\rightarrow}}$ denotes the weak convergence as in Definition 5.

In Theorem 1, we have considered $a_{0}(t)\equiv 0$ . A generalization of Theorem 1 for standard Brownian motion $\{a_{0}(t);t\geq 0\}$ is discussed in Section 4 after the proof of Theorem 1. The process convergence of $w_{p}(t)$ for odd positive integer $p$ is given in Remark 21.

Now we collect an existing result on fluctuations of linear eigenvalue statistics of band Hankel matrices. We first introduce some notations to state the result.

We consider an input sequence $\{x_{i}:i\geq 1\}$ of independent real random variables which satisfy the following moment assumptions:

\mbox{E}[x_{i}]=0,\quad\mbox{E}[x_{i}^{2}]=1,\ \kappa=\mbox{E}[x_{j}^{4}]\,\,\forall\ i\quad\text{and}\quad\sup_{j\geq 1}\mbox{E}\left[\left|x_{j}\right|^{k}\right]=\alpha_{k}<\infty\quad\text{ for }\quad k\geq 3.

(5)

along with $x_{0}\equiv 0$ . For a sequence $b_{n}$ , with $b_{n}\rightarrow\infty$ and $b_{n}/n\rightarrow b$ , we define the band Hankel matrix $H_{n}$ as $H_{n}=P_{n}T_{n}$ where $T_{n}=(\delta_{|i-j|\leq b_{n}}x_{i-j})$ . Furthermore, we define $A_{n}=H_{n}/\sqrt{b_{n}}$ and for each $p\in\mathbb{N},w_{p}$ as

w_{p}:=\frac{\sqrt{b_{n}}}{n}\bigl{\{}{\mbox{Tr}}(A_{n})^{p}-\mbox{E}[{\mbox{Tr}}(A_{n})^{p}]\bigr{\}}.

(6)

In [13], Liu et al. proved the following result on fluctuations of $w_{p}$ for even values of $p$ .

Result 2.

(Theorem 6.4, [13]) Let $H_{n}$ be a real symmetric random Hankel band matrix whose input sequence satisfies (5), and $b_{n}/n\rightarrow b\in[0,1]$ with $b_{n}\rightarrow\infty$ as $n\rightarrow\infty$ .

Then for every even integer $p\geq 2$ , as $n\rightarrow\infty$

\displaystyle w_{p}\stackrel{{\scriptstyle d}}{{\rightarrow}}N(0,\sigma_{p}^{2}),

where $w_{p}$ as defined in (6). Here $\sigma_{p}^{2}$ is appropriate constant, see [13, Theorem 6.4].

The following theorem provide the fluctuations of $w_{p}$ for odd value of $p$ .

Theorem 3.

Suppose $H_{n}$ is a band Hankel matrix with an input sequence $\{x_{i}\}_{i\geq 1}$ of independent real random variables which satisfies (5) and $\{b_{n}\}$ as in Result 2. Then, for odd $p\geq 1$ , as $n\rightarrow\infty$

\displaystyle w_{p}\stackrel{{\scriptstyle d}}{{\rightarrow}}0.

Now we briefly outline the rest of the paper. In Section 2, first we state trace formula of Hankel matrices and then some basic results of process convergence. In Section 3, we introduce some standard partitions and integrals. In Section 4, we prove Theorem 1 by using moment method and some results on process convergence. Finally, we prove Theorem 3 in Section 5.

2. Preliminaries

Here we introduce some results and definitions which will be used in the proof of theorems. First we state the trace formula for band Hankel matrices.

Result 4 (Lemma 6.3, [13]).

Suppose $H_{n}$ is a band Hankel matrix with an input sequence $\{x_{i}\}_{i\geq 1}$ . Then,

\displaystyle{\mbox{Tr}}(H_{n})^{p}=\begin{cases}\displaystyle\sum_{i=1}^{n}\sum_{j_{1},\ldots,j_{p}=-b_{n}}^{b_{n}}\prod_{\ell=1}^{p}x_{j_{\ell}}\prod_{\ell=1}^{p}\chi_{[1,n]}\left(i-\sum_{q=1}^{\ell}(-1)^{q}j_{q}\right)\delta_{0,\sum_{q=1}^{p}(-1)^{q}j_{q}},&p\quad\text{even};\\ \displaystyle\sum_{i=1}^{n}\sum_{j_{1},\ldots,j_{p}=-b_{n}}^{b_{n}}\prod_{\ell=1}^{p}x_{j_{\ell}}\prod_{\ell=1}^{p}\chi_{[1,n]}\left(i-\sum_{q=1}^{l}(-1)^{q}j_{q}\right)\delta_{2i-1-n,\sum_{q=1}^{p}(-1)^{q}j_{q}},&p\quad\text{odd}.\end{cases}

Now we see some standard results on process convergence.

Let $(C_{\infty},\mathcal{C_{\infty}})$ be a probability measure space, where $C_{\infty}:=C[0,\infty)$ is the space of all real-valued continuous functions on $[0,\infty)$ and $\mathcal{C_{\infty}}$ is a $\sigma$ -field generated by open sets of $C_{\infty}$ . For more details about $(C_{\infty},\mathcal{C_{\infty}})$ and $\sigma$ -field, see [16].

Definition 5.

Suppose $\{\bm{X_{n}}\}_{n\geq 1}$ = $\{X_{n}(t);t\geq 0\}_{n\geq 1}$ and $\bm{X}=\{X(t);t\geq 0\}$ are real-valued continuous process on $C[0,\infty)$ . We say $\bm{X_{n}}$ converge to $\bm{X}$ weakly or in distribution, denote by $\bm{X_{n}}\stackrel{{\scriptstyle\mathcal{D}}}{{\rightarrow}}\bm{X}$ if $\mbox{P}_{n}\stackrel{{\scriptstyle\mathcal{D}}}{{\rightarrow}}\mbox{P}$ , that is,

\mbox{P}_{n}f:=\int_{C_{\infty}}fd\mbox{P}_{n}\longrightarrow\mbox{P}f:=\int_{C_{\infty}}fd\mbox{P}\ \mbox{ as }n\to\infty,

for every bounded, continuous real-valued function $f$ on $C_{\infty}$ , where $\mbox{P}_{n}$ and P are the probability measures on $(C_{\infty},\mathcal{C_{\infty}})$ induced by $\bm{X_{n}}$ and $\bm{X}$ , respectively.

Suppose $\{\bm{X_{n}}\}_{n\geq 1}$ = $\{X_{n}(t);t\geq 0\}_{n\geq 1}$ and $\bm{X}$ = $\{X(t);t\geq 0\}$ are sequences of continuous processes on $C[0,\infty)$ . Then $\bm{X_{n}}\stackrel{{\scriptstyle\mathcal{D}}}{{\rightarrow}}\bm{X}$ if and only if:
(i) Finite dimensional convergence: Suppose $0<t_{1}<t_{2}\ldots<t_{r}$ . Then as $n\rightarrow\infty$

(X_{n}(t_{1}),X_{n}(t_{2}),\ldots,X_{n}(t_{r}))\stackrel{{\scriptstyle\mathcal{D}}}{{\rightarrow}}(X(t_{1}),X(t_{2}),\ldots,X(t_{r})),

(ii) Tightness: the sequence $\{\bm{X_{n}}\}_{n\geq 1}$ is tight.

Take $X_{n}(t)$ = $w_{p}(t)$ where $w_{p}(t)$ is as defined in (3). From the above discussion and [9, Theorem I.4.3], it is clear that Theorem 1 follows from the following two propositions:

Proposition 6.

For each $p\geq 2$ , suppose $0<t_{1}<t_{2}\ldots<t_{r}$ . Then as $n\rightarrow\infty$

(w_{p}(t_{1}),w_{p}(t_{2}),\ldots,w_{p}(t_{r}))\stackrel{{\scriptstyle\mathcal{D}}}{{\rightarrow}}(W_{p}(t_{1}),W_{p}(t_{2}),\ldots,W_{p}(t_{r})).

Proposition 7.

For each $p\geq 2$ , there exists positive constants $M$ and $\gamma$ such that

\mbox{E}{|w_{p}(0)|^{\gamma}}\leq M\ \ \ \forall\ n\in\mathbb{N},

and there exists positive constants $\alpha,\ \beta$ and $M_{T}$ , $T=1,2,\ldots,$ such that

\mbox{E}{|w_{p}(t)-w_{p}(s)|^{\alpha}}\leq M_{T}|t-s|^{1+\beta}\ \ \ \forall\ n\in\mathbb{N}\ \mbox{and }t,s\in[0,\ T],(T=1,2,\ldots).

Then $\{w_{p}(t);t\geq 0\}$ is tight.

3. Some partitions and integrals

In this section, we introduce some partitions and integrals which will be required for the proof of Theorem 1.

Definition 8.

Let $J_{1}\in\mathbb{R}^{p},J_{2}\in\mathbb{R}^{q}$ , with $J_{1}=(j_{1},\ldots,j_{p})$ and $J_{2}=(j_{p+1},\ldots,j_{p+q})$ . Consider two components $j_{k_{1}}$ and $j_{k_{2}}$ with $j_{k_{1}}=j_{k_{2}}$ . Then

(i)

$j_{k_{1}}$ and $j_{k_{2}}$ are said to be self-matched in $J_{1}$ if $1\leq k_{1},k_{2}\leq p$ .
(ii)

$j_{k_{1}}$ and $j_{k_{2}}$ are said to be self-matched in $J_{2}$ if $p+1\leq k_{1},k_{2}\leq p+q$ .
(iii)

$j_{k_{1}}$ and $j_{k_{2}}$ are said to be cross-matched if $1\leq k_{1}\leq p<k_{2}\leq p+q$ or $1\leq k_{2}\leq p<k_{1}\leq p+q$ .

Definition 9.

Consider the set $[n]=\{1,2,\ldots,n\}$ . A partition $\pi$ of $[n]$ is called a pair-partition if each equivalency class of $\pi$ has exactly two elements. If $i,j$ belongs to the same equivalency class of $\pi$ , we write $i\sim_{\pi}j$ . The set of all pair-partitions of $[n]$ is denoted by ${\mathcal{P}}_{2}(n)$ . Clearly, ${\mathcal{P}}_{2}(n)=\emptyset$ for odd n.

Definition 10.

Consider an even number $n$ . A pair-partition $\pi\in{\mathcal{P}}_{2}(n)$ is called an odd-even pair partition if for all $i,j$ such that $i\sim_{\pi}j$ and $i\neq j$ , one of $\{i,j\}$ is odd and the other is even. The set of all odd-even pair partitions of $[n]$ is denoted by $\Delta_{2}(n)$ . For odd $n$ , $\Delta_{2}(n)=\emptyset$ .

Definition 11.

Consider $p,q\in\mathbb{N}$ such that $p+q$ is even. We define the set $\Delta_{2}(p,q)\subset{\mathcal{P}}_{2}(p+q)$ in the following way:

(i)

Any $\pi\in\Delta_{2}(p+q)$ is an odd-even pair partition.
(ii)

For every $\pi\in\Delta_{2}(p+q),$ there exists $i,j$ such that $i\leq p<j$ , such that $i\sim_{\pi}j$ .

For $p,q$ such that $p+q$ is odd, $\Delta_{2}(p+q)$ is taken to be the empty set.

Consider, for fixed natural numbers $p$ and $q$ , $J=(j_{1},j_{2},\ldots,j_{p+q})\in\mathbb{R}^{p+q}$ . Then $J$ can be thought of as a 2-tuple $J=(J_{1},J_{2})$ where $J_{1}=(j_{1},j_{2},\ldots,j_{p})$ and $J_{2}=(j_{p+1},j_{p+2},\ldots,j_{p+q})$ . Thus, taking cue from Definition 8, we can introduce the terms self-matched and cross-matched for any vector $J$ . A closer observation yields that the concept of self-matching (and cross-matching) depends only on the relationship between the components and not on the numerical value of the pair. Therefore the concept has a natural extension to any partition $\pi$ of $[p+q]$ .

Definition 12.

Consider ${\mathcal{P}}_{2}(p,q)$ , the subset of ${\mathcal{P}}_{2}(p+q)$ of elements having at least one cross-matched pair. Consider the set of all $\pi\in{\mathcal{P}}_{2}(p,q)$ such that

(i)

every self-matched pair of $\pi$ is an odd-even pair,
(ii)

every cross-matched pair of $\pi$ is either an even-even pair or an odd-odd pair.

The set of all such $\pi$ is denoted by $\tilde{\Delta}_{2}(p,q)$ .

We also define the following important partition of $[p+q]$ for natural numbers $p$ and $q$ .

Definition 13.

Let $\pi=\{V_{1}\ldots,V_{r}\}$ be a partition of $[p+q]$ . We say $\pi\in\Delta_{2,4}(p,q)$ if the following conditions are satisfied:

(i)

there exists such that $|V_{i}|=4$ and $|V_{j}|=2$ for all $j\neq i$ ,
(ii)

for each $j\neq i$ , one element of $V_{j}$ is odd and the other is even,
(iii)

for each $j\neq i$ , $V_{j}\subset\{1,\ldots,p\}$ or $V_{j}\subset\{p+1,\ldots,p+q\}$ ,
(iv)

two elements of $V_{i}$ come from $\{1,\ldots,p\}$ and they form an odd-even pair. The other two elements coming from $\{p+1,\ldots,p+q\}$ also forms an odd-even pair.

For a partition $\pi$ and a set of unknowns $\{x_{i}\}$ , we can construct a new set of unknowns $\{y_{i}\}$ by defining $y_{i}=x_{\pi(i)}$ . Now we define some integrals associated with the partition.

Definition 14.

(i)

For $\pi\in\Delta_{2}(p,q)$ , define

	$\displaystyle f_{I}^{-}\left(\pi\right)=$	$\displaystyle\int_{[0,1]^{2}}dy_{0}dx_{0}\int_{\left[-1,1\right]^{\frac{p+q-2}{2}}}dx_{1}dx_{2}\cdots dx_{(p+q)/2}$
		$\displaystyle\times\delta\left(\sum_{i=1}^{p}(-1)^{i}y_{i}\right)\prod_{j=1}^{p}I_{[0,1]}\left(x_{0}+b\sum_{i=1}^{j}(-1)^{i}y_{i}\right)\prod_{j^{\prime}=p+1}^{p+q}I_{[0,1]}\left(y_{0}+b\sum_{i=p+1}^{j^{\prime}}(-1)^{i}y_{i}\right).$

(ii)

For $\pi\in\tilde{\Delta}_{2}(p,q)$ , define

	$\displaystyle f_{I}^{+}\left(\pi\right)=$	$\displaystyle\int_{[0,1]^{2}}dy_{0}dx_{0}\int_{\left[-1,1\right]^{\frac{p+q-2}{2}}}dx_{1}dx_{2}\cdots dx_{(p+q)/2}$
		$\displaystyle\times\delta\left(\sum_{i=1}^{p}(-1)^{i}y_{i}\right)\prod_{j=1}^{p}I_{[0,1]}\left(x_{0}+b\sum_{i=1}^{j}(-1)^{i}y_{i}\right)\prod_{j^{\prime}=p+1}^{p+q}I_{[0,1]}\left(y_{0}-b\sum_{i=p+1}^{j^{\prime}}(-1)^{i}y_{i}\right).$

(iii)

For $\pi\in\Delta_{2}(p,q)$ , define

	$\displaystyle f_{II}^{-}\left(\pi\right)=$	$\displaystyle\int_{[0,1]^{2}}dy_{0}dx_{0}\int_{\left[-1,1\right]^{\frac{p+q-2}{2}}}dx_{1}\cdots dx_{\frac{p+q}{2}-1}$
		$\displaystyle\times\prod_{j=1}^{p}I_{[0,1]}\left(x_{0}+b\sum_{i=1}^{j}(-1)^{i}y_{i}\right)\prod_{j^{\prime}=p+1}^{p+q}I_{[0,1]}\left(y_{0}+b\sum_{i=p+1}^{j^{\prime}}(-1)^{i}y_{i}\right),$

	$\displaystyle f_{II}^{+}(\pi)=$	$\displaystyle\int_{[0,1]^{2}}dy_{0}dx_{0}\int_{\left[-1,1\right]^{p/2}}dx_{1}\cdots dx_{\frac{p+q}{2}-1}$
		$\displaystyle\times\prod_{j=1}^{p}I_{[0,1]}\left(x_{0}+b\sum_{i=1}^{j}(-1)^{i}y_{i}\right)\prod_{j^{\prime}=p+1}^{p+q}I_{[0,1]}\left(y_{0}-b\sum_{i=p+1}^{j^{\prime}}(-1)^{i}y_{i}\right).$

4. Proof of Theorem 1

First recall that Propositions 6 and 7 yield Theorem 1. In Subsection 4.1, we prove Proposition 6 and in 4.2, we prove Proposition 7. Throughout this section we assume that $b_{n}=o(n)$ , which gives $b=\lim_{n\rightarrow\infty}b_{n}/n=0$ .

4.1. Proof of Proposition 6

We first start with the following two lemmata which will used in the proof of Proposition 6.

Lemma 15.

Suppose $0<t_{1}\leq t_{2}$ , $p$ and $q$ are even natural numbers. For $j\in\mathbb{Z}$ , let $u_{j}=a_{j}(t_{1})$ and $v_{j}=a_{j}(t_{2})-a_{j}(t_{1})$ . Then

	$\displaystyle\displaystyle\mbox{E}\left[w_{p}\left(t_{1}\right)w_{q}\left(t_{2}\right)\right]$	$\displaystyle=b_{n}^{-\frac{p+q}{2}+1}\sum_{r=0}^{q}{q\choose r}\sum_{J=(j_{1},\ldots,j_{p})\in A_{p},\atop J^{\prime}=(j_{1}^{\prime},\ldots,j_{q}^{\prime})\in A_{q}}\left(\mbox{E}[u_{j_{1}}\cdots u_{j_{p}}u_{j_{i}^{\prime}}\cdots u_{j_{r}^{\prime}}v_{j_{r+1}^{\prime}}\cdots v_{j_{q}^{\prime}}]\right.$
		$\displaystyle\qquad\displaystyle\left.-\mbox{E}[u_{j_{1}}\cdots u_{j_{p}}]\mbox{E}[u_{j_{i}^{\prime}}\cdots u_{j_{r}^{\prime}}v_{j_{r+1}^{\prime}}\cdots v_{j_{q}^{\prime}}]\right)+o(1),$		(7)

where for each $p\in\mathbb{N},$ $A_{p}$ is defined as

A_{p}=\left\{\left(j_{1},\ldots,j_{p}\right)\in\left\{\pm 1,\ldots,\pm b_{n}\right\}^{p}:\sum_{k=1}^{p}(-1)^{k}j_{k}=0\right\}.

(8)

Proof.

The proof is similar to the proof of Lemma $4.1$ of [11]. So, we skip the details. ∎

Lemma 16.

If $0<t_{1}\leq t_{2}$ and $p,q\geq 2$ be even natural numbers, then

\lim_{n\to\infty}\mbox{E}[w_{p}(t_{1})w_{q}(t_{2})]=2^{\frac{p+q}{2}}\sum_{r=2,4,\ldots,q}{q\choose r}t_{1}^{\frac{p+r}{2}}\left(t_{2}-t_{1}\right)^{\frac{q-r}{2}}R(p,r)\Gamma\left(\frac{q-r}{2}+1\right),

(9)

where $R(p,r)=\left|\Delta_{2}(p,q)\right|+\left|\tilde{\Delta}_{2}(p,r)\right|+2\left|\Delta_{2,4}(p,r)\right|$ and $\Gamma$ denotes the Gamma function.

Proof.

First we define $U:=A_{n}(t_{1})$ and $V:=A_{n}(t_{2})-A_{n}(t_{1})$ . Further for $0\leq r\leq q$ , we define

	$\displaystyle J=$	$\displaystyle(j_{1},j_{2},\ldots j_{p}),\quad J^{\prime}=(j_{1}^{\prime},j_{2}^{\prime},\ldots,j_{q}^{\prime}),\quad J_{1}^{\prime}=\left(j_{1}^{\prime},\ldots,j_{r}^{\prime}\right),\quad J_{2}^{\prime}=\left(j_{r+1}^{\prime},\ldots,j_{q}^{\prime}\right),$		(10)
	$\displaystyle u_{\mathrm{J}}=$	$\displaystyle u_{j_{1}}\cdots u_{j_{p}},\quad u_{\mathrm{J}_{1}^{\prime}}=u_{j_{1}^{\prime}}\cdots u_{j_{r}^{\prime}}\ \mbox{ and }\ v_{\mathrm{J}_{2}^{\prime}}=v_{j_{r+1}^{\prime}}\cdots v_{j_{q}^{\prime}}.$

Now with the above notations, (15) can be written as

\displaystyle\mbox{E}\left[w_{p}\left(t_{1}\right)w_{q}\left(t_{2}\right)\right]

\displaystyle=b_{n}^{-\frac{p+q}{2}+1}\sum_{r=0}^{q}{q\choose r}\sum_{J,J^{\prime}}\Big{\{}\left(\mbox{E}\left[u_{J}u_{J_{1}^{\prime}}\right]-\mbox{E}\left[u_{J}\right]\mbox{E}\left[u_{J_{1}^{\prime}}\right]\right)\mbox{E}\left[v_{J_{2}^{\prime}}\right]\Big{\}}+o(1),

(11)

where $J\in A_{p}$ and $J^{\prime}\in A_{q}$ . Now for a fixed $r$ , $0\leq r\leq q$ , consider

b_{n}^{-\frac{p+q}{2}+1}\sum_{J,J^{\prime}}\Big{\{}\left(\mbox{E}\left[u_{J}u_{J_{1}^{\prime}}\right]-\mbox{E}\left[u_{J}\right]\mbox{E}\left[u_{J_{1}^{\prime}}\right]\right)\mbox{E}\left[v_{J_{2}^{\prime}}\right]\Big{\}}.

(12)

Consider the term $\left(\mbox{E}[u_{J}u_{J_{1}^{\prime}}]-\mbox{E}[u_{J}]\mbox{E}[u_{J_{1}^{\prime}}]\right)\mbox{E}[v_{J_{2}^{\prime}}]$ . Due to the independence of random variables $\{u_{i}\}$ and $\{v_{i}\}$ , this term is non-zero only when the following three conditions are satisfied:

(a)

there exists at least one common element between $S_{J}$ and $S_{J_{1}^{\prime}}$ , that is, $S_{J}\cap S_{J_{1}^{\prime}}\neq\emptyset$ ,
(b)

every element of $S_{J}\cup S_{J_{1}^{\prime}}$ has cardinality at least 2, and
(c)

every element of $S_{J_{2}^{\prime}}$ has cardinality at least 2,

where for a vector $J=\left(j_{1},j_{2},\ldots,j_{p}\right)\in A_{p},$ the multi-set $S_{J}$ is defined as

S_{J}=\left\{j_{1},j_{2},\ldots,j_{p}\right\}.

(13)

Now consider the map $L:\{\pm 1,\pm 2,\ldots\pm b_{n}\}^{p}\times\{\pm 1,\pm 2,\ldots\pm b_{n}\}^{q}\rightarrow\{\pm 1,\pm 2,\ldots\pm b_{n}\}^{p+q-2}$ defined in the following way: Let $j_{u}\in J$ be the first cross-matched element. If $(-1)^{u}j_{u}=(-1)^{v}j_{v}^{\prime}$ for some $v$ , define $L\in\{\pm 1,\pm 2,\ldots\pm b_{n}\}^{p+q-2}$ by

	$\displaystyle l_{1}$	$\displaystyle=j_{1},\ldots,l_{u-1}=j_{u-1},l_{u}=j_{1}^{\prime},\ldots,l_{u+v-2}=-j_{v-1}^{\prime},$
	$\displaystyle l_{u+v-1}$	$\displaystyle=-j_{v+1}^{\prime},\ldots,l_{u+q-2}=-j_{q}^{\prime},l_{u+q-1}=j_{u+1},\ldots,l_{p+q-2}=j_{p}.$

And if for the first cross-matched element $j_{u}$ , $(-1)^{u}j_{u}=-(-1)^{v}j_{v}^{\prime}$ , then construct $L$ by starting from $(J,-J^{\prime})$ . This process of constructing $L$ from $(J,J^{\prime})$ is called a reduction step and is denoted by $L=J\bigvee_{j_{u}}J^{\prime}$ .

Notice that if $J\in A_{p}$ and $J^{\prime}\in A_{q}$ , then $L\in A_{p+q-2}$ . Furthermore, for every $L\in A_{p+q-2}$ , we can find at most $pq$ pairs $(J,J^{\prime})\in A_{p}\times A_{q}$ such that $L=J\bigvee_{j_{u}}J^{\prime}$ . This is because when $p-1$ elements in $J$ (or similarly $q-1$ elements in $J^{\prime}$ ) are fixed, the value of the left out element becomes automatically fixed.

Claim A.

Non-zero contribution for (11) are due to those $L$ such that each element of $S_{L}$ have multiplicity 2.

Consider $L\in A_{p+q-2}$ obtained from $(J,J^{\prime})$ satisfying the above conditions (a) - (c). Notice that for any such $L$ , there can exist at most one element in $S_{L}$ of multiplicity 1. Since $L\in A_{p+q-2}$ , the element of multiplicity 1 is determined by other elements of $L$ . Furthermore, this case is only possible when the cross-matched element has multiplicity 3 in $S_{J}\cup S_{J^{\prime}}$ . Thus the number of all $(J,J^{\prime})$ which has such $L$ as its image is $O(b_{n}^{\frac{p+q-3}{2}})$ which implies that the contribution of such terms to $\mbox{E}[w_{p}w_{q}]$ is $O(b_{n}^{-\frac{1}{2}})$ . It is also easy to see that the sum of all terms $(J,J^{\prime})$ such that an $S_{L}$ has an element of multiplicity 3, leads to zero contribution in (12). Hence the claim.

The claim along with constraints (b) and (c) implies that each element in $S_{J}\cup S_{J_{1}^{\prime}}$ and $S_{J_{2}^{\prime}}$ are of cardinality 2 and that $S_{J_{1}^{\prime}}$ and $S_{J_{2}^{\prime}}$ have no elements in common. Furthermore, the contribution is maximum when $J\in A_{p},J_{1}^{\prime}\in A_{r}$ and $J_{2}\in A_{q-r}$ . Thus (12) becomes

\noindent\left(b_{n}^{-\frac{p+r}{2}+1}\sum_{J,\mathrm{J}_{1}^{\prime}}\left(\mbox{E}\left[u_{{J}}u_{{J}_{1}^{\prime}}\right]-\mbox{E}\left[u_{{J}}\right]\mbox{E}\left[u_{{J}_{1}^{\prime}}\right]\right)\right)\left(b_{n}^{-\frac{q-r}{2}}\sum_{{J}_{2}^{\prime}}\mbox{E}\left[v_{{J}_{2}^{\prime}}\right]\right)+o(1),

where $J\in A_{p}$ , $J_{1}^{\prime}\in A_{r}$ and $J_{2}^{\prime}\in A_{q-r}$ . Notice that when $r$ is odd, the pair partition mentioned above cannot happen. Hence non-zero contribution occurs only for even $r$ .

Claim A also implies that it is sufficient to fix a particular $\pi\in{\mathcal{P}}_{2}(p+q-2)$ and count the number of $L\in A_{p+q-2}$ corresponding to $\pi$ . Since each element in $S_{L}$ is repeated exactly twice, the number of $L$ that correspond to a particular $\pi$ is of order at most $O(b_{n}^{\frac{p+q-2}{2}})$ . Now since the number of pre-images of a particular $L$ is at most $pq$ , only those $\pi$ from which $O(b_{n}^{\frac{p+q-2}{2}})$ many $L^{\prime}$ s can be generated contribute to $\lim_{n\rightarrow\infty}\mbox{E}[w_{p}w_{q}]$ , which are exactly those $\pi\in\Delta_{2}(p+q-2)$ .

Thus we obtain for $r$ even,

		$\displaystyle\lim_{n\rightarrow\infty}b_{n}^{-\frac{p+r}{2}+1}\sum_{{J},{J}_{1}^{\prime}}\left(\mbox{E}\left[u_{{J}}u_{{J}_{1}^{\prime}}\right]-\mbox{E}\left[u_{{J}}\right]\mbox{E}\left[u_{{J}_{1}^{\prime}}\right]\right)$
		$\displaystyle=\left(\mbox{E}\left[u_{i}^{2}\right]\right)^{\frac{p+r}{2}}\left[\sum_{\pi\in\Delta_{2}(p,r)}f_{I}^{-}(\pi)+\sum_{\pi\in\tilde{\Delta}_{2}(p,r)}f_{I}^{+}(\pi)\right]$
		$\displaystyle\qquad+\left(\left(\mbox{E}\left[u_{i}^{2}\right]\right)^{\frac{p+r-4}{2}}\mbox{E}\left[u_{i}^{4}\right]-\left(\mbox{E}\left[u_{i}^{2}\right]\right)^{\frac{p+r}{2}}\right)\times\left[\sum_{\pi\in{\Delta_{2,4}}(p,r)}f_{II}^{-}(\pi)+\sum_{\pi\in{\Delta}_{2,4}(p,r)}f_{II}^{+}(\pi)\right],$		(14)

where $f_{I}^{-}(\pi),f_{I}^{+}(\pi),f_{II}^{-}(\pi)$ and $f_{II}^{+}(\pi)$ are as in Definition 14. Since we have $a_{i}(t)$ to be Brownian motion entries, $\mbox{E}[u_{i}^{2}]=t_{1}$ and $\mbox{E}[u_{i}^{4}]=3t_{1}^{2}$ for all $i\neq 0$ . Since $b=0$ as $b_{n}=o(n)$ , we get

f_{I}^{-}(\pi)=f_{I}^{+}(\pi)=f_{II}^{-}(\pi)=f_{II}^{+}(\pi)=2^{\frac{p+r}{2}-1}.

Therefore (4.1) will be

\displaystyle\begin{cases}2^{\frac{p+r}{2}}t_{1}^{\frac{p+r}{2}}\left(\left|\Delta_{2}(p,q)\right|+\left|\tilde{\Delta}_{2}(p,r)\right|+2\left|\Delta_{2,4}(p,r)\right|\right)&\text{when}\hskip 5.69054ptr\hskip 5.69054pt\text{is even,}\\ 0&\text{otherwise}.\end{cases}

(15)

Now we proceed to find $b_{n}^{-\frac{q-r}{2}}\sum_{J_{2}^{\prime}}\mbox{E}\left[v_{J_{2}^{\prime}}\right]$ . For that consider a random band Hankel matrix $H_{n}$ with input variables as $\frac{v_{i-j}}{\sqrt{t_{2}-t_{1}}}$ and scaling as $\frac{1}{\sqrt{b_{n}}}$ . Then

\displaystyle\frac{1}{n}\mbox{E}\left[{\mbox{Tr}}H_{n}^{q-r}\right]=\frac{1}{n\left(2b_{n}\right)^{\frac{q-r}{2}}}\sum_{i=1}^{n}\sum_{j_{1},\ldots,j_{q-r}=-b_{n}}^{b_{n}}\mbox{E}\left[w_{j_{1}}\cdots w_{j_{q-r}}\right]\prod_{s=1}^{q-r}I_{[1,n]}\left(i-\sum_{z=1}^{s}(-1)^{z}j_{z}\right)\delta\left(\sum_{z=1}^{q-r}(-1)^{z}j_{z}\right).

Since $b_{n}=o(n)$ , it is easy to see that $\prod_{s=1}^{q-r}I_{[1,n]}\left(i-\sum_{z=1}^{s}(-1)^{z}j_{z}\right)$ converges to 1 as $n$ tends to infinity, for all choices of $j_{1},j_{2},\ldots,j_{q-r}$ . Therefore using dominated convergence theorem, we get

	$\displaystyle\lim_{n\rightarrow\infty}\frac{1}{n}\mbox{E}\left[{\mbox{Tr}}H_{n}^{q-r}\right]$	$\displaystyle=\lim_{n\rightarrow\infty}\frac{1}{\left(2b_{n}\right)^{\frac{q-r}{2}}}\sum_{j_{1},\ldots,j_{q-r}=-b_{n}}^{b_{n}}\mbox{E}\left[w_{j_{1}}\cdots w_{j_{q-r}}\right]\delta\left(\sum_{z=1}^{q-r}(-1)^{z}j_{z}\right)+o(1),$
		$\displaystyle=\lim_{n\rightarrow\infty}\frac{1}{\left[2\left(t_{2}-t_{1}\right)\right]^{\frac{q-r}{2}}}\frac{1}{\left(b_{n}\right)^{\frac{q-r}{2}}}\sum_{J_{2}^{\prime}}\mbox{E}\left[v_{J_{2}^{\prime}}\right]+o(1),$

where $J_{2}^{\prime},v_{J_{2}^{\prime}}$ are as given in (10) and $J_{2}^{\prime}\in A_{q-r}$ . By [14, Theorem 3.2], the empirical measure of $H_{n}$ converge to the distribution given by $f(x)=|x|\exp\left(-x^{2}\right)$ . Hence we get that $\lim_{n\rightarrow\infty}\frac{1}{n}\mbox{E}\left[{\mbox{Tr}}H_{n}^{q-r}\right]$ equals to $\Gamma(\frac{q-r}{2}+1)$ when $r$ is even and 0 when $r$ is odd, where $\Gamma$ denote the Gamma function. Thus

\lim_{n\rightarrow\infty}b_{n}^{-\frac{q-r}{2}}\sum_{J_{2}^{\prime}}\mbox{E}\left[v_{J_{2}^{\prime}}\right]=\left\{\begin{array}[]{ll}0&\text{ if }r\text{ is odd, }\\ \Gamma(\frac{q-r}{2}+1)\left(t_{2}-t_{1}\right)^{\frac{q-r}{2}}2^{\frac{q-r}{2}}&\text{ if }r\text{ is even. }\end{array}\right.

(16)

Now using (15) and (16) in (11), we get that for $p,q$ even

	$\displaystyle\lim_{n\rightarrow\infty}\mbox{E}\left[w_{p}\left(t_{1}\right)w_{q}\left(t_{2}\right)\right]$
	$\displaystyle=\sum_{r=2,4,\ldots,q}{q\choose r}t_{1}^{\frac{p+r}{2}}2^{\frac{p+q}{2}}\left(\left\|\Delta_{2}(p,q)\right\|+\left\|\tilde{\Delta}_{2}(p,r)\right\|+2\left\|\Delta_{2,4}(p,r)\right\|+\right)\Gamma\left(\frac{q-r}{2}+1\right)\left(t_{2}-t_{1}\right)^{\frac{q-r}{2}}$
	$\displaystyle=2^{\frac{p+q}{2}}\sum_{r=2,4,\ldots,q}{q\choose r}t_{1}^{\frac{p+r}{2}}\left(t_{2}-t_{1}\right)^{\frac{q-r}{2}}R(p,r)\Gamma\left(\frac{q-r}{2}+1\right).$

This completes the proof of the lemma. ∎

Now we prove Proposition 6 with the help of above lemmata.

Proof of Proposition 6.

We use Cramér-Wold device and method of moment to prove Proposition 6. It is enough to show that, for $0<t_{1}\leq t_{2}\leq\cdots\leq t_{\ell}$ and for even integers $p_{1},p_{2},\ldots,p_{\ell}\geq 2$ ,

\lim_{n\to\infty}\mbox{E}[w_{p_{1}}(t_{1})w_{p_{2}}(t_{2})\cdots w_{p_{\ell}}(t_{\ell})]=\mbox{E}[W_{p_{1}}(t_{1})W_{p_{2}}(t_{2})\cdots W_{p_{\ell}}(t_{\ell})],

(17)

where $\{W_{p}(t)\}_{p\geq 2}$ is a centered Gaussian family with covariance as in (4). Note that the proof of (17) goes similar to the proof of Theorem 1.4 of [11], once we get the covariance formula, (9). So with the helps of above lemmata and by similar arguments of Theorem 1.4 of [11], we get the result. We skip the details here. ∎

4.2. Proof of Proposition 7

The idea of the proof of Proposition 7 is similar to the proof of Proposition 10 in [16]. Here we outline only the main steps, for the details, see the proof of Proposition 10 of [16].

Proof of Proposition 7.

We prove Proposition 7 for $\alpha=4$ and $\beta=1$ . Suppose $p\geq 2$ is an even fixed positive integer and $t,s\in[0,T]$ , for some fixed $T\in\mathbb{N}$ . Then from (3),

\displaystyle w_{p}(t)-w_{p}(s)

\displaystyle=\frac{\sqrt{b_{n}}}{n}\big{[}{\mbox{Tr}}(A_{n}(t))^{p}-{\mbox{Tr}}(A_{n}(s))^{p}-\mbox{E}[{\mbox{Tr}}(A_{n}(t))^{p}-{\mbox{Tr}}(A_{n}(s))^{p}]\big{]}.

(18)

For $0<s<t$ , using binomial expansion, we get

{\mbox{Tr}}(A_{n}(t))^{p}-{\mbox{Tr}}(A_{n}(s))^{p}={\mbox{Tr}}[(A^{\prime}_{n}(t-s))^{p}]+\sum_{d=1}^{p-1}\binom{p}{d}{\mbox{Tr}}[(A^{\prime}_{n}(t-s))^{d}(A_{n}(s))^{p-d}],

(19)

where $A^{\prime}_{n}(t-s)$ is a band Hankel matrix with entries $\{a_{n}(t)-a_{n}(s)\}_{n\geq 0}$ . Now by using the trace formula of band Hankel matrices from Result 4, we get

	$\displaystyle{\mbox{Tr}}[(A^{\prime}_{n}(t-s))^{p}]$	$\displaystyle=\frac{1}{{b_{n}}^{\frac{p}{2}}}\sum_{i=1}^{n}\sum_{A_{p}}a^{\prime}_{J_{p}}(t-s)I(i,J_{p}),$		(20)
	$\displaystyle{\mbox{Tr}}[(A^{\prime}_{n}(t-s))^{d}(A_{n}(s))^{p-d}]$	$\displaystyle=\frac{1}{{b_{n}}^{\frac{p}{2}}}\sum_{i=1}^{n}\sum_{A_{p}}a^{\prime}_{J_{d}}(t-s)a_{J_{p-d}}(s)I(i,J_{p}),$

where

	$\displaystyle a^{\prime}_{J_{p}}(t-s)$	$\displaystyle=(a_{j_{1}}(t)-a_{j_{1}}(s))(a_{j_{2}}(t)-a_{j_{2}}(s))\cdots(a_{j_{p}}(t)-a_{j_{p}}(s)),$
	$\displaystyle a^{\prime}_{J_{d}}(t-s)$	$\displaystyle=(a_{j_{1}}(t)-a_{j_{1}}(s))(a_{j_{2}}(t)-a_{j_{2}}(s))\cdots(a_{j_{d}}(t)-a_{j_{d}}(s)),$
	$\displaystyle a_{J_{p-d}}(s)$	$\displaystyle=a_{j_{d+1}}(s)a_{j_{d+2}}(s)\cdots a_{j_{p}}(s).$

Note that, $a_{n}(t)-a_{n}(s)$ has same distribution as $a_{n}(t-s)$ and $a_{n}(t)$ has same distribution as $N(0,t)$ . Therefore by using (20) in (19), we get

		$\displaystyle{\mbox{Tr}}(A_{n}(t))^{p}-{\mbox{Tr}}(A_{n}(s))^{p}$
		$\displaystyle\mathbin{\overset{D}{\kern 0.0pt\sim}}\frac{1}{{b_{n}}^{\frac{p}{2}}}\sum_{i=1}^{n}\Big{[}\sum_{A_{p}}\Big{(}(t-s)^{\frac{p}{2}}x_{j_{1}}x_{j_{2}}\cdots x_{j_{p}}+\sum_{d=1}^{p-1}\binom{p}{d}(t-s)^{\frac{d}{2}}{s}^{\frac{p-d}{2}}x_{j_{1}}x_{j_{2}}\cdots x_{j_{d}}y_{j_{d+1}}y_{j_{d+2}}\cdots y_{j_{p}}\Big{)}\Big{]}I(i,J_{p})$
		$\displaystyle=\sqrt{\frac{t-s}{{b_{n}}^{p}}}\sum_{i=1}^{n}\sum_{A_{p}}Z_{J_{p}}I(i,J_{p}),\mbox{ say},$		(21)

where $\{x_{i}\}$ and $\{y_{i}\}$ are independent normal random variables with mean zero, and variances $(t-s)$ and $s$ , respectively. If $x_{1}$ and $x_{2}$ have same distribution, then we denote it as $x_{1}\mathbin{\overset{D}{\kern 0.0pt\sim}}x_{2}$ . Finally, by using (4.2) in (18), we get

\displaystyle\mbox{E}[w_{p}(t)-w_{p}(s)]^{4}

\displaystyle=\frac{(t-s)^{2}}{n^{4}{b_{n}}^{2p-2}}\sum_{i_{1},i_{2},i_{3},i_{4}=1}^{n}\sum_{A_{p},A_{p},A_{p},A_{p}}\mbox{E}\Big{[}\prod_{k=1}^{4}(Z_{J^{k}_{p}}-\mbox{E}Z_{J^{k}_{p}})I(i_{k},J^{k}_{p})\Big{]},

(22)

where $J^{k}_{p}$ are vectors from $A_{p}$ , for each $k=1,2,3,4$ .

We introduce the terminology: $J^{k}_{p}$ and $J^{\ell}_{p}$ are connected if $S_{J^{k}_{p}}\cap S_{J^{\ell}_{p}}\neq\emptyset$ , where $S_{J^{i}_{p}}$ is as given in (13). Now depending on connectedness between $J^{k}_{p}$ ’s, the following three cases arise in (22): Case I. At least one of $J^{k}_{p}$ for $k=1,2,3,4$ , is not connected with the remaining ones:

Since one $J^{k}_{p}$ is not connected with the others therefore due to the independence of entries, we get $\mbox{E}[w_{p}(t)-w_{p}(s)]^{4}=0$ . Case II. $J^{1}_{p}$ is connected with only one of $J^{2}_{p},J^{3}_{p},J^{4}_{p}$ and the remaining two of $J^{2}_{p},J^{3}_{p},J^{4}_{p}$ are connected only with each other: Without loss of generality, we assume $J^{1}_{p}$ is connected only with $J^{2}_{p}$ and $J^{3}_{p}$ is connected only with $J^{4}_{p}$ . Under this situation the terms in the right hand side of (22) can be written as

		$\displaystyle\frac{(t-s)^{2}}{n^{4}{b_{n}}^{2p-2}}\sum_{i_{1},i_{2},i_{3},i_{4}=1}^{n}\sum_{A_{p},A_{p}}\mbox{E}\big{[}(Z_{J^{1}_{p}}-\mbox{E}Z_{J^{1}_{p}})(Z_{J^{2}_{p}}-\mbox{E}Z_{J^{2}_{p}})\big{]}I(i_{1},J^{1}_{p})I(i_{2},J^{2}_{p})$
		$\displaystyle\qquad\qquad\qquad\qquad\qquad\sum_{A_{p},A_{p}}\mbox{E}\big{[}(Z_{J^{3}_{p}}-\mbox{E}Z_{J^{3}_{p}})(Z_{J^{4}_{p}}-\mbox{E}Z_{J^{4}_{p}})\big{]}I(i_{3},J^{3}_{p})I(i_{4},J^{4}_{p})=\theta,\mbox{ say}.$		(23)

Now from the definition of $Z_{J_{p}}$ , given in (4.2) and $\mbox{E}(x_{i})=\mbox{E}(y_{i})=0$ , we get

		$\displaystyle\sum_{A_{p},A_{p}}\big{\|}\mbox{E}\big{[}(Z_{J^{1}_{p}}-\mbox{E}Z_{J^{1}_{p}})(Z_{J^{2}_{p}}-\mbox{E}Z_{J^{2}_{p}})\big{]}I(i_{1},J^{1}_{p})I(i_{2},J^{2}_{p})\big{\|}$
		$\displaystyle\qquad=\sum_{(J^{1}_{p},J^{2}_{p})\in B_{P_{2}}}\big{\|}\mbox{E}\big{[}(Z_{J^{1}_{p}}-\mbox{E}Z_{J^{1}_{p}})(Z_{J^{2}_{p}}-\mbox{E}Z_{J^{2}_{p}})\big{]}I(i_{1},J^{1}_{p})I(i_{2},J^{2}_{p})\big{\|}\leq\sum_{(J^{1}_{p},J^{2}_{p})\in B_{P_{2}}}\alpha,$		(24)

where $\alpha$ is a positive constant and last inequality arises as $x_{i}\mathbin{\overset{D}{\kern 0.0pt\sim}}N(0,t-s)$ , $y_{i}\mathbin{\overset{D}{\kern 0.0pt\sim}}N(0,s)$ , and $B_{P_{2}}$ is defined as

	$\displaystyle B_{P_{2}}=\{(J^{1}_{p},J^{2}_{p})\in A_{p}\times A_{p}$	$\displaystyle:S_{J^{1}_{p}}\cap S_{J^{2}_{p}}\neq\emptyset\mbox{ and each entries of }S_{J^{1}_{p}}\cup S_{J^{2}_{p}}$
		$\displaystyle\quad\mbox{ has multiplicity greater than or equal to two}\},$

with $S_{J^{i}_{p}}$ is as given in (13). It is easy to observe that $|B_{P_{2}}|=O({b_{n}}^{p-1})$ .

Now from (4.2) and (4.2), we get

\displaystyle|\theta|\leq\frac{(t-s)^{2}}{n^{4}{b_{n}}^{2p-2}}\sum_{i_{1},i_{2},i_{3},i_{4}=1}^{n}\alpha^{2}|B_{P_{2}}|^{2}=(t-s)^{2}\alpha^{2}O(1)\leq M_{1}(t-s)^{2},

(25)

where $M_{1}$ is a positive constant depending only on $p$ and $T$ .
Case III. $J^{1}_{p},J^{2}_{p},J^{3}_{p},J^{4}_{p}$ are connected: Since $\mbox{E}(x_{i})=0$ and $\mbox{E}(y_{i})=0$ , therefore

\displaystyle\sum_{A_{p},A_{p},A_{p},A_{p}}

\displaystyle\mbox{E}\big{[}\prod_{k=1}^{4}(Z_{J^{k}_{p}}-\mbox{E}Z_{J^{k}_{p}})\big{]}=\sum_{(J^{1}_{p},J^{2}_{p},J^{3}_{p},J^{4}_{p})\in B_{P_{4}}}\mbox{E}\big{[}\prod_{k=1}^{4}(Z_{J^{k}_{p}}-\mbox{E}Z_{J^{k}_{p}})\big{]},

where $B_{P_{4}}$ is defined as,

	$\displaystyle B_{P_{4}}=\{(J^{1}_{p},J^{2}_{p},J^{3}_{p},J^{4}_{p})\in$	$\displaystyle A_{p}\times A_{p}\times A_{p}\times A_{p}:S_{J^{1}_{p}}\cap S_{J^{2}_{p}}\cap S_{J^{3}_{p}}\cap S_{J^{4}_{p}}\neq\emptyset\mbox{ and each entries of }$
		$\displaystyle\qquad S_{J^{1}_{p}}\cup S_{J^{2}_{p}}\cup S_{J^{3}_{p}}\cup S_{J^{4}_{p}}\mbox{ has multiplicity greater than or equal to two}\}.$

Observe that $|B_{P_{4}}|=o({b_{n}}^{2p-2})$ . Hence

$\displaystyle\frac{(t-s)^{2}}{n^{4}{b_{n}}^{2p-2}}\sum_{i_{1},i_{2},i_{3},i_{4}=1}^{n}\sum_{A_{p},A_{p},A_{p},A_{p}}\Big{\|}\mbox{E}\big{[}\prod_{k=1}^{4}(Z_{J^{k}_{p}}-\mbox{E}Z_{J^{k}_{p}})\big{]}I(i_{k},J^{k}_{p})\Big{\|}$	$\displaystyle\leq\frac{(t-s)^{2}}{n^{4}{b_{n}}^{2p-2}}\sum_{i_{1},i_{2},i_{3},i_{4}=1}^{n}\beta o({b_{n}}^{2p-2})$
	$\displaystyle=\beta(t-s)^{2}o(1)$
	$\displaystyle\leq M_{2}(t-s)^{2},$	(26)

where $M_{2}$ is a positive constant depending only on $p,T$ .

Now combining (25) and (4.2), it follows that there exists a positive constant $M_{T}$ , depending only on $p,T$ such that

\displaystyle\mbox{E}[w_{p}(t)-w_{p}(s)]^{4}

\displaystyle\leq M_{T}(t-s)^{2}\ \ \ \forall\ n\in\mathbb{N}\ \mbox{and }t,s\in[0,\ T].

(27)

This completes the proof of Proposition 7 with $\alpha=4$ and $\beta=1$ . ∎

Remark 17.

For any fixed $N\in\mathbb{N}$ , (27) implies

\{w_{p}(t);t\geq 0,1\leq p\leq N\}\stackrel{{\scriptstyle\mathcal{D}}}{{\rightarrow}}\{W_{p}(t);t\geq 0,1\leq p\leq N\}\quad\mbox{ as }n\rightarrow\infty.

But the process convergence of $\{w_{p}(t);t\geq 0,p\geq 2\}$ does not follow from the proof of Theorem 1, because the constant $M_{T}$ (depends on $p$ ) of (27) may tends to infinity as $p\rightarrow\infty$ .

The following remark provides an application of Theorem 1 in the study of the limiting law of $\sup_{0\leq t\leq T}|w_{p}(t)|$ :

Remark 18.

Suppose $p\geq 2$ is an even positive integer and $T$ is a positive real number. Then as $n\to\infty$

\sup_{0\leq t\leq T}|w_{p}(t)|\stackrel{{\scriptstyle\mathcal{D}}}{{\rightarrow}}\sup_{0\leq t\leq T}|W_{p}(t)|,

(28)

where $\{W_{p}(t);t\geq 0\}$ is as defined in Theorem 1. The convergence of (28) follows from the continuous mapping theorem and Theorem 1.

Remark 19.

Theorem 1 and Remark 18 can also be generalised for even degree polynomial test function. The proofs will be similar to the proof of Theorem 1 and Remark 18, respectively.

In the following remark, we discuss the process convergence when the entry $\{a_{0}(t);t\geq 0\}$ is the standard Brownian motion.

Remark 20.

Let $\{a_{0}(t);t\geq 0\}$ be a standard Brownian motion and independent of $\{a_{n}(t);t\geq 0\}_{n\geq 1}$ . For $p\geq 1$ , define

\displaystyle\tilde{A}_{n}(t)

\displaystyle:=A_{n}(t)+\frac{a_{0}(t)}{\sqrt{b_{n}}}I_{n}\ \mbox{ and }\ \tilde{w}_{p}(t):=\frac{\sqrt{b_{n}}}{n}\big{(}{\mbox{Tr}}(\tilde{A}_{n}(t))^{p}-\mbox{E}{\mbox{Tr}}(\tilde{A}_{n}(t))^{p}\big{)},

where $A_{n}(t)$ is the Hankel matrix as defined in (2) and $I_{n}$ is an $n\times n$ identity matrix.

Suppose $p\geq 2$ is a positive even integer. Then as $n\to\infty$

\{\tilde{w}_{p}(t);t\geq 0\}\stackrel{{\scriptstyle\mathcal{D}}}{{\rightarrow}}\{\tilde{W}_{p}(t);t\geq 0\},

(29)

where $\{\tilde{W}_{p}(t);t\geq 0\}$ is a Gaussian process and

\tilde{W}_{p}(t)=W_{p}(t)+pt^{\frac{p-1}{2}}R_{p-1}a_{0}(t),

with $\{W_{p}(t);t\geq 0\}$ as in Theorem 1 and $R_{k}=\lim\limits_{n\to\infty}\mbox{E}\big{[}\frac{1}{n}{\mbox{Tr}}\big{(}A_{n}(1)\big{)}^{k}\big{]}$ .

Note that, to prove (29), it is enough to prove Proposition 6 and Proposition 7 for the process $\{\tilde{w}_{p}(t);t\geq 0\}$ . The idea of proof is same as the proof of Theorem 1. One can also see, [16, Theorem 4], where Maurya and Saha have derived similar type of result for band Toeplitz matrices.

In the following remark, we discuss the process convergence of $\{w_{p}(t);t\geq 0\}$ for odd $p\geq 1$ .

Remark 21.

By the similar ideas as used in the proof of Theorem 3, one can derive the following result for any odd integer $p\geq 1$ and $t\geq 0$ ,

w_{p}(t)\stackrel{{\scriptstyle d}}{{\rightarrow}}0\mbox{ as }n\to\infty,

which implies the finite dimensional convergence of $\{w_{p}(t);t\geq 0\}$ for odd $p\geq 1$ . The proof of tightness of $\{w_{p}(t);t\geq 0\}$ for odd $p\geq 1$ goes similar to the proof of tightness of $\{w_{p}(t);t\geq 0\}$ for even $p\geq 2$ (Proof of Proposition 7) and hence we conclude that for odd $p$ ,

\{w_{p}(t);t\geq 0\}\stackrel{{\scriptstyle\mathcal{D}}}{{\rightarrow}}0\mbox{ as }n\to\infty.

5. Proof of Theorem 3

We prove Theorem 3 by showing that $\mbox{\rm Var}(w_{p})\rightarrow 0$ as $n\rightarrow\infty$ .

Proof of Theorem 3: .

First note from (6) that for $p=1$ , $w_{1}=0$ and the result is trivial.

Using the trace formula from Result 4, we get for every odd $p\geq 3$ ,

w_{p}=\frac{\sqrt{b_{n}}}{n}\bigl{\{}{\mbox{Tr}}(A_{n})^{p}-\mbox{E}[{\mbox{Tr}}(A_{n})^{p}]\bigr{\}}=\frac{1}{n}b_{n}^{-\frac{p-1}{2}}\sum_{i=1}^{n}\sum_{J\in A_{p,i}}I_{J}\left(x_{J}-\mbox{E}\left[x_{J}\right]\right),

(30)

where $I_{J}=\prod_{\ell=1}^{p}\chi_{[1,n]}\left(i-\sum_{q=1}^{\ell}j_{q}\right)$ , $x_{J}=\prod_{\ell=1}^{p}x_{j_{\ell}}$ and for $1\leq i\leq n$ , define

A_{p,i}=\left\{\left(j_{1},\ldots,j_{p}\right)\in\left\{\pm 1,\ldots,\pm b_{n}\right\}^{p}:\sum_{k=1}^{p}(-1)^{k}j_{k}=2i-1-n\right\}.

Since $\mbox{E}(w_{p})=0$ therefore $\mbox{\rm Cov}(w_{p},w_{q})=\mbox{E}(w_{p}w_{q})$ and hence from (30), we get

\displaystyle\mbox{\rm Cov}(w_{p},w_{q})=\mbox{E}[w_{p}w_{q}]

\displaystyle=\frac{1}{n^{2}}b_{n}^{-\frac{p+q}{2}+1}\sum_{i_{1},i_{2}=1}^{n}\sum_{J_{1},J_{2}}\big{\{}\mbox{E}\left[x_{J_{1}}x_{J_{2}}\right]-\mbox{E}[x_{J_{1}}]\mbox{E}[x_{J_{2}}]\big{\}}I_{J_{1}}I_{J_{2}},

(31)

where $J_{1}\in A_{p,i_{1}}$ and $J_{2}\in A_{q,i_{2}}$ .

Notice that the summand in (31) will be non-zero only if each element in $S_{J_{1}}\cup S_{J_{2}}$ have multiplicity at least two and $S_{J_{1}}\cap S_{J_{2}}\neq\emptyset$ , where for a vector $J=\left(j_{1},j_{2},\ldots,j_{p}\right)\in A_{p,i},$ the multi-set $S_{J}$ is defined as

S_{J}=\left\{j_{1},j_{2},\ldots,j_{p}\right\}.

Since $p$ and $q$ are odd, there exist at least one element each in $S_{J_{1}}$ and $S_{J_{2}}$ with odd multiplicity in $S_{J_{i}}$ . Notice that the number of terms with at least one component having odd multiplicity greater than 1 in $S_{J_{i}}$ , is at most $O(b_{n}^{\frac{p+q-3}{2}})$ , leading to a zero contribution in (31) as $n\rightarrow\infty$ . A similar argument also shows that terms with at least one $j_{k}$ having multiplicity greater than 2 in $S_{J_{1}}\cup S_{J_{2}}$ also lead to zero contribution in (31) as $n\rightarrow\infty$ . Thus we restrict ourselves to the case where the multiplicity of every element in $S_{J_{1}}\cup S_{J_{2}}$ is 2. Since there exist elements of multiplicity $1$ in both $S_{J_{1}}$ and $S_{J_{2}}$ , independence of the input sequence implies that $\mbox{E}[x_{J_{1}}]=0=\mbox{E}[x_{J_{2}}]$ . Therefore from (31),

\displaystyle\lim_{n\rightarrow\infty}\mbox{\rm Cov}(w_{p},w_{q})=\displaystyle\lim_{n\rightarrow\infty}\frac{1}{n^{2}}b_{n}^{-\frac{p+q}{2}+1}\sum_{i_{1},i_{2}=1}^{n}\sum_{\pi\in{\mathcal{P}}_{2}(p+q)}\sum_{J_{\pi}}\mbox{E}\left[x_{J_{1}}x_{J_{2}}\right]I_{J_{1}}I_{J_{2}},

(32)

where ${\mathcal{P}}_{2}(p+q)$ is the set of all pair partitions of $\{1,2,\ldots,p+q\}$ and $J_{\pi}$ is the set of all vectors $(J_{1},J_{2})$ corresponding to $\pi$ with $J_{1}\in A_{p,i_{1}}$ and $J_{2}\in A_{q,i_{2}}$ .

The rest of the proof is done by considering the following cases:

Case I. $\mathbf{i_{1}+i_{2}=n+1}$ : Consider $J_{1}\in A_{p,i_{1}},J_{2}\in A_{q,i_{2}}$ . Since $p$ is odd, there exist a component in $J_{1}$ with multiplicity 1 in $S_{J_{1}}$ , whose value would be determined by the other $p-1$ terms. Hence the number of terms $(J_{1},J_{2})\in A_{p,i_{1}}\times A_{q,i_{2}}$ such that $\mbox{E}[x_{J_{1}}x_{J_{2}}]$ is non-zero is of order $O\left(b_{n}^{\frac{p+q-2}{2}}\right)$ . Hence, the sum of such terms in (32) is of order $O\left(\frac{1}{n^{2}}\times n\times\frac{1}{b_{n}^{\frac{p+q-2}{2}}}\times b_{n}^{\frac{p+q-2}{2}}\right)=O\left(\frac{1}{n}\right)$ .

Case II. $\mathbf{i_{1}+i_{2}\neq n+1}$ : For a pair-partition $\pi\in{\mathcal{P}}_{2}(p+q)$ , consider an associated $J$ such that $J=(j_{1},j_{2},\ldots,j_{p+q})=(J_{1},J_{2})$ where $J_{1}=(j_{1},j_{2},\ldots,j_{p})$ and $J_{2}=(j_{p+1},j_{p+2},\ldots,j_{p+q});J_{1}\in A_{p,i_{1}}$ , $J_{2}\in A_{q,i_{2}}$ and $\mbox{E}[x_{J_{1}}x_{J_{2}}]$ is non-zero.

subcase (i). $i_{1}\neq i_{2}$ :

Without loss of generality assume that $2i_{1}-n-1\neq 0$ . Since $\sum_{i=1}^{p}(-1)^{k}j_{k}=2i_{1}-n-1$ , $\sum_{i=1}^{p+q}(-1)^{k}j_{k}=2(i_{1}+i_{2})-2n-2$ and both of the right hand sides are non-zero, there is a loss of two degree of freedom. Thus the sum of such terms in (32) is $O\left(\frac{1}{n^{2}}\times n^{2}\times\frac{1}{b_{n}^{\frac{p+q-2}{2}}}\times b_{n}^{\frac{p+q-4}{2}}\right)=O\left(\frac{1}{b_{n}}\right)$ .

subcase (ii). $i_{1}=i_{2}$ :

By a similar argument, as given in Case I, we get that the order of the summation of such terms in (32) would be $O\left(\frac{1}{n^{2}}\times n\times\frac{1}{b_{n}^{\frac{p+q-2}{2}}}b_{n}^{\frac{p+q-2}{2}}\right)=O\left(\frac{1}{n}\right)$ .

Hence on combining Case I and Case II, we get that for all odd integers $p,q\geq 2$

\displaystyle\lim_{n\rightarrow\infty}\mbox{\rm Cov}(w_{p},w_{q})=0,

which shows $\mbox{\rm Var}(w_{p})=0$ and hence $w_{p}\stackrel{{\scriptstyle d}}{{\longrightarrow}}0$ . This completes the proof of Theorem 3. ∎

Remark 22.

Theorem 3 can also be generalised for odd degree polynomial test functions.

Acknowledgement: We express our most sincere thanks to Prof. Koushik Saha for reading the initial and final draft and providing helpful advice.

References

[1] Kartick Adhikari and Koushik Saha, Universality in the fluctuation of eigenvalues of random circulant matrices, Statist. Probab. Lett. 138 (2018), 1–8. MR 3788711
[2] N. I. Akhiezer, The classical moment problem and some related questions in analysis, Translated by N. Kemmer, Hafner Publishing Co., New York, 1965. MR 0184042
[3] Z. D. Bai, Methodologies in spectral analysis of large dimensional random matrices, a review, Statistica Sinica 9 (1999), no. 3, 611–662.
[4] Z. D. Bai and Jack W. Silverstein, CLT for linear spectral statistics of large-dimensional sample covariance matrices, Ann. Probab. 32 (2004), no. 1A, 553–605. MR 2040792
[5] A. Basak and A. Bose, Limiting spectral distributions of some band matrices, Periodica Mathematica Hungarica 63 (2011), 113–150.
[6] Arup Bose, Shambhu Nath Maurya, and Koushik Saha, Process convergence of fluctuations of linear eigenvalue statistics of random circulant matrices, Random Matrices: Theory and Applications 0 (0), no. 0, 2150032.
[7] Wlodzimierz Bryc, Amir Dembo, and Tiefeng Jiang, Spectral measure of large random hankel, markov and toeplitz matrices, The Annals of Probability 34 (2006), no. 1, 1–38.
[8] Christopher Hammond and Steven Miller, Distribution of eigenvalues for the ensemble of real symmetric toeplitz matrices, Journal of Theoretical Probability 18 (2005), 537–566.
[9] Nobuyuki Ikeda and Shinzo Watanabe, Stochastic differential equations and diffusion processes, North-Holland Mathematical Library, vol. 24, North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1981. MR 637061
[10] I. Jana, K. Saha, and A. Soshnikov, Fluctuations of linear eigenvalue statistics of random band matrices, Theory Probab. Appl. 60 (2016), no. 3, 407–443. MR 3568789
[11] Yiting Li and Xin Sun, On fluctuations for random band Toeplitz matrices, Random Matrices Theory Appl. 4 (2015), no. 3, 1550012, 28. MR 3385706
[12] D. Liu and Zheng-Dong Wang, Limit distribution of eigenvalues for random hankel and toeplitz band matrices, Journal of Theoretical Probability 24 (2009), 988–1001.
[13] Dang-Zheng Liu, Xin Sun, and Zheng-Dong Wang, Fluctuations of eigenvalues for random Toeplitz and related matrices, Electron. J. Probab. 17 (2012), no. 95, 22. MR 2994843
[14] Dang-Zheng Liu and Zheng-Dong Wang, Limit distribution of eigenvalues for random Hankel and Toeplitz band matrices, J. Theoret. Probab. 24 (2011), no. 4, 988–1001. MR 2851241
[15] A. Lytova and L. Pastur, Central limit theorem for linear eigenvalue statistics of random matrices with independent entries, Ann. Probab. 37 (2009), no. 5, 1778–1840. MR 2561434
[16] Shambhu Nath Maurya and Koushik Saha, Process convergence of fluctuations of linear eigenvalue statistics of band Toeplitz matrices, Statist. Probab. Lett. 166 (2020), 108875, 11. MR 4122111
[17] Jet Wimp, Padé-type approximation and general orthogonal polynomials (claude brezinski), SIAM Review 23 (1981), no. 3, 403–406.