Asymptotically efficient estimation for diffusion processes with nonsynchronous observations

For $l\in\{1,2\}$, let the observation times $\{S_{i}^{n,l}\}_{i=0}^{M_{l}}$ be strictly increasing random times with respect to $i$, and satisfy $S_{0}^{n,l}=0$ and $S_{M_{l}}^{n,l}=nh_{n}$, where $M_{l}$ is a random positive integer depending on $n$. We assume that $\{S_{i}^{n,l}\}_{0\leq i\leq M_{l},l=1,2}$ is independent of $\mathcal{F}_{T}$ and $\alpha$. We consider nonsynchronous observations of $X$, that is, we observe $\{S_{i}^{n,l}\}_{0\leq i\leq M_{l},l=1,2}$ and $\{X^{l}_{S^{n,l}_{i}}\}_{0\leq i\leq M_{l},l=1,2}$.

We denote by $\lVert\cdot\rVert$ the operator norm of a matrix, and by $\top$ the transpose operator for a matrix or a vector. We often regard a $p$-dimensional vector $v$ as a $p\times 1$ matrix. For $j\in\mathbb{N}$ and a vector $\kappa=(\kappa_{1},\cdots,\kappa_{j})$, we denote $\partial_{\kappa}^{k}=(\frac{\partial^{k}}{\partial\kappa_{i_{1}}\cdots\partial\kappa_{i_{k}}})_{i_{1},\cdots i_{k}=1}^{j}$. For a set $A$ in a topological space, let ${\rm clos}(A)$ denote the closure of $A$. For a matrix $A$, $[A]_{ij}$ denotes its $(i,j)$ element. For a vector $v=(v_{j})_{j=1}^{K}$, ${\rm diag}(v)$ denotes a $k\times k$ diagonal matrix with elements $[{\rm diag}(v)]_{jj}=v_{j}$.

Let $M=M_{1}+M_{2}$. For $1\leq i\leq M$, let

$${\varphi(i)=\left\{\begin{array}[]{ll}i&{\rm if}\ i\leq M_{1}\\ i-M_{1}&{\rm if}\ i>M_{1}\end{array}\right.\quad\psi(i)=\left\{\begin{array}[]{ll}1&{\rm if}\ i\leq M_{1}\\ 2&{\rm if}\ i>M_{1}\end{array}\right.}$$

For a two-dimensional stochastic process $(U_{t})_{t\geq 0}=((U_{t}^{1},U_{t}^{2}))_{t\geq 0}$, let $\Delta_{i}^{l}U=U^{l}_{S^{n,l}_{i}}-U^{l}_{S^{n,l}_{i-1}}$ , and let $\Delta^{l}U=(\Delta_{i}^{l}U)_{1\leq i\leq M_{l}}$ and $\Delta_{i}U=\Delta_{\varphi(i)}^{\psi(i)}U$ for $1\leq i\leq M$. Let $\Delta U=((\Delta^{1}U)^{\top},(\Delta^{2}U)^{\top})^{\top}$. Let $|K|=b-a$ for an interval $K=(a,b]$. Let $I_{i}^{l}=(S_{i-1}^{n,l},S_{i}^{n,l}]$ for $1\leq i\leq M_{l}$, and let $I_{i}=I_{\varphi(i)}^{\psi(i)}$ for $1\leq i\leq M$. We denote a unit matrix of size $k$ by $\mathcal{E}_{k}$.

Let $\tilde{\Sigma}_{i}^{l}(\sigma)=\int_{I_{i}^{l}}[b_{t}b_{t}^{\top}(\sigma)]_{ll}dt$ and $\tilde{\Sigma}_{i,j}^{1,2}(\sigma)=\int_{I_{i}^{1}\cap I_{j}^{2}}[b_{t}b_{t}^{\top}(\sigma)]_{12}dt$. By setting $\tilde{\mathcal{D}}={\rm diag}(\{\tilde{\Sigma}_{i}\}_{1\leq i\leq M})$,

$${G=\bigg{\{}\frac{|I_{i}^{1}\cap I_{j}^{2}|}{|I_{i}^{1}|^{1/2}|I_{j}^{2}|^{1/2}}\bigg{\}}_{1\leq i\leq M_{1},1\leq j\leq M_{2}},\quad\rho_{ij}(\sigma)=\frac{\tilde{\Sigma}_{i,j}^{1,2}}{\sqrt{\tilde{\Sigma}_{i}^{1}}\sqrt{\tilde{\Sigma}_{j}^{2}}}(\sigma),\quad\tilde{G}(\sigma)=\{\rho_{ij}(\sigma)[G]_{ij}\}_{1\leq i\leq M_{1},1\leq j\leq M_{2}},}$$

we can calculate the covariance matrix of $\Delta X$ as

$${S_{n}(\sigma)=\tilde{\mathcal{D}}^{1/2}\left(\begin{array}[]{cc}\mathcal{E}_{M_{1}}&\tilde{G}(\sigma)\\ \tilde{G}^{\top}(\sigma)&\mathcal{E}_{M_{2}}\end{array}\right)\tilde{\mathcal{D}}^{1/2}.}$$

As we will see later, we can ignore the drift term when we consider estimation of $\sigma$ because the drift term converges to zero very fast. Therefore, we first construct an estimator for $\sigma$, and then construct an estimator for $\theta$. Such adaptive estimation can speed up the calculation.

We define the quasi-likelihood function $H_{n}^{1}(\sigma)$ for $\sigma$ as follows.

$$\begin{split}{H_{n}^{1}(\sigma)&=-\frac{1}{2}\Delta X^{\top}S_{n}^{-1}(\sigma)\Delta X-\frac{1}{2}\log\det S_{n}(\sigma).}\end{split}$$

Then, the maximum-likelihood-type estimator for $\sigma$ is defined by

$${\hat{\sigma}_{n}\in{\rm argmax}_{\sigma\in{\rm clos}(\Theta_{1})}H_{n}^{1}(\sigma).}$$

We consider estimation for $\theta$ in the next. Let $V(\theta)=(V_{t}(\theta))_{t\geq 0}$ be a two-dimensional stochastic process defined by $V_{t}(\theta)=(\int_{0}^{t}\mu^{1}_{s}(\theta)^{\top}ds,\int_{0}^{t}\mu^{2}_{s}(\theta)^{\top}ds)^{\top}$. Let $\bar{X}(\theta)=\Delta X-\Delta V(\theta)$. We define the quasi-likelihood function $H_{n}^{2}(\theta)$ for $\theta$ as follows.

$${H_{n}^{2}(\theta)=-\frac{1}{2}\bar{X}(\theta)^{\top}S_{n}^{-1}(\hat{\sigma}_{n})\bar{X}(\theta).}$$

Then, the maximum-likelihood-type estimator for $\theta$ is defined by

$${\hat{\theta}_{n}\in{\rm argmax}_{\theta\in{\rm clos}(\Theta_{2})}H_{n}^{2}(\theta).}$$

The quasi-(log-)likelihood function $H_{n}^{1}$ is defined in the same way as that in [16]. Since $\Delta X$ follows normal distribution, we can construct such a Gaussian quasi-likelihood function even for the nonsynchronous data. When the coefficients are random, though the distribution of $\Delta X$ is not Gaussian, such Gaussian-type quasi-likelihood function is still valid due to the local Gaussian property of diffusion processes. The Gaussian mean that comes from the drift part is ignored when we construct the quasi-likelihood $H_{n}^{1}$. When we estimate the parameter $\theta$ for the drift part, we substruct the mean in $\bar{X}(\theta)$ to construct the quasi-likelihood function $H_{n}^{2}$. Since the effect of the drift term on the estimation of $\sigma$ is small, it works well to estimate $\sigma$ in this way and then plug in $\hat{\theta}_{n}$ to $S_{n}$ to construct the estimator for $\theta$. Thus, we can speed up the calculation by separating the estimation for $\sigma$ and $\theta$.

In this section, we state the assumptions of our main results, and state the asymtotic normality of the estimator.

For $m\in\mathbb{N}$, an open subset $U\subset\mathbb{R}^{m}$ is said to admit Sobolev’s inequality if for any $p>m$, there exists a positve constant $C$ depending $U$ and $p$ such that $\sup_{x\in U}|u(x)|\leq C\sum_{k=0,1}(\int|\partial_{x}^{k}u(x)|^{p})^{1/p}$ for any $u\in C^{1}(U)$. This is the case when $U$ has a Lipschitz boundary. We assume that $\Theta$, $\Theta_{1}$, and $\Theta_{2}$ admit Sobolev’s inequality.

Let $\Sigma_{t}(\sigma)=b_{t}b_{t}^{\top}(\sigma)$, and let

$${\rho_{t}(\sigma)=\frac{[\Sigma_{t}]_{12}}{[\Sigma_{t}]_{11}^{1/2}[\Sigma_{t}]_{22}^{1/2}}(\sigma),\quad B_{l,t}(\sigma)=\frac{[\Sigma_{t}(\sigma_{0})]_{ll}}{[\Sigma_{t}(\sigma)]_{ll}}.}$$

Let $\rho_{t,0}=\rho_{t}(\sigma_{0})$ and $r_{n}=\max_{i,l}|I_{i}^{l}|$. Let $\mathfrak{S}$ be the set of all partitions $(s_{k})_{k=0}^{\infty}$ of $[0,\infty)$ satisfying $\sup_{k\geq 1}|s_{k}-s_{k-1}|\leq 1$ and $\inf_{k\geq 1}|s_{k}-s_{k-1}|>0$. For $(s_{k})_{k=0}^{\infty}\in\mathfrak{S}$, let $M_{l,k}=\#\{i;\sup I_{i}^{l}\in(s_{k-1},s_{k}]\}$ and $q_{n}=\max\{k;s_{k}\leq nh_{n}\}$, and let $\mathcal{E}_{(k)}^{l}$ be an $M_{l}\times M_{l}$ matrix satisfying $[\mathcal{E}_{(k)}^{l}]_{ij}=1$ if $i=j$ and $\sup I_{i}^{l}\in(s_{k-1},s_{k}]$, and otherwise $[\mathcal{E}_{(k)}^{l}]_{ij}=0$.

Assumption (A1).

There exist positive constants $c_{1}$ and $c_{2}$ such that $c_{1}\mathcal{E}_{2}\leq\Sigma_{t}(\sigma)\leq c_{2}\mathcal{E}_{2}$ for any $t\in[0,\infty)$ and $\sigma\in\Theta_{1}$. For $k\in\{0,1,2,3,4\}$, $\partial_{\theta}^{k}\mu_{t}(\theta)$ and $\partial_{\sigma}^{k}b_{t}(\sigma)$ exist and are continuous with respect to $(t,\sigma,\theta)$ on $[0,\infty)\times{\rm clos}(\Theta_{1})\times{\rm clos}(\Theta_{2})$. For any $\epsilon>0$, there exist $\delta>0$ and $K>0$ such that

$${|\partial_{\theta}^{k}\mu_{t}(\theta)|+|\partial_{\sigma}^{k}b_{t}(\sigma)|\leq K,\quad|\partial_{\theta}^{k}\mu_{t}(\theta)-\partial_{\theta}^{k}\mu_{s}(\theta)|+|\partial_{\sigma}^{k}b_{t}(\sigma)-\partial_{\sigma}^{k}b_{s}(\sigma)|\leq\epsilon}$$

for any $k\in\{0,1,2,3,4\}$, $\sigma\in\Theta_{1}$, $\theta\in\Theta_{2}$, and $t,s\geq 0$ satisfying $|t-s|<\delta$.

Assumption (A2).

$r_{n}\overset{P}{\to}0$ as $n\to\infty$.

Assumption (A3).

For any $l\in\{1,2\}$, $i_{1}\in\mathbb{Z}_{+}$, $i_{2}\in\{0,1\}$, $i_{3}\in\{0,1,2,3,4\}$, $k_{1},k_{2}\in\{0,1,2\}$ satisfying $k_{1}+k_{2}=2$, and any polynomial function $F(x_{1},\cdots,x_{14})$ of degree equal to or less than $4$, there exist continuous functions $\Phi_{i_{1},i_{2}}^{1,F}(\sigma)$, $\Phi_{l,i_{3}}^{2}(\sigma)$ and $\Phi^{3,k_{1},k_{2}}_{i_{1},i_{3}}(\theta)$ on ${\rm clos}(\Theta_{1})$ and ${\rm clos}(\Theta_{2})$ such that

$${\frac{1}{T}\int_{0}^{T}F((\partial_{\sigma}^{k}B_{l,t}(\sigma))_{0\leq k\leq 4,l=1,2},(\partial_{\sigma}^{k^{\prime}}\rho_{t}(\sigma))_{k^{\prime}=1}^{4})\rho_{t}(\sigma)^{i_{1}}\rho_{t,0}^{i_{2}}dt\to\Phi_{i_{1},i_{2}}^{1,F}(\sigma),}$$

$${\frac{1}{T}\int_{0}^{T}\partial_{\sigma}^{i_{3}}\log B_{l,t}(\sigma)dt\to\Phi_{l,i_{3}}^{2}(\sigma),\quad\frac{1}{T}\int_{0}^{T}\partial_{\theta}^{i_{3}}(\phi_{1,t}^{k_{1}}\phi_{2,t}^{k_{2}})(\theta)\rho_{t,0}^{i_{1}}dt\to\Phi^{3,k_{1},k_{2}}_{i_{1},i_{3}}(\theta)}$$

as $T\to\infty$ for $\sigma\in{\rm clos}(\Theta_{1})$, $\theta\in{\rm clos}(\Theta_{2})$, where $\phi_{l,t}(\theta)=[\Sigma_{t}(\sigma_{0})]_{ll}^{-1/2}(\mu_{t}^{l}(\theta)-\mu_{t}^{l}(\theta_{0}))$.

Assumption (A1) and the Ascoli–Arzelà theorem yield that the convergences in (A3) can be replaced by uniform convergence with respect to $\sigma$ and $\theta$. Assumption (A3) is satisfied if $\mu_{t}(\theta)$ and $b_{t}(\sigma)$ are independent of $t$, or are periodic functions with respect to $t$ having a common period (when the period does not depend on $\sigma$ nor $\theta$).

Let $\mathfrak{I}_{l}=(|I_{i}^{l}|^{1/2})_{i=1}^{M_{l}}$.

Assumption (A4).

There exist positive constants $a_{0}^{1}$ and $a_{0}^{2}$ such that $\{h_{n}M_{l,q_{n}+1}\}_{n=1}^{\infty}$ is $P$-tight and

$${\max_{1\leq k\leq q_{n}}|h_{n}M_{l,k}-a_{0}^{l}(s_{k}-s_{k-1})|\overset{P}{\to}0}$$

for $l\in\{1,2\}$ and any partition $(s_{k})_{k=0}^{\infty}\in\mathfrak{S}$. Moreover, for any $p\in\mathbb{N}$, there exists a nonnegative constant $a_{p}^{1}$ such that

$${\max_{1\leq k\leq q_{n}}|h_{n}{\rm tr}(\mathcal{E}_{(k)}^{1}(GG^{\top})^{p})-a_{p}^{1}(s_{k}-s_{k-1})|\overset{P}{\to}0}$$

as $n\to\infty$ for any partition $(s_{k})_{k=0}^{\infty}\in\mathfrak{S}$.

Assumption (A5).

For $p\in\mathbb{Z}_{+}$, there exist nonnegative constants $f_{p}^{1,1}$, $f_{p}^{1,2}$, and $f_{p}^{2,2}$ such that

$$\begin{split}{\max_{1\leq k\leq q_{n}}|\mathfrak{I}_{1}\mathcal{E}_{(k)}^{1}(GG^{\top})^{p}\mathfrak{I}_{1}-f_{p}^{1,1}(s_{k}-s_{k-1})|&\overset{P}{\to}0,\\ \max_{1\leq k\leq q_{n}}|\mathfrak{I}_{1}\mathcal{E}_{(k)}^{1}(GG^{\top})^{p}G\mathfrak{I}_{2}-f_{p}^{1,2}(s_{k}-s_{k-1})|&\overset{P}{\to}0,\\ \max_{1\leq k\leq q_{n}}|\mathfrak{I}_{2}\mathcal{E}_{(k)}^{2}(G^{\top}G)^{p}\mathfrak{I}_{2}-f_{p}^{2,2}(s_{k}-s_{k-1})|&\overset{P}{\to}0}\end{split}$$

as $n\to\infty$ for any partition $(s_{k})_{k=0}^{\infty}\in\mathfrak{S}$.

Assumption (A4) corresponds to [A3^′] in Ogihara and Yoshida [16]. The functionals in (A4) and (A5) appear in $H_{n}^{1}$ and $H_{n}^{2}$, and hence, we cannot specify the limits of $H_{n}^{1}$ and $H_{n}^{2}$ unless we assume existence of the limits of these functionals. It is difficult to directly check (A4) and (A5) for general sampling scheme. We study sufficient conditions for these conditions in Section 2.4.

Assumption (A6).

The constant $a_{1}^{1}$ in (A4) is positive, and there exist positive constants $c_{3}$ and $c_{4}$ such that

$$\begin{split}{\limsup_{T\to\infty}\bigg{(}\frac{1}{T}\int_{0}^{T}\lVert\Sigma_{t}(\sigma)-\Sigma_{t}(\sigma_{0})\rVert^{2}dt\bigg{)}&\geq c_{3}|\sigma-\sigma_{0}|^{2},\\ \limsup_{T\to\infty}\bigg{(}\frac{1}{T}\int_{0}^{T}|\mu_{t}(\theta)-\mu_{t}(\theta_{0})|^{2}dt\bigg{)}&\geq c_{4}|\theta-\theta_{0}|^{2}}\end{split}$$

for any $\sigma\in{\rm clos}(\Theta_{1})$ and $\theta\in{\rm clos}(\Theta_{2})$.

Assumption (A6) is necessary to identify the parameter $\sigma$ and $\theta$ from the data. If $a_{1}^{1}=0$, then we have $a_{p}^{1}=0$ for any $p\in\mathbb{N}$. This implies that the non-diagonal components of the covariance matrix $S_{n}$ are negligible in the limit. Then, we cannot consistently estimate the parameter in $\rho_{t}(\sigma)$. This is why we need the assumption $a_{1}^{1}>0$ (see Proposition 3.2 and the following discussion to obtain the consistency).

Let $\mathcal{A}(\rho)=\sum_{p=1}^{\infty}a_{p}^{1}\rho^{2p}$ for $\rho\in(-1,1)$, and let $\partial_{\sigma}^{k}B_{l,t,0}=\partial_{\sigma}^{k}B_{l,t}(\sigma_{0})$. Let

$${\gamma_{1,t}=\mathcal{A}(\rho_{t,0})\bigg{(}\frac{\partial_{\sigma}\rho_{t,0}}{\rho_{t,0}}-\partial_{\sigma}B_{1,t,0}-\partial_{\sigma}B_{2,t,0}\bigg{)}^{2}-\partial_{\rho}\mathcal{A}(\rho_{t,0})\frac{(\partial_{\sigma}\rho_{t,0})^{2}}{\rho_{t,0}}-2\sum_{l=1}^{2}(a_{0}^{l}+\mathcal{A}(\rho_{t,0}))(\partial_{\sigma}B_{l,t,0})^{2},}$$

and let $\Gamma_{1}=\lim_{T\to\infty}T^{-1}\int_{0}^{T}\gamma_{1,t}dt$, which exists under (A1), (A3) and (A4). Let

$${\Gamma_{2}=\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}\sum_{p=0}^{\infty}\rho_{t,0}^{2p}\bigg{\{}\sum_{l=1}^{2}f_{p}^{ll}(\partial_{\theta}\phi_{l,t})^{2}(\theta_{0})-2\rho_{t,0}f_{p}^{12}\partial_{\theta}\phi_{1,t}\partial_{\theta}\phi_{2,t}(\theta_{0})\bigg{\}}dt,}$$

which exists under (A1), (A3) and (A5). Let $T_{n}=nh_{n}$ and

$${\Gamma=\left(\begin{array}[]{cc}\Gamma_{1}&0\\ 0&\Gamma_{2}\end{array}\right).}$$

Next, to discuss the optimality of the estimator, we discuss local asymptotic normality of the statistical model. In this section, local asymptotic normality of our model is shown, and the maximum-likelihood-type estimator is shown to be asymptotically efficient.

Let $\mathbb{N}$ be the set of all positive integers. Let $\alpha_{0}\in\Theta$, $\Theta\subset\mathbb{R}^{d}$, and $\{P_{\alpha,n}\}_{\alpha\in\Theta}$ be a family of probability measures defined on a measurable space $(\mathcal{X}_{n},\mathcal{A}_{n})$ for $n\in\mathbb{N}$, where $\Theta$ is an open subset of $\mathbb{R}^{d}$. As usual we shall refer to $dP_{\alpha_{2},n}/dP_{\alpha_{1},n}$ the derivative of the absolutely continuous component of the measure $P_{\alpha_{2},n}$ with respect to measure $P_{\alpha_{1},n}$ at the observation $x$ as the likelihood ratio. The following definition of local asymptotic normality is Definition 2.1 in Chapter II of Ibragimov and Has’minskiĭ [9].

Let $\Theta=\Theta_{1}\times\Theta_{2}$. For $\alpha\in\Theta$, let $P_{\alpha,n}$ be the probability measure generated by the observation $\{S_{i}^{n,l}\}_{i,l}$ and $\{X_{S_{i}^{n,l}}^{(\alpha),l}\}_{i,l}$.

The proof is left to Section 3.6. Theorem 11.2 in Chapter II of Ibragimov and Has’minskiĭ [9] gives lower bounds of estimation errors for any regular estimator of parameters under the LAN property. Then, the optimal asymptotic variance of $\epsilon_{n}^{-1}(T_{n}-\alpha_{0})$ for regular estimator $T_{n}$ is $\mathcal{E}_{d}$. Therefore, Theorems 2.2 ensures that our estimator $(\hat{\sigma}_{n},\hat{\theta}_{n})$ is asymptotically efficient in this sense under the assumptions of the theorem (we can show that $(\hat{\sigma}_{n},\hat{\theta}_{n})$ is regular by the proof of Theorem 2.2, (3.49), (3.9), (3.31), (3.35) and Theorem 2 in [10]).

It is not easy to directly check Assumptions (A4) and (A5) for general random sampling scheme. In this section, we study tractable sufficient conditions for these assumptions. The proofs of the results in this section are left to Section 3.6.

Let $q>0$ and $\mathcal{N}^{n,l}_{t}=\sum_{i=1}^{M_{l}}1_{\{S_{i}^{n,l}\leq t\}}$. We consider the following conditions for point process $\mathcal{N}_{t}^{n,l}$.

Assumption (B1-$q$).

$${\sup_{n\geq 1}\max_{l\in\{1,2\}}\sup_{0\leq t\leq(n-1)h_{n}}E[(\mathcal{N}^{n,l}_{t+h_{n}}-\mathcal{N}^{n,l}_{t})^{q}]<\infty.}$$

Assumption (B2-$q$).

$${\limsup_{u\to\infty}\sup_{n\geq 1}\max_{l\in\{1,2\}}\sup_{0\leq t\leq nh_{n}-uh_{n}}u^{q}P(\mathcal{N}^{n,l}_{t+uh_{n}}-\mathcal{N}^{n,l}_{t}=0)<\infty.}$$

For example, let $(\bar{\mathcal{N}}_{t}^{1},\bar{\mathcal{N}}_{t}^{2})$ be two independent homogeneous Poisson processes with positive intensities $\lambda_{1}$ and $\lambda_{2}$, respectively, and $\mathcal{N}^{n,l}_{t}=\bar{\mathcal{N}}_{h_{n}^{-1}t}^{l}$. Then (B1-$q$) obviously holds for any $q>0$. Moreover, (B2-q) holds for any $q>0$ since

$${\limsup_{u\to\infty}\sup_{n\geq 1}\max_{l\in\{1,2\}}\sup_{0\leq t\leq nh_{n}-uh_{n}}u^{q}P(\mathcal{N}^{n,l}_{t+uh_{n}}-\mathcal{N}^{n,l}_{t}=0)=\lim_{u\to\infty}u^{q}e^{-(\lambda_{1}\wedge\lambda_{2})u}=0.}$$

To give sufficient conditions for (A4) and (A5), we consider mixing properties of $\mathcal{N}^{n,l}$. That is, we assume condtions for the following mixing coefficient $\alpha_{k}^{n}$. Let

$${\mathcal{G}_{i,j}^{n}=\sigma(\mathcal{N}^{n,l}_{t}-\mathcal{N}^{n,l}_{s};ih_{n}\leq s<t\leq jh_{n},l=1,2)\quad(0\leq i,j\leq n),}$$

and let

$${\alpha_{k}^{n}=0\vee\sup_{1\leq i,j\leq n-1,j-i\geq k}\sup_{A\in\mathcal{G}_{0,i}^{n}}\sup_{B\in\mathcal{G}_{j,n}^{n}}|P(A\cap B)-P(A)P(B)|.}$$

In the following, let $(\bar{\mathcal{N}}_{t}^{l})_{t\geq 0}$ be an exponential $\alpha$-mixing point process for $l\in\{1,2\}$. Assume that the distribution of $(\bar{\mathcal{N}}_{t+t_{k}}^{l}-\bar{\mathcal{N}}_{t+t_{k-1}}^{l})_{1\leq k\leq K,l=1,2}$ does not depend on $t\geq 0$ for any $K\in\mathbb{N}$ and $0\leq t_{0}<t_{1}<\cdots<t_{K}$.

By the above results, we obtain simple tractable sufficient conditions for the assumptions of the sampling scheme.

For a real number $a$, $[a]$ denotes the maximum integer which is not greater than $a$. Let $\Pi=\Pi_{n}=\{S_{i}^{n,l}\}_{1\leq i\leq M_{l},l\in\{1,2\}}$. We denote $|x|^{2}=\sum_{i_{1},\cdots,i_{k}}|x_{i_{1},\cdots,i_{k}}|^{2}$ for $x=\{x_{i_{1},\cdots,i_{k}}\}_{i_{1},\cdots,i_{k}}$ with $k\in\mathbb{N}$. $C$ denotes generic positive constant whose value may vary depending on context. We often omit the parameters $\sigma$ and $\theta$ in general functions $f(\sigma)$ and $g(\theta)$.