---
author:
- |
    [Yingyin Lu Zuoxiang Peng[^1] ]{}\
    [School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China]{}
title: Maxima and minima of homogeneous Gaussian random fields over continuous time and uniform grids
---

> [**Abstract.**]{}  In this paper, for centered homogeneous Gaussian random fields the joint limiting distributions of normalized maxima and minima over continuous time and uniform grids are investigated. It is shown that maxima and minima are asymptotic dependent for strongly dependent homogeneous Gaussian random field with the choice of sparse grid, Pickands’ grid or dense grid, while for the weakly dependent Gaussian random field maxima and minima are asymptotically independent.
>
> [**Keywords.**]{}  Maximum and minimum; joint limit distribution; continuous time; uniform grid; homogeneous Gaussian random field.
>
> [**AMS 2000 Subject Classification.**]{}   Primary 60G70; Secondary 60G15.

Introduction {#sec1}
============

\[sec1\] Let $\{ X(t), t\geq 0\}$ be a stationary Gaussian process with mean zero, variance one and the correlation function $r(t)$ some $\alpha \in (0,2]$, $$\begin{aligned}
\label{eq1.1}
r(t)=1-|t|^{\alpha}+o\left(|t|^{\alpha}\right),\;\ t\to 0.\end{aligned}$$ Assume that $$\begin{aligned}
\label{eq1.2}
r(t)\log t\to 0\;\ \mbox{as}\;\ t\to \infty.\end{aligned}$$ Under conditions and , the limit distribution theory on the maximum of $\{ X(t), t\geq 0\}$ up to time $T$, $$M_{T}=\max\{ X(t) , t\in [0,T]\}$$ is well developed that $$\begin{aligned}
\P\{ a_{T}\left(M_{T}-b_{T}\right)\leq x\} \to \Lambda(x)\end{aligned}$$ as $T\to\infty$, where $ \Lambda(x)= \exp\left(-e^{-x}\right) $ is the standard distribution and $$\begin{aligned}
a_{T}=\sqrt{2 \log T},\;\ b_{T}=\sqrt{2 \log T} + \frac{\log\Big[(2\pi)^{-1/2} H_{\alpha} (2\log T)^{-1/2+1/\alpha}\Big]}{\sqrt{2 \log T}}.\end{aligned}$$ Here $ H_{\alpha} $ is the well-known Pickands’ constant, which is defined by $ H_{\alpha}=\lim_{\lambda\to\infty} H_{\alpha}(\lambda)/\lambda $ with $$\begin{aligned}
\label{eq.1.3}
H_{\alpha}(\lambda)=\E\exp\left( \max_{t\in [0,\lambda]} \sqrt{2} B_{\alpha/2}(t)-t^{\alpha}\right)\end{aligned}$$ and $ B_{H}$ is a fractional Brownian motion with $ \E B_{H}^{2}(t)=|t|^{2H}$. It is well known that $ 0<H_{\alpha}<\infty $, Leadbetter et al. (1983), Pickands (1969) and Piterbarg (1996). The limit distribution theorem about $ M_{T}$ extended by Mittal and Ylvisaker (1975) and McCormick and Qi (2000), and Dȩbicki et al. (2013) extended the results to homogeneous Gaussian random fields.

The joint limiting distributions of $M_{T}$ and its discrete time maximum $M_{T}^{\delta}=\max\{ X(t),t\in [0,T]\cap \Re(\delta)\}$ was first studied by Piterbarg (2004) under the conditions and , where the uniform is given by $\Re=\Re(\delta)=\{k\delta,\;\ k=0,1,2\ldots\}$ with $\delta \left( 2\log T \right)^{1/\alpha} \to D\in[0,\infty] $ as $T\to \infty.$ For more details, see Piterbarg (2004). The results multivariate stationary Gaussian processes can be found in Tan and Hashorva (2014, 2015), Tan and Wang (2013) and Tan and Tang (2014) for strongly dependent Gaussian processes. Further, Turkman (2012) considered the non-Gaussian processes and Chen and Tan (2016) studied the asymptotic of $M_{T}$, $M_{T}^{\delta}$ and the partial sum of dependent Gaussian processes.

For the joint asymptotic behaviors of $M_{T}$ and $M_{T}^{\delta}$ of homogeneous Gaussian random field, Tan and Wang (2015) considered the following model. Let $\{ X(\t), \t\geq \mathbf{0}\}$ be a homogeneous Gaussian field with mean zero, variance one and covariance function $ r(\mathbf{t})=\Cov(X(\mathbf{t}),X(\mathbf{0})) $ satisfies the following conditions, for $ d\geq 2 $:

$\mathbf{A1}: r(\t)=1-\sum_{i=1}^{d}|t_{i}|^{\alpha_{i}}(1+o(1))$, as $t_{i}\to 0$, with $\alpha_{i}\in (0,2] $ ;

$\mathbf{A2}: r(\t) < 1$, for $\t \neq \mathbf{0} $ ;

$\mathbf{A3}: \lim_{\T\to \infty} r(\T)\log(\prod_{i=1}^{d} T_{i})=r\in[0,\infty)$, where $\T\to \infty$ means $T_{i}\to\infty, i=1,2,\cdots,d$. If $T_{i}=0$, $r(T_{1},\cdots,T_{i-1},0,T_{i+1},\cdots, T_{d})\log(\prod_{j\neq i}^{d}T_{j})$ is bounded.

Define $$\mathbf{M}_{\T}=\left( M_{\T}, M_{\T}^{\boldsymbol{\delta}}\right)=\left( \max_{\t\in \I_{\T}} X(\t),\max_{\t\in \I_{\T}\bigcap \prod_{i=1}^{d} \Re(\delta_{i})} X(\t) \right),$$ where $ \I_{\T}=\prod_{i=1}^{d}[0,T_{i}]$, and $ \I_{\T}\bigcap \prod_{i=1}^{d} \Re(\delta_{i}) $ means $ \prod_{i=1}^{d} \{[0,T_{i}] \bigcap \Re(\delta_{i})\} $, and the uniform $ \Re(\delta_{i})= \{k \delta_{i}, k\in N\} $ is given by $$\delta_{i}\left( 2\log\prod_{i=1}^{d} T_{i}\right)^{1/\alpha_{i}} \to D_{i},\;\ i=1,2,\cdots,d.$$ and if all $ D_{i}=\infty $, the grid is sparse. The grid is called a Pickands grid if all $ D_{i}\in (0,\infty) $. Under conditions $\mathbf{A1}-\mathbf{A3}$, Tan and Wang (2015) derived the limiting distribution of $ \M_{\T} $ when the uniform grid is sparse grid, Pickands’ grid and dense grid, respectively.

In this paper, our focus is on the joint limit distributions of maxima and minima of aforementioned homogeneous Gaussian random fields. Davis (1979) established the joint limiting distribution of maxima and minima of weakly dependent stationary sequences, and weakly dependent stationary Gaussian processes was studied by Berman (1971). For the asymptotic distributions of maxima and minima on bivariate Hüsler-Reiss models, see Liao and Peng (2015) and Lu and Peng (2017).

to the definition of maxima $ \M_{\T} $, define the minima $ \m_{\T}$ as follows: $$\mathbf{m}_{\T}=\left( m_{\T}, m_{\T}^{\boldsymbol{\delta}}\right)=\left( \min_{\t\in \I_{\T}} X(\t),\min_{\t\in \I_{\T}\bigcap \prod_{i=1}^{d} \Re(\delta_{i})} X(\t) \right)$$ and let $$\begin{aligned}
\begin{cases}
b_{\T}^{\boldsymbol{\delta}}=a_{\T}+a_{\T}^{-1}\log\left((2\pi)^{-1/2}(\prod_{i=1}^{d} \delta_{i}^{-1})(a_{\T})^{-1}\right)\\
b_{\a,\T}=a_{\T}+a_{\T}^{-1}\log\left((2\pi)^{-1/2}(\prod_{i=1}^{d} H_{a_{i},\alpha_{i}})(a_{\T})^{\sum_{i=1}^{d} \frac{2}{\alpha_{i}}-1}\right)\\
b_{\T}=a_{\T}+a_{\T}^{-1}\log\left((2\pi)^{-1/2}(\prod_{i=1}^{d} H_{\alpha_{i}})(a_{\T})^{\sum_{i=1}^{d} \frac{2}{\alpha_{i}}-1}\right),
\end{cases}\end{aligned}$$ where $a_{\T}=\sqrt{2\log(\prod_{i=1}^{d} T_{i})}$ and $H_{a_{i},\alpha_{i}}=\lim_{\lambda_{i}\to\infty}\frac{H_{a_{i},\alpha_{i}}(\lambda_{i})}{\lambda_{i}}\in(0,\infty)$ with $$H_{a_{i},\alpha_{i}}(\lambda_{i})=\E\exp\left( \max_{k_{i}a_{i}\in[0,\lambda_{i}]} \sqrt{2} B_{\alpha_{i}/2}^{(i)}(k_{i}a_{i})-(k_{i}a_{i})^{\alpha_{i}}\right),\; i=1,2,\cdots,d,$$ where $B_{\alpha_{1}/2}^{(1)}(\cdot),\cdots, B_{\alpha_{d}/2}^{(d)}(\cdot)$ are independent fractional Brownian motions, cf. Piterbarg (2004) and Tan and Wang (2015). Further, let the bivariate normalizing constants $\mathbf{u}_{\T}$ and $\mathbf{v}_{\T}$ be given by $$\begin{aligned}
\label{eq4.2}
\begin{cases}
 \mathbf{u}_{\T}=\left( u_{\T}(x_{2}), u_{\T}^{\boldsymbol{\delta}}(y_{2})\right)=\left(b_{\T}+\frac{x_{2}}{a_{\T}},b_{\T}^{*}+\frac{y_{2}}{a_{\T}}\right) \\
 \mathbf{v}_{\T}=\left( v_{\T}(x_{1}), v_{\T}^{\boldsymbol{\delta}}(y_{1})\right)=\left(-b_{\T}+\frac{x_{1}}{a_{\T}},-b_{\T}^{*}+\frac{y_{1}}{a_{\T}}\right),
 \end{cases}
 \end{aligned}$$ where $ b_{\T}^{*}=b_{\T}^{\boldsymbol{\delta}}$ for sparse grids, $ b_{\T}^{*}=b_{\a,\T} $ for Pickands grids and $ b_{\T}^{*}=b_{\T} $ for dense grids.

Throughout this paper, let $ \phi(x) $ and $ \Phi(x) $ denote respectively the density function and distribution function of a standard normal random variable, and $ \Psi(x)=1-\Phi(x) $, and operations of vectors componentwise operating. For example, for vectors $ \t=(t_{1},t_{2},\cdots,t_{d}) $ and $ \s=(s_{1},s_{2},\cdots,s_{d}) $, operations of $ \s\leq \t $, $\s-\t$, $ \s \t$, and $ \s ^{\t}$ mean $ s_{i}\leq t_{i}, i=1,2,\cdots,d $, $(s_{1}-t_{1},s_{2}-t_{2},\cdots,s_{d}-t_{d})$, $ (s_{1}t_{1},s_{2}t_{2},\cdots,s_{d}t_{d}) $ , and $ (s_{1}^{t_{1}},s_{2}^{t_{2}},\cdots,s_{d}^{t_{d}}) $, respectively. Let $C$ be positive constant with values varying from place to place.

The contents of this paper are organized as follows. Section \[sec2\] presents the main results and Section \[sec3\] gives some auxiliary lemmas. The proofs of the main results will be given in Section \[sec4\].

Main results {#sec2}
============

\[th1\] Let $X(\t)$ be a centered homogenous Gaussian field with and covariance function $r(\t)$ satisfying $\mathbf{A1}-\mathbf{A3}$. Then for any sparse grids $ \Re(\delta_{i}) $, $ i=1,2,\cdots,d $, $$\begin{aligned}
&&\lim_{\T\to\infty}\P\left(\mathbf M_{\T}\leq\mathbf u_{\T},\mathbf m_{\T}\leq\mathbf v_{\T}\right)\nonumber\\
&=&\int_{-\infty}^{+\infty}\Big[1-\exp\left(-e^{x_{1}+r+\sqrt{2r}z}\right)-\exp\left(-e^{y_{1}+r+\sqrt{2r}z}\right)
+\exp\left(-e^{x_{1}+r+\sqrt{2r}z}-e^{y_{1}+r+\sqrt{2r}z}\right)\Big]\nonumber\\
&&\times\exp\Big\{-\left(e^{-x_{2}-r+\sqrt{2r}z}+e^{-y_{2}-r+\sqrt{2r}z}\right)\Big\}d\Phi(z),\end{aligned}$$ where $\mathbf{u}_{\T}$, $\mathbf{v}_{\T}$ are given by .

\[th2\] Let $X(\t)$ be a centered homogenous Gaussian field with and covariance function $r(\t)$ satisfying $\mathbf{A1}-\mathbf{A3}$ and $\mathbf{u}_{\T}$, $\mathbf{v}_{\T}$ be given by . and $$\begin{aligned}
&&\lim_{\T\to\infty}\P\left(\mathbf M_{\T}\leq\mathbf u_{\T},\mathbf m_{\T}\leq\mathbf v_{\T}\right)\\
&=&\int_{-\infty}^{+\infty}\Big[1-\exp\Big(-e^{x_{1}+r+\sqrt{2r}z}\Big)-\exp\Big(-e^{y_{1}+r+\sqrt{2r}z}\Big)
+ \exp\Big(-e^{x_{1}+r+\sqrt{2r}z}-e^{y_{1}+r+\sqrt{2r}z}\\
&&-H_{\a,\boldsymbol{\alpha}}^{-x_{1}+\log( \prod_{i=1}^{d}H_{\alpha_{i}}),-y_{1}+\log(\prod_{i=1}^{d} H_{a_{i},\alpha_{i}})}e^{r+\sqrt{2r}z}\Big)\Big]
\times \exp\Big\{-\Big(e^{-x_{2}-r+\sqrt{2r}z}+e^{-y_{2}-r+\sqrt{2r}z}\\
&&-H_{\a,\boldsymbol{\alpha}}^{x_{2}+\log( \prod_{i=1}^{d}H_{\alpha_{i}}),y_{2}+\log(\prod_{i=1}^{d} H_{a_{i},\alpha_{i}})}e^{-r+\sqrt{2r}z} \Big)\Big\}d\Phi(z).\end{aligned}$$

\[th3\] Let $X(\t)$ be a centered homogenous Gaussian field with and covariance function $r(\t)$ satisfying $\mathbf{A1}-\mathbf{A3}$. Then for any dense grids $ \Re(\delta_{i}) $, $ i=1,2,\cdots,d $, we have $$\begin{aligned}
&&\lim_{\T\to\infty}\P\left(\mathbf M_{\T}\leq\mathbf u_{\T},\mathbf m_{\T}\leq\mathbf v_{\T}\right)\\
&=&\int_{-\infty}^{+\infty}\Big[1-\exp\left(-e^{x_{1}+r+\sqrt{2r}z}\right)-\exp\left(-e^{y_{1}+r+\sqrt{2r}z}\right)
+\exp\left(-e^{\max(x_{1},y_{1})+r+\sqrt{2r}z}\right)\Big]\\
&&\times\exp\Big\{-\left(e^{-\min(x_{2},y_{2})-r+\sqrt{2r}z}\right)\Big\}d\Phi(z),\end{aligned}$$ where $\mathbf{u}_{\T}$, $\mathbf{v}_{\T}$ are given by .

Similar to the weakly dependent stationary Gaussian sequences and processes, for homogeneous Gaussian random fields, Theorems \[th1\]-\[th3\] shows that $\mathbf M_{\T}$ and $\mathbf m_{\T}$ are asymptotically independent if $r=0$.

Auxiliary results {#sec3}
=================

For simplicity, let $u=a_{\T}=\sqrt{2\log(\prod_{i=1}^{d}T_{i})} $, so for $ i=1,2,\cdots,d$, if $\delta_{i}u^{2/\alpha_{i}} \to 0$, ; if $\delta_{i}u^{2/\alpha_{i}} \to \infty$ , the grid is sparse, and $\delta_{i}u^{2/\alpha_{i}} \to D_{i}\in (0,\infty)$ for the Pickands grid.

For the sparse grids, let $ \delta_{i}=\delta_{i}(u)=l_{i}(u)u^{-2/\alpha_{i}}, i=1,2,\cdots,d $, where $ l_{i}(u)\to\infty $ as $ u\to\infty $ with $ \delta_{i}(u)\leq \delta_{0} $ for some $ \delta_{0}>0 $. Denote $ \I_{(\boldsymbol{\delta})}=\prod_{i=1}^{d}[-\delta_{i},\delta_{i}] $ and $$\label{eqpux}
P(u,x)=\P\left( X(\mathbf{0})>u, \max_{\t\in\I_{(\boldsymbol{\delta})}} X(\t)> u+\frac{v^2+x}{u}\right),$$ where $ v^{2}= (\sum_{i=1}^{d} {2}/{\alpha_{i}})\log u + \log (\prod_{i=1}^{d} \delta_{i}).$ The following Lemmas \[lem1\]-\[lem3\] extended Lemmas 1, 2 and 4 of Piterbarg (2004) from stationary Gaussian processes to homogenous Gaussian random fields.

\[lem1\] Suppose $ \Re(\delta_{i})$, $ i=1,2,\cdots,d $ are all sparse grids and the conditions $ \mathbf{A1}-\mathbf{A2} $ hold, then $P(u,x)$ given by satisfies $P(u,x)= o(\Psi(u))$ as $ u\to\infty $.

Now, define $$P_{\S}(u,x)=\P\left( \max_{\t\in\I_{\S}\cap\prod_{i=1}^{d} \Re(\delta_{i})} X(\t)> u, \max_{\t\in\I_{\S}} X(\t)> u+\frac{v^2+x}{u}\right)$$ and for small $ \epsilon>0 $ $$\begin{aligned}
\label{eq3.1}
\delta(\epsilon)=\inf_{\max\{|t_{i}|,i=1,2,\cdots,d\}>\epsilon} (1-r(\t)) >0.\end{aligned}$$

\[lem2\] Suppose $ \Re(\delta_{i})$, $ i=1,2,\cdots,d,$ are all sparse grids and the conditions $ \mathbf{A1}-\mathbf{A2} $ hold. Let $ S_{i}=S_{i}(u)\geq 2\delta_{i} $, $ i=1,2,\cdots,d $, for all $u$, and $ \prod_{i=1}^{d}S_{i}u^{\frac{2}{\alpha_{i}}}=o\left(\exp(u^{2}\delta(\epsilon))\right) $ as $ u\to\infty $. Then there exists an $ \epsilon>0 $ such that $$\begin{aligned}
\label{eq3.2}
\P\left(\max_{\t\in\I_{\S}} X(\t)> u+\frac{v^2+x}{u}\right) \sim \left(\prod_{i=1}^{d}S_{i}\delta_{i}^{-1}H_{\alpha_{i}}\right) e^{-x} \Psi(u),\end{aligned}$$

$$\begin{aligned}
\label{eq3.3}
\P\left( \max_{\t\in\I_{\S}\cap\prod_{i=1}^{d} \Re(\delta_{i})} X(\t)> u\right)\sim \left(\prod_{i=1}^{d}S_{i}\delta_{i}^{-1}\right) \Psi(u)\end{aligned}$$

as $ u\to\infty $ and $$P_{\S}(u,x)=o\left(\P\left( \max_{\t\in\I_{\S}\cap\prod_{i=1}^{d} \Re(\delta_{i})} X(\t)> u\right)+\P\left( \max_{\t\in\I_{\S}} X(\t)> u+\frac{v^2+x}{u}\right)\right)$$ as $ u\to\infty $, so that $$\begin{aligned}
\label{eq3.4}
&& 1-\P\left( \max_{\t\in\I_{\S}\cap\prod_{i=1}^{d} \Re(\delta_{i})} X(\t)\leq u, \max_{\t\in\I_{\S}} X(\t)\leq u+\frac{v^2+x}{u}\right)\nonumber\\
&&\sim\P\left( \max_{\t\in\I_{\S}\cap\prod_{i=1}^{d} \Re(\delta_{i})} X(\t)> u\right)+\P\left( \max_{\t\in\I_{\S}} X(\t)> u+\frac{v^2+x}{u}\right)\end{aligned}$$ as $ u\to\infty $.

In the following Lemma, we can prove that the maxima asymptotically coincide when the grids all are dense.

\[lem3\] Let $ S_{i}=S_{i}(u) $ with $ (\prod_{i=1}^{d} S_{i}u^{-2/\alpha_{i}})\to\infty $ and $ \prod_{i=1}^{d} S_{i}=O(\exp(\kappa u^{2})) $ with $ \kappa\in (0,1/2) $ as $ u\to\infty $. For any dense grids $ \Re(\delta_{i})=\{ a_{i}ku^{-2/\alpha_{i}},\;\  k\in N\} $, $ i=1, 2, \cdots, d$ , we have $$\begin{aligned}
\label{eq4.1}
\P\left( \max_{\t\in\I_{\S}\cap\prod_{i=1}^{d} \Re(\delta_{i})} X(\t)\leq u\right)-\P\left( \max_{\t\in\I_{\S}} X(\t)\leq u\right)\leq \rho_{\a} \left(\prod_{i=1}^{d} H_{\alpha_{i}} S_{i} u^{\frac{2}{\alpha_{i}}}\right) \Psi(u),\end{aligned}$$ where $ \rho_{\a}\to 0 $ as $ \a\to 0$.

[**Proof.**]{} This lemma follows from the Lemma D.1 and Lemma 15.3 of Piterbarg (1996), we can also find the detailed proof in the Lemma 12.3.2 of Leadbetter (1983).

\[sec3\] Following Tan and Wang (2015), define $ \rho(\T)=r/ \log(\prod_{i=1}^{d} T_{i}) $ and let $ 1>a>b>0 $ be constants. Dividing $ [0,T_{i}] $ into intervals with length $ T_{i}^{a} $ alternating with shorter intervals with length $ T_{i}^{b}$, $ i=1,2,\cdots,d $. Then the number of the long intervals is at most . Denote $ \O_{\i}=\prod_{j=1}^{d}[(i_{j}-1)(T_{j}^{a}+T_{j}^{b}), (i_{j}-1)(T_{j}^{a}+T_{j}^{b})+T_{j}^{a}] $, $ \E_{\i}=\prod_{j=1}^{d}[(i_{j}-1)(T_{j}^{a}+T_{j}^{b}), i_{j}(T_{j}^{a}+T_{j}^{b})] $, $ \i=\mathbf{1},\cdots,\mathbf{n} $ and $\O=\bigcup_{\i}\O_{\i}$. Let $ \{ X_{\i}(\t),\t\geq \mathbf{0}\} $, $ \i \geq 1 $ be independent copies of $ \{X(\t),\t\geq \mathbf{0}\} $ and $\{ \eta(\t),\t\geq \mathbf{0}\} $ be such that $ \eta(\t)=X_{\i}(\t) $ for $ \t\in \E_{\i} $.

Define $$\xi_{\T}(\t)=(1-\rho(\T))^{1/2}\eta(\t)+\rho^{1/2}(\T)U,\;\ \t \in \mathbf{I}_{\T},$$ where $ U $ is a standard normal variable independent of $ \{ \eta(\t),\t\geq 0\} $. Then $ \varrho(\s,\t) $, covariance function of $ \{\xi_{\T}(\t),\;\ \t \in \I_{\T}\} $, is $$\begin{aligned}
\varrho(\s,\t) &=& \left\{ {{\begin{array}{*{20}c}
 {r(\t,\s)+\left(1-r(\t,\s)\right)\rho(\T),\;\ \s\in\E_{\i},\;\ \t\in\E_{\j},\;\ \i=\mathbf{j};} \hfill \\
 {\rho(\T),\quad\quad\quad\quad\qquad\quad\quad\quad\quad \s\in\E_{\i},\;\ \t\in\E_{\j},\;\ \i\neq\j. } \hfill \\
\end{array} }} \right.\\\end{aligned}$$ Let $$\mathbf{ M}_{\O}=\left( M_{\O},M_{\O}^{\boldsymbol{\delta}}\right)=\left( \max_{\t\in \O} X(\t),\max_{\t\in \O \bigcap \prod_{i=1}^{d} \Re(\delta_{i})} X(\t) )\right) ,$$ $$\mathbf{ m}_{\O}=\left( m_{\O},m_{\O}^{\boldsymbol{\delta}}\right)=\left( \min_{\t\in \O} X(\t),\min_{\t\in \O \bigcap \prod_{i=1}^{d} \Re(\delta_{i})} X(\t) )\right) .$$

\[lem4\] Suppose $ \Re(\delta_{i}), i=1,2,\cdots,d $ are all sparse grids or all Pickands grids. Then for the $\mathbf{u}_{\T}$, $\mathbf{v}_{\T}$ given by we have $$\begin{aligned}
 |\P\left( \mathbf{v}_{\T}<\mathbf{m}_{\T}\leq\mathbf{M}_{\T}\leq\mathbf{u}_{\T}\right)-\P\left( \mathbf{v}_{\T}<\mathbf{m}_{\O}\leq\mathbf{M}_{\O}\leq\mathbf{u}_{\T}\right)| \to 0
 \end{aligned}$$ as $ \T\to\infty $.

[**Proof.**]{} Note that the homogeneous Gaussian fields $ \{X(\t),\t\geq \mathbf{0}\} $ and $ \{-X(\t),\t\geq \mathbf{0}\} $ have the same distribution so that $$\begin{aligned}
&&|\P\left( \mathbf{v}_{\T}<\mathbf{m}_{\T}\leq\mathbf{M}_{\T}\leq\mathbf{u}_{\T}\right)-\P\left( \mathbf{v}_{\T}<\mathbf{m}_{\O}\leq\mathbf{M}_{\O}\leq\mathbf{u}_{\T}\right)| \\
&&\leq \P\left( \max_{\t\in\I_{\T}\backslash \mathbf{O}} X(\t)> u_{\T}(x_{2})\right)+ \P\left( \max_{\t\in(\I_{\T}\backslash \mathbf{O})\cap\prod_{i=1}^{d} \Re(\delta_{i})} X(\t)> u_{\T}^{\boldsymbol{\delta}}(y_{2})\right)\\
&&+\P\left( \max_{\t\in\I_{\T}\backslash \mathbf{O}} (-X(\t))> u_{\T}(-x_{1})\right)+ \P\left( \max_{\t\in(\I_{\T}\backslash \mathbf{O})\cap\prod_{i=1}^{d} \Re(\delta_{i})} (-X(\t))> u_{\T}^{\boldsymbol{\delta}}(-y_{1})\right).\end{aligned}$$ By arguments similar to Lemma 3.1 of Tan and Wang (2015) (denote by $ mes(\cdot) $ the Lebesgue measure), we have $$\begin{aligned}
 \P\left( \max_{\t\in\I_{\T}\backslash \mathbf{O}} X(\t)> u_{\T}(x_{2})\right)= O(1) mes(\mathbf{I}_{\T}\backslash \mathbf{O})(( u_{\T}(x_{2}))^{\sum_{i=1}^{d} \frac{2}{\alpha_{i}}}\Psi( u_{\T}(x_{2}))
 \to 0
 \end{aligned}$$ as $\T\to\infty$. Combining , we can get the assertion of this lemma.

\[lem5\] Denote the grids $ \Re(q_{i})$ with $ q_{i}=\gamma_{i}u^{-\frac{2}{\alpha_{i}}} $, $ \gamma_{i}>0 $ and $ i=1,2,\cdots,d $. Assume that $\Re(\delta_{i}), i=1,2,\cdots,d,$ are all sparse grids or all Pickands grids. Then for ${\u}_{\T}$ and $\v_{\T}$ given by we have $$\begin{aligned}
\label{eq5.1}
&&\Big| \P\Big( v_{\T}(x_{1}) < \min_{\t\in \O} X(\t) \leq \max_{\t\in \O} X(\t)\leq u_{\T}(x_{2}), \nonumber\\ && \qquad v_{\T}^{\boldsymbol{\delta}}(y_{1}) < \min_{\t\in \O \cap\prod_{i=1}^{d} \Re(\delta_{i})} X(\t) \leq \max_{\t\in \O\cap\prod_{i=1}^{d} \Re(\delta_{i})} X(\t)\leq  u_{\T}^{\boldsymbol{\delta}}(y_{2}) \Big) \nonumber \\
&&\qquad\qquad-\P\Big( v_{\T}(x_{1}) < \min_{\t\in \O\cap\prod_{i=1}^{d} \Re(q_{i})} X(\t) \leq \max_{\t\in \O\cap \prod_{i=1}^{d} \Re(q_{i})} X(\t)\leq u_{\T}(x_{2}), \nonumber\\ &&\qquad\qquad\qquad v_{\T}^{\boldsymbol{\delta}}(y_{1}) < \min_{\t\in \O \cap\prod_{i=1}^{d} \Re(\delta_{i})} X(\t) \leq \max_{\t\in \O\cap\prod_{i=1}^{d} \Re(\delta_{i})} X(\t)\leq  u_{\T}^{\boldsymbol{\delta}}(y_{2})\Big) \Big| \nonumber \\
&&\to 0\end{aligned}$$ as $ \T\to\infty $ and $ \boldsymbol{\gamma}=(\gamma_{1},\gamma_{2},\cdots,\gamma_{d})\to 0 $.

[**Proof.**]{} It follows from Lemma \[lem3\] and the fact $ \{X(\t),\t\geq \mathbf{0}\} =^{d} \{-X(\t),\t\geq \mathbf{0}\} $ that can be bounded by $$\begin{aligned}
&& \P\left( \max_{\t\in \O \cap\prod_{i=1}^{d} \Re(q_{i})} X(\t)\leq u_{\T}(x_{2}), \max_{\t\in \O} X(\t) > u_{\T}(x_{2}) \right)\\
&&+ \P\left( \max_{\t\in \O\cap \prod_{i=1}^{d} \Re(q_{i})} (-X(\t))> u_{\T}(-x_{1}), \max_{\t\in \O} (-X(\t)) < u_{\T}(-x_{1}) \right)\\
&\to& 0\end{aligned}$$ as $ \T\to\infty $ and $ \boldsymbol{\gamma}=(\gamma_{1},\gamma_{2},\cdots,\gamma_{d})\to 0 $. The result follows.

\[lem6\] Suppose $ \Re(\delta_{i}), i=1,2,\cdots,d $ are all sparse grids or all Pickands grids, then for ${\u}_{\T}$ and ${\v}_{\T}$ given by we have $$\begin{aligned}
\label{eq6.1}
&&\Big|\P\Big( v_{\T}(x_{1}) < \min_{\t\in \O\cap\prod_{i=1}^{d} \Re(q_{i})} X(\t) \leq \max_{\t\in \O\cap \prod_{i=1}^{d} \Re(q_{i})} X(\t)\leq u_{\T}(x_{2}), \nonumber\\
&&\qquad v_{\T}^{\boldsymbol{\delta}}(y_{1}) < \min_{\t\in \O \cap\prod_{i=1}^{d} \Re(\delta_{i})} X(\t) \leq \max_{\t\in \O\cap\prod_{i=1}^{d} \Re(\delta_{i})} X(\t)\leq  u_{\T}^{\boldsymbol{\delta}}(y_{2})\Big) \nonumber\\
&&\qquad\qquad-\P\Big( v_{\T}(x_{1}) < \min_{\t\in \O\cap\prod_{i=1}^{d} \Re(q_{i})} \xi_{\T}(\t) \leq \max_{\t\in \O \cap\prod_{i=1}^{d} \Re(q_{i})} \xi_{\T}(\t)\leq u_{\T}(x_{2}), \nonumber\\
&&\qquad\qquad\qquad v_{\T}^{\boldsymbol{\delta}}(y_{1}) < \min_{\t\in \O \cap\prod_{i=1}^{d} \Re(\delta_{i})} \xi_{\T}(\t) \leq \max_{\t\in \O\cap\prod_{i=1}^{d} \Re(\delta_{i})} \xi_{\T}(\t)\leq  u_{\T}^{\boldsymbol{\delta}}(y_{2})) \Big| \nonumber\\
&&\to 0\end{aligned}$$ as $ \T\to\infty $.

[**Proof.**]{} It follows from the Normal Comparison Lemma that the left hand side of can be bounded by $$\begin{aligned}
&& C\sum_{\mbox{\tiny$\begin{array}{c} \k\q\in \O_{\i}, \l\q\in \O_{\j} \\ \k\q\neq \l\q, \mathbf{1}\leq \i,\j\leq \mathbf{n}\end{array}$}} | r(\k\q,\l\q)-\varrho(\k\q,\l\q)|\int_{0}^{1}\frac{1}{\sqrt{1-r^{h}(\k\q,\l\q)}}\times \\
&& \Big[\exp\left(-\frac{u_{\T}^{2}(x_{2})}{1+r^{h}(\k\q,\l\q)}\right)
+\exp\left(-\frac{v_{\T}^{2}(x_{1})}{1+r^{h}(\k\q,\l\q)}\right)\Big] dh \\
&& +C\sum_{\mbox{\tiny$\begin{array}{c} \k\boldsymbol{\delta}\in \O_{\i}, \l\boldsymbol{\delta}\in \O_{\j} \\ \k\boldsymbol{\delta}\neq \l\boldsymbol{\delta}, \mathbf{1}\leq \i,\j\leq \mathbf{n}\end{array}$}} | r(\k\boldsymbol{\delta},\l\boldsymbol{\delta})-\varrho(\k\boldsymbol{\delta},\l\boldsymbol{\delta})|\int_{0}^{1}\frac{1}{\sqrt{1-r^{h}(\k\boldsymbol{\delta},\l\boldsymbol{\delta})}}\times \\
&& \Big[\exp\left(-\frac{\left(u_{\T}^{\boldsymbol{\delta}}(y_{2})\right)^{2}}{1+r^{h}(\k\boldsymbol{\delta},\l\boldsymbol{\delta})}\right)
+\exp\left(-\frac{\left(v_{\T}^{\boldsymbol{\delta}}(y_{1})\right)^{2}}{1+r^{h}(\k\boldsymbol{\delta},\l\boldsymbol{\delta})}\right)\Big] dh \\
 && +C\sum_{\mbox{\tiny$\begin{array}{c} \k\q\in \O_{\i}, \l\boldsymbol{\delta}\in \O_{\j} \\ \k\q\neq \l\boldsymbol{\delta}, \mathbf{1}\leq \i,\j\leq \mathbf{n}\end{array}$}} | r(\k\q,\l\boldsymbol{\delta})-\varrho(\k\q,\l\boldsymbol{\delta})|\int_{0}^{1}\frac{1}{\sqrt{1-r^{h}(\k\q,\l\boldsymbol{\delta})}}\times \\
&& \Big[\exp\left(-\frac{u_{\T}^{2}(x_{2})+\left(u_{\T}^{\boldsymbol{\delta}}(y_{2})\right)^{2}}{2\left(1+r^{h}(\k\q,\l\boldsymbol{\delta})\right)}\right)
+\exp\left(-\frac{v_{\T}^{2}(x_{1})+\left(v_{\T}^{\boldsymbol{\delta}}(y_{1})\right)^{2}}{2\left(1+r^{h}(\k\q,\l\boldsymbol{\delta})\right)}\right)\\
&&+\exp\left(-\frac{v_{\T}^{2}(x_{1})+\left(u_{\T}^{\boldsymbol{\delta}}(y_{2})\right)^{2}}{2\left(1+r^{h}(\k\q,\l\boldsymbol{\delta})\right)}\right)
+\exp\left(-\frac{u_{\T}^{2}(x_{2})+\left(v_{\T}^{\boldsymbol{\delta}}(y_{1})\right)^{2}}{2\left(1+r^{h}(\k\q,\l\boldsymbol{\delta})\right)}\right)\Big] dh\\
&&:= F_{1}+F_{2}+F_{3},\end{aligned}$$ where with $ h\in [0,1]$.

Next we will show that $ F_{1}\to 0 $, $ F_{2}\to 0 $ and $ F_{3}\to\ 0 $ as $ \T\to \infty $, respectively. For $ F_{1} $, we first consider the case that $ \k\q $, $ \l\q $ in the same interval firstly, and split $ F_{1} $ into the following two parts: $$\begin{aligned}
\label{eq6.2}
\sum_{\mbox{\tiny$\begin{array}{c} \k\q, \l\q \in \O_{\i}, \k\q\neq\l\q, \i=\mathbf{1,2,\cdots,n} \\ \max\{|l_{j}q_{j}-k_{j}q_{j}|,j=1,2,\cdots,d\}\leq \epsilon\end{array}$}} + \sum_{\mbox{\tiny$\begin{array}{c} \k\q, \l\q \in \O_{\i}, \k\q\neq\l\q, \i=\mathbf{1,2,\cdots,n} \\ \max\{|l_{j}q_{j}-k_{j}q_{j}|,j=1,2,\cdots,d\} > \epsilon\end{array}$}} := J_{\T,1} +J_{\T,2}.\end{aligned}$$ Following condition $ \mathbf{A1} $, we can choose small enough $ \epsilon >0 $ such that $ \max\{|l_{j}q_{j}-k_{j}q_{j}|,j=1,2,\cdots,d\}\leq \epsilon $ and for all $  |t_{i}|\leq \epsilon < 2^{-1/\alpha_{i}} $, $$\begin{aligned}
\label{eq6.3}
\frac{1}{2} \left(\sum_{i=1}^{d} |t_{i}|^{\alpha_{i}}\right)\leq 1-r(\t) \leq 2 \left(\sum_{i=1}^{d} |t_{i}|^{\alpha_{i}}\right),\end{aligned}$$ then, by definition of $ \xi_{\T}(\t) $, we have $ \varrho(\k\q,\l\q)-r(\k\q,\l\q)=\rho(\T)\left( 1-r(\k\q,\l\q)\right) $ and $ \varrho(\k\q,\l\q) \sim r(\k\q,\l\q) $ for sufficiently large $ \T $. With $v_{\T}$ and $u_{\T}$ given by we have $$\begin{aligned}
\label{eq6.4}
v_{\T}^{2}(x_{1})=u^{2}-\left[1-\left(\sum_{i=1}^{d} \frac{2}{\alpha_{i}}\right)\right]\log\left(\frac{u^{2}}{2}\right)+O(1),\quad
u_{\T}^{2}(x_{2})=u^{2}-\left[1-\left(\sum_{i=1}^{d} \frac{2}{\alpha_{i}}\right)\right]\log\left(\frac{u^{2}}{2}\right)+O(1).\end{aligned}$$ Hence, $$\begin{aligned}
\label{eq6.5}
J_{\T,1}&\leq& C \sum_{\mbox{\tiny$\begin{array}{c} \k\q, \l\q \in \O_{\i}, \k\q\neq\l\q, \i=\mathbf{1,2,\cdots,n} \\ \max\{|l_{j}q_{j}-k_{j}q_{j}|,j=1,2,\cdots,d\}\leq \epsilon\end{array}$}} | r(\k\q,\l\q)-\varrho(\k\q,\l\q)| \frac{1}{\sqrt{1-r(\k\q,\l\q)}}
\exp\left(-\frac{u_{\T}^{2}(x_{2})}{1+r(\k\q,\l\q)}\right) \nonumber\\
&\leq & C (\prod_{i=1}^{d} T_{i}q_{i}^{-1}) \rho(\T) \sum_{j=1}^{d}  \sum_{ 0\leq k_{j}q_{j} \leq \epsilon} \sqrt{1-r(\k\q)}  \exp\left(-\frac{u_{\T}^{2}(x_{2})}{1+r(\k\q)}\right) \nonumber\\
&\leq & C (\prod_{i=1}^{d} T_{i}q_{i}^{-1}) \rho(\T)\exp(-\frac{u_{\T}^{2}(x_{2})}{2})\sum_{j=1}^{d}  \sum_{ 0\leq k_{j}q_{j} \leq \epsilon} \sqrt{1-r(\k\q)}  \exp\left(-\frac{(1-r(\k\q))u_{\T}^{2}(x_{2})}{2(1+r(\k\q))}\right) \nonumber\\
&\leq & C u^{-1} \sum_{j=1}^{d}  \sum_{ 0\leq k_{j}q_{j} \leq \epsilon} \left[\sum_{i=1}^{d} (k_{i}q_{i})^{\alpha_{i}}\right]^{1/2}\exp\left(-\frac{1}{4}\left[\sum_{i=1}^{d} (k_{i}q_{i})^{\alpha_{i}}\right]  \left(\log \prod_{i=1}^{d} T_{i}\right) \right) \nonumber\\
&\leq & C u^{-1} \sum_{j=1}^{d}  \sum_{ 0\leq k_{j}q_{j} \leq \epsilon} \exp\left(-\frac{1}{4}\left[\sum_{i=1}^{d} (k_{i}q_{i})^{\alpha_{i}}\right]  \left(\log \prod_{i=1}^{d} T_{i}\right) \right) \nonumber\\
&\leq & C u^{-1} \sum_{j=1}^{d} \sum_{k_{j}=0}^{\infty} e^{-\frac{1}{4}(k_{j}\gamma_{j})^{\alpha_{j}}} \nonumber\\
&\leq & C u^{-1}\to0\end{aligned}$$ as $ u\to\infty $.

Let $\varpi(\t,\s)=\max\{ |r(\t,\s)|,|\varrho(\t,\s)| \} $ and $$\theta(z)=\sup_{\mbox{\tiny$\begin{array}{c} \mathbf{0}\leq \s,\t\leq \T \\ \max\{|s_{i}-t_{i}|,j=1,2,\cdots,d\}> z\end{array}$}}\{ \varpi(\t,\s) \}.$$ For $ J_{\T,2} $, by the fact that $ u_{\T}^{2}(x_{2})\sim 2 \log \prod_{i=1}^{d} T_{i}=u^{2}$ we have $$\begin{aligned}
\label{eq6.6}
J_{\T,2}&\leq& C \sum_{\mbox{\tiny$\begin{array}{c} \k\q, \l\q \in \O_{\i}, \k\q\neq\l\q, \i=\mathbf{1,2,\cdots,n} \\ \max\{|l_{j}q_{j}-k_{j}q_{j}|,j=1,2,\cdots,d\}> \epsilon\end{array}$}} | r(\k\q,\l\q)-\varrho(\k\q,\l\q)| \exp\left(-\frac{u_{\T}^{2}(x_{2})}{1+\varpi(\k\q,\l\q)}\right)\nonumber\\
&\leq & C (\prod_{i=1}^{d} T_{i}q_{i}^{-1}) \sum_{j=1}^{d} \sum_{\mbox{\tiny$\begin{array}{c} 0\leq k_{j}q_{j}\leq T_{j}^{a} \\ \max\{k_{j}q_{j},j=1,2,\cdots,d\}> \epsilon\end{array}$}} \exp\left(-\frac{u_{\T}^{2}(x_{2})}{1+\theta(\epsilon)}\right)   \nonumber\\
&\leq & C \exp\left[\left(a-\frac{1-\theta(\epsilon)}{1+\theta(\epsilon)}\right)\frac{u^{2}}{2}+\sum_{i=1}^{d} \frac{4}{\alpha_{i}}\log u\right]\to0\end{aligned}$$ as $ u\to\infty $ since $ a<\frac{1-\theta(\epsilon)}{1+\theta(\epsilon)} $. Combining with , and , we show that $F_1\to0$ for the first case that $ \k\q $, $ \l\q $ in the same interval.

Second, we consider the case that $ \k\q\in \O_{\i} $, $ \l\q\in \O_{\j} $, $ \i\neq \j $. Note that the distance between the points in any two rectangles $ \O_{\i} $ and $ \O_{\j} $ are larger than $ \min\{ T_{i}^{b}, i=1,2,\cdots,d \} $ and $ \varrho(\k\q,\l\q)=\rho(\T) $ for $ \k\q\in \O_{\i} $, $ \l\q\in \O_{\j} $, $ \i\neq \j $. Then, $ F_{1} $ can be bounded by $$\begin{aligned}
\label{eq6.7}
C \sum_{\mbox{\tiny$\begin{array}{c} \k\q\in \O_{\i},\l\q\in \O_{\j} \\ \mathbf{1\leq i\neq j\leq n}\end{array}$}} | r(\k\q,\l\q)-\rho(\T)| \exp\left(-\frac{u_{\T}^{2}(x_{2})}{1+\varpi(\k\q,\l\q)}\right).\end{aligned}$$ Split into two parts, the first for $ \min\{|k_{j}q_{j}-l_{j}q_{j}|,j=1,2,\cdots,d\}> 0 $, the second for $ \min\{|k_{j}q_{j}-l_{j}q_{j}|,j=1,2,\cdots,d\}= 0 $ and denote them $ S_{\T,1} $, $ S_{\T,2} $, respectively. Let $ \beta $ be such that $ 0<b<a<\beta< \frac{1-\theta(\epsilon)}{1+\theta(\epsilon)} $ for all sufficiently large $ \T $.

For $ S_{\T,1} $, we can also divide it into the following two parts: $$S_{\T,1}=C\sum_{\mbox{\tiny$\begin{array}{c} \k\q\in \O_{\i},\l\q\in \O_{\j},\mathbf{1\leq i\neq j\leq n} \\ \prod_{m=1}^{d}|k_{m}q_{m}-l_{m}q_{m}|\leq {\prod_{i=1}^{d} T_{i}}^{\beta}\end{array}$}}+C\sum_{\mbox{\tiny$\begin{array}{c} \k\q\in \O_{\i},\l\q\in \O_{\j},\mathbf{1\leq i\neq j\leq n} \\ \prod_{m=1}^{d}|k_{m}q_{m}-l_{m}q_{m}|> \prod_{i=1}^{d} T_{i}^{\beta}\end{array}$}} :=S_{\T,11}+S_{\T,12} ,$$ then, by the same arguments as used in , we have $$\begin{aligned}
\label{eq6.8}
S_{\T,11}&\leq & C (\prod_{i=1}^{d} T_{i}q_{i}^{-1}) \sum_{j=1}^{d} \sum_{\mbox{\tiny$\begin{array}{c} 0\leq k_{j}q_{j}\leq T_{j} \\ \prod_{m=1}^{d}k_{m}q_{m}\leq {\prod_{i=1}^{d} T_{i}}^{\beta} \end{array}$}} \exp\left(-\frac{u_{\T}^{2}(x_{2})}{1+\theta(\epsilon)}\right)   \nonumber\\
&\leq & C \exp\left[\left(\beta-\frac{1-\theta(\epsilon)}{1+\theta(\epsilon)}\right)\frac{u^{2}}{2}+\sum_{i=1}^{d} \frac{4}{\alpha_{i}}\log u\right],\nonumber\\
&\to& 0\end{aligned}$$ as $u\to\infty  $ since $\beta< \frac{1-\theta(\epsilon)}{1+\theta(\epsilon)}$.

To deal with $ S_{\T,12} $, we can define $ w_{1}(\t)=\max\{ |r(\t)|,|\rho(\T)| \}$ and $$\theta_{1}(\mathbf{z})=\sup_{\mbox{\tiny$\begin{array}{c} \mathbf{0}\leq \t\leq \T \\ \prod_{i=1}^{d}|t_{i}|> \prod_{i=1}^{d} z_{i} \end{array}$}}\{w_{1}(\t)\}.$$ Denote $ \T^{\beta}=(T_{1}^{\beta},T_{1}^{\beta},\cdots,T_{d}^{\beta}) $ and by the condition $ \mathbf{A3} $, we have $ \theta_{1}(\T^{\beta}) \leq Cu^{-2}$ for sufficiently large $ \T $ and $ \prod_{j=1}^{d} k_{j}q_{j} >{\prod_{i=1}^{d} T_{i}}^{\beta}=e^{\beta u^{2}/2}$, then, using , we have $$\begin{aligned}
\label{eq6.9}
&&\frac{(\prod_{i=1}^{d} T_{i}q_{i}^{-1})^{2}}{\log\left(\prod_{i=1}^{d} T_{i}\right)}\exp\left(-\frac{u_{\T}^{2}(x_{2})}{1+\theta_{1}(\T^{\beta})}\right)\nonumber\\
&\le& O(1)\exp\left[\frac{C}{C/u^{2}+\beta/2}+2C\left(\sum_{i=1}^{d}2/\alpha_{i}-1\right)\frac{\log u}{C+\beta u^{2}/2}\right]=O(1)\end{aligned}$$ as $ u\to\infty $. Therefore, we have $$\begin{aligned}
\label{eq6.10}
S_{\T,12}&\leq& C (\prod_{i=1}^{d} T_{i}q_{i}^{-1}) \sum_{\mbox{\tiny$\begin{array}{c} \mathbf{0}\leq \k\q \leq \T \\ \prod_{m=1}^{d}k_{m}q_{m}> {\prod_{i=1}^{d} T_{i}}^{\beta} \end{array}$}} |r(\k\q)-\rho(\T) |\exp\left(-\frac{u_{\T}^{2}(x_{2})}{1+\theta_{1}(\T^{\beta})}\right)   \nonumber\\
&\leq & C\frac{(\prod_{i=1}^{d} T_{i}q_{i}^{-1})^{2}}{\log\left(\prod_{i=1}^{d} T_{i}\right)}\exp\left(-\frac{u_{\T}^{2}(x_{2})}{1+\theta_{1}(\T^{\beta})}\right) \frac{\log\left(\prod_{i=1}^{d} T_{i}\right)}{(\prod_{i=1}^{d} T_{i}q_{i}^{-1})} \sum_{\mbox{\tiny$\begin{array}{c} \mathbf{0}\leq \k\q \leq \T \\ \prod_{m=1}^{d}k_{m}q_{m}> {\prod_{i=1}^{d} T_{i}}^{\beta} \end{array}$}} |r(\k\q)-\rho(\T) | \nonumber\\
&\leq & C \left(\prod_{i=1}^{d} q_{i} T_{i}^{-1}\right) \sum_{\mbox{\tiny$\begin{array}{c} \mathbf{0}\leq \k\q \leq \T \\ \prod_{m=1}^{d}k_{m}q_{m}> {\prod_{i=1}^{d} T_{i}}^{\beta} \end{array}$}} \left|r(\k\q)\log\left(\prod_{j=1}^{d} k_{j}q_{j}\right)-r \right| \nonumber\\
&+&  C \left(\prod_{i=1}^{d} q_{i} T_{i}^{-1}\right) \sum_{\mbox{\tiny$\begin{array}{c} \mathbf{0}\leq \k\q \leq \T \\ \prod_{m=1}^{d}k_{m}q_{m}> {\prod_{i=1}^{d} T_{i}}^{\beta} \end{array}$}} \left| 1-\frac{\log\prod_{i=1}^{d} T_{i}}{\log\prod_{j=1}^{d} k_{j}q_{j}} \right|.\end{aligned}$$ Note that $$\begin{aligned}
\label{eq6.11}
&&\left(\prod_{i=1}^{d} q_{i} T_{i}^{-1}\right) \sum_{\mbox{\tiny$\begin{array}{c} \mathbf{0}\leq \k\q \leq \T \\ \prod_{m=1}^{d}k_{m}q_{m}> {\prod_{i=1}^{d} T_{i}}^{\beta} \end{array}$}} \left| 1-\frac{\log\prod_{i=}^{d} T_{i}}{\log\prod_{j=1}^{d} k_{j}q_{j}} \right| \nonumber\\
&\leq & \frac{\prod_{i=1}^{d} q_{i} T_{i}^{-1}}{\log\prod_{i=}^{d} T_{i}^{\beta}} \sum \left|\log\left( \frac{\prod_{i=1}^{d} T_{i}}{\prod_{j=1}^{d} k_{j}q_{j}}\right)\right| \nonumber\\
&=& O\left( {u^{-2}}\int_{0}^{1}\cdots\int_{0}^{1} |\log\prod_{i=1}^{d} x_{i}| dx_{1}\cdots dx_{d}\right)\nonumber\\
&\to& 0\end{aligned}$$ as $ u\to\infty $. Combining condition $ \mathbf{A3} $, and -, we can get $ S_{\T,1}\to 0 $ as $ \T\to\infty $.

Now, for $ S_{\T,2} $, we only prove the case that $ k_{1}q_{1}= l_{1}q_{1} $, and $ \min\{|k_{j}q_{j}-l_{j}q_{j}|,j=2,\cdots,d\}> 0  $. Other cases can be proved by the similar arguments. If $ \prod_{i=2}^{d} T_{i}\leq \prod_{i=1}^{d} T_{i}^{\beta}=e^{\beta u^{2}/2} $, where $ \beta< \frac{1-\theta(\epsilon)}{1+\theta(\epsilon)} $, by $$\begin{aligned}
S_{\T,2}&\leq & C (\prod_{i=1}^{d} T_{i}q_{i}^{-1})\sum_{j=2}^{d} \sum_{\mbox{\tiny$\begin{array}{c} 0\leq k_{j}q_{j}\leq T_{j} \\ k_{1}q_{1}=0\end{array}$}} |r(\k\q)-\rho(\T) |\exp\left(-\frac{u_{\T}^{2}(x_{2})}{1+\theta(\epsilon)}\right)   \nonumber\\
&\leq & C \exp\left[\left(\beta- \frac{1-\theta(\epsilon)}{1+\theta(\epsilon)}\right)u^{2}/2+ \left(\frac{1}{\alpha_{1}}+\sum_{i=2}^{d} \frac{2}{\alpha_{i}}\right)\log (u^{2}/2)\right] \nonumber\\
&\to& 0\end{aligned}$$ as $u\to\infty$. If $ \prod_{i=2}^{d} T_{i}> \prod_{i=1}^{d} T_{i}^{\beta}=e^{\beta u^{2}/2} $, split $ S_{\T,2} $ as follows: $$S_{\T,2}=C\sum_{\mbox{\tiny$\begin{array}{c} \k\q\in \O_{\i},\l\q\in \O_{\j},\mathbf{1\leq i\neq j\leq n} \\ \prod_{m=2}^{d}|k_{m}q_{m}-l_{m}q_{m}|\leq {\prod_{i=1}^{d} T_{i}}^{\beta}\\ k_{1}=l_{1}\end{array}$}}+C\sum_{\mbox{\tiny$\begin{array}{c} \k\q\in \O_{\i},\l\q\in \O_{\j},\mathbf{1\leq i\neq j\leq n} \\ \prod_{m=2}^{d}|k_{m}q_{m}-l_{m}q_{m}|> \prod_{i=1}^{d} T_{i}^{\beta}\\ k_{1}=l_{1}\end{array}$}} :=S_{\T,21}+S_{\T,22}.$$ For $ S_{\T,21} $, similar to , we have $$\begin{aligned}
S_{\T,21}&\leq & C (\prod_{i=1}^{d} T_{i}q_{i}^{-1}) \sum_{\mbox{\tiny$\begin{array}{c} \prod_{j=2}^{d} k_{j}q_{j}\leq \prod_{i=1}^{d}T_{i}^{\beta} \\ k_{1}q_{1}=0\end{array}$}} |r(\k\q)-\rho(\T) |\exp\left(-\frac{u_{\T}^{2}(x_{2})}{1+\theta(\epsilon)}\right)   \nonumber\\
&\leq & C \exp\left[\left(\beta- \frac{1-\theta(\epsilon)}{1+\theta(\epsilon)}\right)u^{2}/2+\left(\frac{1}{\alpha_{1}}+\sum_{i=2}^{d} \frac{2}{\alpha_{i}}\right)\log u^{2}/2\right] \nonumber\\
&\to& 0\end{aligned}$$ as $ u\to\infty $. To deal with $ S_{\T,22} $, let $ w_{2}(\t)=\max\{ |r(0,t_{2},\cdots,t_{d})|,|\rho(\T)| \}$ and $$\theta_{2}(\mathbf{z})=\sup_{\mbox{\tiny$\begin{array}{c} \mathbf{0}\leq \t\leq \T \\ \prod_{i=2}^{d}|t_{i}|> \prod_{i=1}^{d} z_{i} \end{array}$}}\{w_{2}(\t)\}.$$ By condition $ \mathbf{A3} $, we have $ \theta_{2}(\T^{\beta}) \leq C u^{-2} $ for sufficiently large $\T $ and $ \prod_{j=2}^{d} k_{j}q_{j} > \prod_{i=1}^{d} T_{i}^{\beta}=e^{\beta u^{2}/2} $, then, by , $$\begin{aligned}
\frac{(\prod_{i=1}^{d} T_{i}q_{i}^{-1})^{2}}{\log\left(\prod_{i=1}^{d} T_{i}\right)}\exp\left(-\frac{u_{\T}^{2}(x_{2})}{1+\theta_{2}(\T^{\beta})}\right)=O(1),\end{aligned}$$ thus we have $$\begin{aligned}
\label{eq1}
S_{\T,22}&\leq & C (\prod_{i=1}^{d} q_{i}T_{i}^{-1})\log\left(\prod_{i=1}^{d} T_{i}\right) \sum_{\mbox{\tiny$\begin{array}{c} \prod_{j=2}^{d} k_{j}q_{j}> \prod_{i=1}^{d}T_{i}^{\beta} \\ k_{1}q_{1}=0\end{array}$}} \left| r(0,k_{2}q_{2},\cdots,k_{d}q_{d})-\rho(\T)  \right|\nonumber\\
&\leq & C (\prod_{i=1}^{d} q_{i}T_{i}^{-1})\log\left(\prod_{i=1}^{d} T_{i}\right) \sum_{\prod_{i=2}^{d} T_{i}\geq\prod_{j=2}^{d} k_{j}q_{j}> \prod_{i=1}^{d}T_{i}^{\beta}} \left| r(0,k_{2}q_{2},\cdots,k_{d}q_{d})+\rho(\T)  \right| \nonumber\\
&\leq& C\frac{q_{1}}{T_{1}},\end{aligned}$$ which implies $ S_{\T,22}\to 0 $ as $ \T\to\infty $. Therefore, we showed that $ S_{\T,2}\to 0 $ as $ \T\to\infty $. Combining with $ S_{\T,1}\to0$, we showed that $ F_{1}\to 0 $ as $ \T\to\infty $ for the second case.

Arguments similar to the proof of $F_{1}\to 0$, we can show that $ F_{2}\to 0 $ as $ \T\to\infty $. Details are omitted here. The reminder is to show that $ F_{3}\to 0 $.

If $ \k\q $, $ \l\boldsymbol{\delta} $ in the same interval $ \O_{\i} $, split $ F_{3} $ into two parts as $$\begin{aligned}
\label{eq6.12}
\sum_{\mbox{\tiny$\begin{array}{c} \k\q, \l\boldsymbol{\delta}\in \O_{\i}, \k\q\neq\l\boldsymbol{\delta}, \i=\mathbf{1,2,\cdots,n} \\ \max\{|l_{j}\delta_{j}-k_{j}q_{j}|,j=1,2,\cdots,d\}\leq \epsilon\end{array}$}} + \sum_{\mbox{\tiny$\begin{array}{c} \k\q, \l\boldsymbol{\delta} \in \O_{\i}, \k\q\neq\l\boldsymbol{\delta}, \i=\mathbf{1,2,\cdots,n} \\ \max\{|l_{j}\delta_{j}-k_{j}q_{j}|,j=1,2,\cdots,d\} > \epsilon\end{array}$}} := W_{\T,1} +W_{\T,2}.\end{aligned}$$ Following condition $\mathbf{A1}$, we can choose small enough $ \epsilon >0 $ and $ \max\{|l_{j}\delta_{j}-k_{j}q_{j}|,j=1,2,\cdots,d\}\leq \epsilon $. By definition of $ \xi_{\T}(\t) $, we have $ \varrho(\k\q,\l\boldsymbol{\delta})=r(\k\q,\l\boldsymbol{\delta})+\rho(\T)\left( 1-r(\k\q,\l\boldsymbol{\delta})\right) \sim r(\k\q,\l\boldsymbol{\delta}) $ for sufficient large $ \T $. If $ \Re(\delta_{i}), i=1,2,\cdots,d $ are all Pickands grids, by the arguments similar to the proof of $ F_{1} $, we can show $F_{3}\to0$. If $ \Re(\delta_{i}), i=1,2,\cdots,d $ are all sparse grids, it follows from that $$\begin{aligned}
 u_{\T}^{2}:&=&\frac{1}{2}\left(u_{\T}^{2}(x_{2})+(u_{\T}^{\boldsymbol{\delta}}(y_{2}))^{2}\right)=u^{2}-\left(1-\sum_{i=1}^{d} \frac{1}{\alpha_{i}}\right)\log (u^{2}/2)+ \log(\prod_{i=1}^{d} \delta_{i}^{-1})+O(1).
 \end{aligned}$$ Similarly for $ \frac{1}{2}\left(v_{\T}^{2}(x_{1})+(v_{\T}^{\boldsymbol{\delta}}(y_{1}))^{2}\right) $, $ \frac{1}{2}\left(u_{\T}^{2}(x_{2})+(v_{\T}^{\boldsymbol{\delta}}(y_{1}))^{2}\right) $ and $ \frac{1}{2}\left(v_{\T}^{2}(x_{1})+(u_{\T}^{\boldsymbol{\delta}}(y_{2}))^{2}\right) $. In view of , we have $$\begin{aligned}
 \label{eq6.13}
 W_{\T,1}&\leq&C \sum_{\mbox{\tiny$\begin{array}{c} \k\q, \l\boldsymbol{\delta} \in \O_{\i}, \k\q\neq\l\boldsymbol{\delta}, \i=\mathbf{1,2,\cdots,n} \\ \max\{|l_{j}\delta_{j}-k_{j}q_{j}|,j=1,2,\cdots,d\}\leq \epsilon\end{array}$}}  |r(\k\q,\l\boldsymbol{\delta})-\varrho(\k\q,\l\boldsymbol{\delta})| \frac{1}{\sqrt{1-r(\k\q,\l\boldsymbol{\delta})}}
\exp\left(-\frac{u_{\T}^{2}}{1+r(\k\q,\l\boldsymbol{\delta})}\right) \nonumber\\
&\leq & C  \rho(\T)   \exp\left(-\frac{u_{\T}^{2}}{2}\right)
 \sum_{\mbox{\tiny$\begin{array}{c} \k\q, \l\boldsymbol{\delta} \in \O_{\i}, \k\q\neq\l\boldsymbol{\delta}, \i=\mathbf{1,2,\cdots,n} \\ \max\{|l_{j}\delta_{j}-k_{j}q_{j}|,j=1,2,\cdots,d\}\leq \epsilon\end{array}$}} \sqrt{1-r(\k\q,\l\boldsymbol{\delta})}  \exp\left(-\frac{(1-r(\k\q,\l\boldsymbol{\delta}))u_{\T}^{2}}{2(1+r(\k\q,\l\boldsymbol{\delta}))}\right) \nonumber\\
&\leq& C (\prod_{i=1}^{d} T_{i}^{-1}\delta_{i}^{1/2})\left(\log \prod_{i=1}^{d} T_{i}\right)^{-\frac{1}{2}-\sum_{i=1}^{d} \frac{1}{2\alpha_{i}}}\nonumber\\
&\times&\sum_{\mbox{\tiny$\begin{array}{c} \k\q, \l\boldsymbol{\delta} \in \O_{\i}, \k\q\neq\l\boldsymbol{\delta}, \i=\mathbf{1,2,\cdots,n} \\ \max\{|l_{j}\delta_{j}-k_{j}q_{j}|,j=1,2,\cdots,d\}\leq \epsilon\end{array}$}} \sqrt{\sum_{j=1}^{d}|l_{j}\delta_{j}-k_{j}q_{j}|^{\alpha_{j}}}  \exp\left(-\frac{u_{\T}^{2}}{8}\left(\sum_{j=1}^{d}|l_{j}\delta_{j}-k_{j}q_{j}|^{\alpha_{j}}\right)\right).\end{aligned}$$ Recall that $ q_{i}=\gamma_{i}u^{-2/\alpha_{i}}$, and $ \Re(\delta_{i}) $, $ i=1,2,\cdots,d $ are sparse grids, then after some calculation, can be bounded by $$\begin{aligned}
\label{eq6.14}
&& C (\prod_{i=1}^{d} T_{i}^{-1}\delta_{i}^{1/2})\left(\log \prod_{i=1}^{d} T_{i}\right)^{-\frac{1}{2}-\sum_{i=1}^{d} \frac{1}{2\alpha_{i}}}\left(\prod_{i=1}^{d} T_{i} \delta_{i}^{-1}\right)\nonumber\\
&\times&\sum_{j=1}^{d}  \sum_{ 0\leq k_{j}q_{j} \leq \epsilon} \exp\left(-\frac{1}{4}\left[\sum_{i=1}^{d} (k_{i}q_{i})^{\alpha_{i}}\right]  \left(\log \prod_{i=1}^{d} T_{i}\right) \right) \nonumber\\
&\leq& Cu^{-1}\left(\prod_{i=1}^{d}\delta_{i}^{-1/2}u^{-1/\alpha_{i}}\right) \sum_{j=1}^{d} \sum_{k_{j}=0}^{\infty} e^{-\frac{1}{4}(k_{j}\gamma_{j})^{\alpha_{j}}}
\to 0\end{aligned}$$ as $ u\to\infty $.

Noting that $ u_{\T}\sim \left(2\log\prod_{i=1}^{d} T_{i}\right)^{1/2}=u $, we have $$\begin{aligned}
\label{eq6.15}
W_{\T,2}&\leq& C \sum_{\mbox{\tiny$\begin{array}{c} \k\q, \l\boldsymbol{\delta} \in \O_{\i}, \k\q\neq\l\boldsymbol{\delta}, \i=\mathbf{1,2,\cdots,n} \\ \max\{|l_{j}\delta_{j}-k_{j}q_{j}|,j=1,2,\cdots,d\}> \epsilon\end{array}$}} | r(\k\q,\l\boldsymbol{\delta})-\varrho(\k\q,\l\boldsymbol{\delta})| \exp\left(-\frac{u_{\T}^{2}}{1+\varpi(\k\q,\l\boldsymbol{\delta})}\right)\nonumber\\
&\leq & C (\prod_{i=1}^{d} T_{i}\delta_{i}^{-1}) \sum_{j=1}^{d} \sum_{\mbox{\tiny$\begin{array}{c} 0\leq k_{j}q_{j}\leq T_{j}^{a} \\ \max\{k_{j}q_{j},j=1,2,\cdots,d\}> \epsilon\end{array}$}} \exp\left(-\frac{u_{\T}^{2}}{1+\theta(\epsilon)}\right)   \nonumber\\
&\leq & C \left(\prod_{i=1}^{d} \delta_{i}^{-1}u^{-2/\alpha_{i}}\right)\exp\left[(a-\frac{1-\theta(\epsilon)}{1+\theta(\epsilon)})u^{2}/2+{\sum_{i=1}^{d} \frac{4}{\alpha_{i}}}\log u\right]\to 0\end{aligned}$$ as $ u\to\infty $.

Next, we consider the case that $ \k\q\in \O_{\i} $, $ \l\boldsymbol{\delta}\in \O_{\j} $, $ \i\neq \j $. Note that the distance between the points in any two rectangles $ \O_{\i} $ and $ \O_{\j} $ is larger than $ \min\{ T_{i}^{b}, i=1,2,\cdots,d \} $ and $ \varrho(\k\q,\l\boldsymbol{\delta})=\rho(\T)$. Then, $ F_{3} $ is at most $$\begin{aligned}
\label{eq6.16}
C \sum_{\mbox{\tiny$\begin{array}{c} \k\q\in \O_{\i},\l\boldsymbol{\delta}\in \O_{\j} \\ \mathbf{1\leq i\neq j\leq n}\end{array}$}} | r(\k\q,\l\boldsymbol{\delta})-\rho(\T)| \exp\left(-\frac{u_{\T}^{2}}{1+\varpi(\k\q,\l\boldsymbol{\delta})}\right).\end{aligned}$$ Split into two parts, the first for $ \min\{|k_{j}q_{j}-l_{j}\delta_{j}|,j=1,2,\cdots,d\}> 0 $, and the second for $ \min\{|k_{j}q_{j}-l_{j}\delta_{j}|,j=1,2,\cdots,d\}= 0 $ and denote them as $ H_{\T,1} $, $ H_{\T,2} $, respectively. Then, $$H_{\T,1}=C\sum_{\mbox{\tiny$\begin{array}{c} \k\q\in \O_{\i},\l\boldsymbol{\delta}\in \O_{\j},\mathbf{1\leq i\neq j\leq n} \\ \prod_{m=1}^{d}|k_{m}q_{m}-l_{m}\delta_{m}|\leq {\prod_{i=1}^{d} T_{i}}^{\beta}\end{array}$}}+C\sum_{\mbox{\tiny$\begin{array}{c} \k\q\in \O_{\i},\l\boldsymbol{\delta}\in \O_{\j},\mathbf{1\leq i\neq j\leq n} \\ \prod_{m=1}^{d}|k_{m}q_{m}-l_{m}q_{m}|> \prod_{i=1}^{d} T_{i}^{\beta}\end{array}$}} :=H_{\T,11}+H_{\T,12} .$$ , we have $$\begin{aligned}
\label{eq6.17}
H_{\T,11}&\leq & C (\prod_{i=1}^{d} T_{i}\delta_{i}^{-1}) \sum_{j=1}^{d} \sum_{\mbox{\tiny$\begin{array}{c} 0\leq k_{j}q_{j}\leq T_{j} \\ \prod_{m=1}^{d} k_{m}q_{m}\leq {\prod_{i=1}^{d} T_{i}}^{\beta} \end{array}$}} \exp\left(-\frac{u_{\T}^{2}}{1+\theta(\epsilon)}\right)   \nonumber\\
&\leq &  C \left(\prod_{i=1}^{d} \delta_{i}^{-1}u^{-2/\alpha_{i}}\right)\exp\left[(\beta-\frac{1-\theta(\epsilon)}{1+\theta(\epsilon)})u^{2}/2+{\sum_{i=1}^{d} \frac{4}{\alpha_{i}}}\log u\right]\nonumber\\
&\to& 0\end{aligned}$$ as $u\to\infty  $.

For $ H_{\T,12} $, by , we have $$\begin{aligned}
&&\frac{(\prod_{i=1}^{d} T_{i}^{2}q_{i}^{-1} \delta_{i}^{-1})}{\log\left(\prod_{i=1}^{d} T_{i}\right)}\exp\left(-\frac{u_{\T}^{2}}{1+\theta_{1}(\T^{\beta})}\right)\nonumber\\
&\leq& \frac{(\prod_{i=1}^{d}T_{i}^{2}q_{i}^{-1} \delta_{i}^{-1}) }{\log\left(\prod_{i=1}^{d} T_{i}\right)}\exp\left(-\frac{u_{\T}^{2}}{1+C/\log\left(\prod_{i=1}^{d} T_{i}^{\beta}\right)}\right)\nonumber\\
&=&O(1)\left(\prod_{i=1}^{d} T_{i}\right)^{ 2C/\left(C+\log\left(\prod_{i=1}^{d} T_{i}^{\beta}\right)\right)} \times\left(\prod_{i=1}^{d}\delta_{i} \right)^{(-C)/\left(C+\log\left(\prod_{i=1}^{d} T_{i}^{\beta}\right)\right)} \nonumber \\
&&\times \left( \log \prod_{i=1}^{d} T_{i} \right)^{\left((\sum_{i=1}^{d} \frac{1}{\alpha_{i}}-1)C\right)/\left(C+\log\left(\prod_{i=1}^{d} T_{i}^{\beta}\right)\right)}\nonumber\\
&=&O(1)\exp\left[\frac{C}{C/u^{2}+\beta/2}+2C\left(\sum_{i=1}^{d}2/\alpha_{i}-1\right)\frac{\log u}{C+\beta u^{2}/2}\right]\left(\prod_{i=1}^{d}\delta_{i} u^{\frac{2}{\alpha_{i}}} \right)^{(-C)/\left(C+\beta u^{2}/2\right)}\\
&=&O(1)\end{aligned}$$ as $u\to\infty$. Then, $$\begin{aligned}
\label{eq6.18}
H_{\T,12}&\leq & C\frac{(\prod_{i=1}^{d} T_{i}^{2}q_{i}^{-1} \delta_{i}^{-1})}{\log\left(\prod_{i=1}^{d} T_{i}\right)}\exp\left(-\frac{u_{\T}^{2}}{1+\theta_{1}(\T^{\beta})}\right) \frac{\log\left(\prod_{i=1}^{d} T_{i}\right)}{(\prod_{i=1}^{d} T_{i}^{2}q_{i}^{-1} \delta_{i}^{-1})} \nonumber\\ &\times&\sum_{\mbox{\tiny$\begin{array}{c} \k\q\in \O_{\i},\l\boldsymbol{\delta}\in \O_{\j},\mathbf{1\leq i\neq j\leq n} \\ \prod_{m=1}^{d}|k_{m}q_{m}-l_{m}q_{m}|> \prod_{i=1}^{d} T_{i}^{\beta}\end{array}$}} |r(\k\q-\l\boldsymbol{\delta})-\rho(\T) | \nonumber\\
&\leq& C \left(\prod_{i=1}^{d} \delta_{i} q_{i} T_{i}^{-2}\right)\sum_{\mbox{\tiny$\begin{array}{c} \k\q\in \O_{\i},\l\boldsymbol{\delta}\in \O_{\j},\mathbf{1\leq i\neq j\leq n} \\ \prod_{m=1}^{d}|k_{m}q_{m}-l_{m}q_{m}|> \prod_{i=1}^{d} T_{i}^{\beta}\end{array}$}} \left|r(\k\q-\l\boldsymbol{\delta})\log\left(\prod_{m=1}^{d} k_{m}q_{m}-l_{m}\delta_{m}\right)-r \right| \nonumber\\
&+& C \left(\prod_{i=1}^{d} \delta_{i} q_{i} T_{i}^{-2}\right)\sum_{\mbox{\tiny$\begin{array}{c} \k\q\in \O_{\i},\l\boldsymbol{\delta}\in \O_{\j},\mathbf{1\leq i\neq j\leq n} \\ \prod_{m=1}^{d}|k_{m}q_{m}-l_{m}\delta_{m}|> \prod_{i=1}^{d} T_{i}^{\beta}\end{array}$}} \left| 1-\frac{\log\prod_{i=1}^{d} T_{i}}{\log\prod_{j=1}^{d} (k_{m}q_{m}-l_{m}\delta_{m})} \right|.\end{aligned}$$ By condition $ \mathbf{A3} $, the first term on the right hand side of tends to $ 0 $ as $ \T\to\infty $, and the second term also tends to $ 0 $ by the same arguments of , then $ H_{\T,12}\to 0 $ as $ \T\to\infty $. Combining , we can get that $ H_{\T,1}\to 0 $ as $ \T\to\infty $.

Now, we consider $ H_{\T,2} $, we only prove the case that $ k_{1}q_{1}= l_{1}\delta_{1} $, and $ \min\{|k_{j}q_{j}-l_{j}\delta_{j}|,j=2,\cdots,d\}> 0  $. By the similar way, the rest cases can be proved. If $ \prod_{i=2}^{d} T_{i}\leq \prod_{i=1}^{d} T_{i}^{\beta}=e^{\beta u^{2}/2} $, by $$\begin{aligned}
\label{eq6.19}
H_{\T,2}&\leq & C  \sum_{\mbox{\tiny$\begin{array}{c} \k\q\in \O_{\i}, \l\boldsymbol{\delta}\in\O_{\j}, \mathbf{1\leq i\neq j\leq n} \\ k_{1}q_{1}=l_{1}\delta_{1}\end{array}$}} |r(\k\q-\l\boldsymbol{\delta})-\rho(\T) |\exp\left(-\frac{u_{\T}^{2}(x_{2})}{1+\theta(\epsilon)}\right)   \nonumber\\
&\leq &  C \prod_{i=1}^{d} \delta_{i}^{-1}u^{-2/\alpha_{i}}\exp\left[(\beta-\frac{1-\theta(\epsilon)}{1+\theta(\epsilon)})u^{2}/2+{\sum_{i=1}^{d} \frac{4}{\alpha_{i}}}\log u\right]\to 0\end{aligned}$$ as $ u\to\infty $. If $ \prod_{i=2}^{d} T_{i}> \prod_{i=1}^{d} T_{i}^{\beta}=e^{\beta u^{2}/2} $, we have $$H_{\T,2}=C\sum_{\mbox{\tiny$\begin{array}{c} \k\q\in \O_{\i},\l\boldsymbol{\delta}\in \O_{\j},\mathbf{1\leq i\neq j\leq n} \\ \prod_{m=2}^{d}|k_{m}q_{m}-l_{m}q_{m}|\leq {\prod_{i=1}^{d} T_{i}}^{\beta}\\ k_{1}q_{1}=l_{1}\delta_{1}\end{array}$}}+C\sum_{\mbox{\tiny$\begin{array}{c} \k\q\in \O_{\i},\l\boldsymbol{\delta}\in \O_{\j},\mathbf{1\leq i\neq j\leq n} \\ \prod_{m=2}^{d}|k_{m}q_{m}-l_{m}q_{m}|> \prod_{i=1}^{d} T_{i}^{\beta}\\ k_{1}q_{1}=l_{1}\delta_{1}\end{array}$}} :=H_{\T,21}+H_{\T,22} .$$ For $ H_{\T,21} $, it follows from that $$\begin{aligned}
H_{\T,21}&\leq & C \sum_{\mbox{\tiny$\begin{array}{c} \k\q\in \O_{\i},\l\boldsymbol{\delta}\in \O_{\j},\mathbf{1\leq i\neq j\leq n} \\ \prod_{m=2}^{d}|k_{m}q_{m}-l_{m}q_{m}|\leq {\prod_{i=1}^{d} T_{i}}^{\beta}\\ k_{1}q_{1}=l_{1}\delta_{1}\end{array}$}}  |r(\k\q-\l\boldsymbol{\delta})-\rho(\T) |\exp\left(-\frac{u_{\T}^{2}(x_{2})}{1+\theta(\epsilon)}\right)   \nonumber\\
&\to& 0\;\ as\;\ \T\to\infty.\end{aligned}$$ For $ H_{\T,22} $, , we have $$\frac{(\prod_{i=1}^{d} T_{i}^{2}q_{i}^{-1}\delta_{i}^{-1})}{\log\left(\prod_{i=1}^{d} T_{i}\right)}\exp\left(-\frac{u_{\T}^{2}}{1+\theta_{2}(\T^{\beta})}\right)= O(1),$$ hence, $$\begin{aligned}
H_{\T,22}&\leq & C \sum_{\mbox{\tiny$\begin{array}{c} \k\q\in \O_{\i},\l\boldsymbol{\delta}\in \O_{\j},\mathbf{1\leq i\neq j\leq n} \\ \prod_{m=2}^{d}|k_{m}q_{m}-l_{m}q_{m}|> \prod_{i=1}^{d} T_{i}^{\beta}\\ k_{1}q_{1}=l_{1}\delta_{1}\end{array}$}} \left| r(0,k_{2}q_{2},\cdots,k_{d}q_{d})-\rho(\T)  \right| \exp\left(-\frac{u_{\T}^{2}}{1+\theta_{2}(\T^{\beta})}\right)\\
&= & C \frac{(\prod_{i=1}^{d} T_{i}^{2}q_{i}^{-1}\delta_{i}^{-1})}{\log\left(\prod_{i=1}^{d} T_{i}\right)}\exp\left(-\frac{u_{\T}^{2}}{1+\theta_{2}(\T^{\beta})}\right)
\times\frac{\log(\prod_{i=1}^{d} T_{i})}{(\prod_{i=1}^{d} T_{i}^{2}q_{i}^{-1}\delta_{i}^{-1})} \\ &\times&\sum_{\mbox{\tiny$\begin{array}{c} \k\q\in \O_{\i},\l\boldsymbol{\delta}\in \O_{\j},\mathbf{1\leq i\neq j\leq n} \\ \prod_{m=2}^{d}|k_{m}q_{m}-l_{m}q_{m}|> \prod_{i=1}^{d} T_{i}^{\beta}\\ k_{1}q_{1}=l_{1}\delta_{1}\end{array}$}} \left| r(0,k_{2}q_{2},\cdots,k_{d}q_{d})-\rho(\T)  \right|\nonumber\\
&\leq& C\frac{\log(\prod_{i=1}^{d} T_{i})}{(\prod_{i=1}^{d} T_{i}^{2}q_{i}^{-1}\delta_{i}^{-1})} \sum_{\mbox{\tiny$\begin{array}{c} \k\q\in \O_{\i},\l\boldsymbol{\delta}\in \O_{\j},\mathbf{1\leq i\neq j\leq n} \\ \prod_{m=2}^{d}|k_{m}q_{m}-l_{m}q_{m}|> \prod_{i=1}^{d} T_{i}^{\beta}\\ k_{1}q_{1}=l_{1}\delta_{1}\end{array}$}} \left( |r(0,k_{2}q_{2},\cdots,k_{d}q_{d})|+|\rho(\T)|  \right)\\
&\leq & C (\prod_{i=1}^{d} T_{i}^{-2}q_{i}\delta_{i}) \left(\prod_{i=1}^{d} T_{i}\delta_{i}^{-1}\right)\left(\prod_{i=2}^{d} T_{i}q_{i}^{-1}\right)\nonumber\\
&\leq& C\frac{q_{1}}{T_{1}},\end{aligned}$$ which implies $ H_{\T,22}\to 0 $ as $ \T\to\infty $. Furthermore, $ H_{\T,2}\to 0 $, as $ \T\to\infty $. Combining , , and , we can get $ F_{3}\to 0 $ as $ \T\to\infty $. The proof is complete.

\[lem7\] Suppose $ \Re(\delta_{i}), i=1,2,\cdots,d $ are all sparse grids or all Pickands grids and ${\u}_{\T}$, $\v_{\T}$ are given by . Then for the grids $ \Re(q_{i})$ with $ q_{i}=\gamma_{i}u^{-2/\alpha_{i}} $ and $ \gamma_{i}>0 $, $i=1,2,\cdots,d $, we have $$\begin{aligned}
\label{eq7.1}
&&\Big|\P\Big( v_{\T}(x_{1}) < \min_{\t\in \O\cap\prod_{i=1}^{d} \Re(q_{i})}\xi_{\T}(\t) \leq \max_{\t\in \O\cap \prod_{i=1}^{d} \Re(q_{i})} \xi_{\T}(\t)\leq u_{\T}(x_{2}), \nonumber\\
&& \qquad v_{\T}^{\boldsymbol{\delta}}(y_{1}) < \min_{\t\in \O \cap\prod_{i=1}^{d} \Re(\delta_{i})} \xi_{\T}(\t) \leq \max_{\t\in \O\cap\prod_{i=1}^{d} \Re(\delta_{i})} \xi_{\T}(\t)\leq  u_{\T}^{\boldsymbol{\delta}}(y_{2})\Big) \nonumber\\
&&\qquad\qquad-\int_{-\infty}^{+\infty} \prod_{j=1}^{d} \prod_{i_{j}=1}^{n_{j}}\P\Big( v_{\T}^{'}(x_{1})<\min_{\t\in\O_{\i}} \eta(\t)\leq \max_{\t\in\O_{\i}} \eta(\t)\leq u_{\T}^{'}(x_{2}),\nonumber\\
&& \qquad\qquad\qquad v_{\T}^{*}(y_{1})<\min_{\t\in\O_{\i}\cap\prod_{i=1}^{d} \Re(\delta_{i})} \eta(\t)\leq \max_{\t\in\O_{\i}\cap\prod_{i=1}^{d} \Re(\delta_{i})} \eta(\t)\leq u_{\T}^{*}(y_{2}) \Big) \phi(z) dz \Big| \nonumber\\
&&\to 0\end{aligned}$$ as $\T\to\infty$ and all $ \gamma_{i}\downarrow 0 $, where $$\begin{aligned}
\label{eq7.2}
\begin{cases}
v_{\T}^{'}(x_{1})=\frac{-b_{\T}+x_{1}/a_{\T}-\rho^{1/2}(\T)z}{(1-\rho(\T))^{1/2}}=\frac{x_{1}-r-\sqrt{2r}z}{a_{\T}}-b_{\T}+o(a_{\T}^{-1});\\
u_{\T}^{'}(x_{2})=\frac{b_{\T}+x_{2}/a_{\T}-\rho^{1/2}(\T)z}{(1-\rho(\T))^{1/2}}=\frac{x_{2}+r-\sqrt{2r}z}{a_{\T}}+b_{\T}+o(a_{\T}^{-1});\\
v_{\T}^{*}(y_{1})=\frac{-b_{\T}^{*}+y_{1}/a_{\T}-\rho^{1/2}(\T)z}{(1-\rho(\T))^{1/2}}=\frac{y_{1}-r-\sqrt{2r}z}{a_{\T}}-b_{\T}^{*}+o(a_{\T}^{-1});\\
u_{\T}^{*}(y_{2})=\frac{b_{\T}^{*}+y_{2}/a_{\T}-\rho^{1/2}(\T)z}{(1-\rho(\T))^{1/2}}=\frac{y_{2}+r-\sqrt{2r}z}{a_{\T}}+b_{\T}^{*}+o(a_{\T}^{-1}).
\end{cases}\end{aligned}$$

[**Proof.**]{} By the definition of $ \{\xi_{\T}(\t),  \t\geq \mathbf{0} \} $ and $\{ \eta(\t),\t\geq \mathbf{0}\} $, we have $$\begin{aligned}
\label{eq7.3}
&&\P\Big( v_{\T}(x_{1}) < \min_{\t\in \O\cap\prod_{i=1}^{d} \Re(q_{i})}\xi_{\T}(\t) \leq \max_{\t\in \O\cap \prod_{i=1}^{d} \Re(q_{i})} \xi_{\T}(\t)\leq u_{\T}(x_{2}), \nonumber\\
&&\qquad v_{\T}^{\boldsymbol{\delta}}(y_{1}) < \min_{\t\in \O \cap\prod_{i=1}^{d} \Re(\delta_{i})} \xi_{\T}(\t) \leq \max_{\t\in \O\cap\prod_{i=1}^{d} \Re(\delta_{i})} \xi_{\T}(\t)\leq  u_{\T}^{\boldsymbol{\delta}}(y_{2})\Big) \nonumber\\
&=&\int_{-\infty}^{+\infty} \P\Big( v_{\T}^{'}(x_{1}) < \min_{\t\in \O\cap\prod_{i=1}^{d} \Re(q_{i})}\eta(\t) \leq \max_{\t\in \O \cap\prod_{i=1}^{d} \Re(q_{i})} \eta(\t)\leq u_{\T}^{'}(x_{2}), \nonumber\\ && v_{\T}^{*}(y_{1}) < \min_{\t\in \O \cap\prod_{i=1}^{d} \Re(\delta_{i})} \eta(\t) \leq \max_{\t\in \O\cap\prod_{i=1}^{d} \Re(\delta_{i})} \eta(\t)\leq  u_{\T}^{*}(y_{2}) \Big) \phi(z) dz \nonumber\\
&=&\int_{-\infty}^{+\infty} \prod_{j=1}^{d} \prod_{i_{j}=1}^{n_{j}}\P\Big( v_{\T}^{'}(x_{1}) < \min_{\t\in \O_{\i}\cap\prod_{i=1}^{d} \Re(q_{i})}\eta(\t) \leq \max_{\t\in \O_{\i}\cap \prod_{i=1}^{d} \Re(q_{i})} \eta(\t)\leq u_{\T}^{'}(x_{2}), \nonumber\\
 && \qquad v_{\T}^{*}(y_{1}) < \min_{\t\in \O_{\i} \cap\prod_{i=1}^{d} \Re(\delta_{i})} \eta(\t) \leq \max_{\t\in \O_{\i}\cap\prod_{i=1}^{d} \Re(\delta_{i})} \eta(\t)\leq  u_{\T}^{*}(y_{2}) \Big) \phi(z) dz.\end{aligned}$$ By Lemma \[lem3\] and the dominated convergence theorem, we have $$\begin{aligned}
&&\Big|\int_{-\infty}^{+\infty} \prod_{j=1}^{d} \prod_{i_{j}=1}^{n_{j}}\P\Big( v_{\T}^{'}(x_{1}) < \min_{\t\in \O_{\i}\cap\prod_{i=1}^{d} \Re(q_{i})}\eta(\t) \leq \max_{\t\in \O_{\i}\cap \prod_{i=1}^{d} \Re(q_{i})} \eta(\t)\leq u_{\T}^{'}(x_{2}), \nonumber\\ && v_{\T}^{*}(y_{1}) < \min_{\t\in \O_{\i} \cap\prod_{i=1}^{d} \Re(\delta_{i})} \eta(\t) \leq \max_{\t\in \O_{\i}\cap\prod_{i=1}^{d} \Re(\delta_{i})} \eta(\t)\leq  u_{\T}^{*}(y_{2}) \Big) \phi(z) dz \nonumber\\
&&-\int_{-\infty}^{+\infty} \prod_{j=1}^{d} \prod_{i_{j}=1}^{n_{j}}\P\Big( v_{\T}^{'}(x_{1})<\min_{\t\in\O_{\i}} \eta(\t)\leq \max_{\t\in\O_{\i}} \eta(\t)\leq u_{\T}^{'}(x_{2}),\nonumber\\ && v_{\T}^{*}(y_{1}) < \min_{\t\in \O_{\i} \cap\prod_{i=1}^{d} \Re(\delta_{i})} \eta(\t) \leq \max_{\t\in \O_{\i}\cap\prod_{i=1}^{d} \Re(\delta_{i})} \eta(\t)\leq  u_{\T}^{*}(y_{2}) \Big) \phi(z) dz \Big| \\
&&\to 0\end{aligned}$$ as $\T\to\infty$ and all $ \gamma_{i}\downarrow 0 $. Combining , we finish the proof.

\[lem8\] Let $ X(\t) $ be a centered homogeneous Gaussian field with and covariance function $ r(\t) $ satisfying $ \mathbf{A1}-\mathbf{A3} $. Then for any sparse grids, $$\begin{aligned}
&&\lim_{\T\to\infty}\P\left(\mathbf{v}_{\T}<\mathbf{m}_{\T}\leq \mathbf{M}_{\T}\leq \mathbf{u}_{\T}\right)\\
&=&\int_{-\infty}^{+\infty} \exp\Big( -(e^{-x_{2}-r+\sqrt{2r}z}
+e^{-y_{2}-r+\sqrt{2r}z}+e^{x_{1}+r+\sqrt{2r}z}+e^{y_{1}+r+\sqrt{2r}z})\Big) \phi(z) dz,\end{aligned}$$ where $\mathbf{u}_{\T}$, $\mathbf{v}_{\T}$ are given by .

[**Proof.**]{} Noting that $\{\eta(\t),\t\geq \mathbf{0}\}=^{d} \{-\eta(\t),\t\geq \mathbf{0}\} $, we have $$\begin{aligned}
\label{eq8.1}
&&1-\P\Big(v_{\T}^{'}(x_{1})<\min_{\t\in\T_{\T^{a}}} \eta(\t)\leq \max_{\t\in\I_{\T^{a}}} \eta(\t)\leq u_{\T}^{'}(x_{2}),\nonumber\\
&&\qquad v_{\T}^{*}(y_{1}) < \min_{\t\in \I_{\T^{a}} \cap\prod_{i=1}^{d} \Re(\delta_{i})} \eta(\t) \leq \max_{\t\in \I_{\T^{a}}\cap\prod_{i=1}^{d} \Re(\delta_{i})} \eta(\t)\leq  u_{\T}^{*}(y_{2}) \Big) \nonumber\\
&=& 1-\P\left(\max_{\t\in\I_{\T^{a}}} \eta(\t)\leq u_{\T}^{'}(x_{2}), \max_{\t\in \I_{\T^{a}}\cap\prod_{i=1}^{d} \Re(\delta_{i})} \eta(\t)\leq  u_{\T}^{*}(y_{2}) \right)\nonumber\\
&&+1-\P\left(\max_{\t\in\I_{\T^{a}}} -\eta(\t)<-v_{\T}^{'}(x_{1}),\max_{\t\in \I_{\T^{a}} \cap\prod_{i=1}^{d} \Re(\delta_{i})} -\eta(\t)< -v_{\T}^{*}(y_{1}) \right)\nonumber\\
&& -\P\left(\max_{\t\in\I_{\T^{a}}} \eta(\t)> u_{\T}^{'}(x_{2}),\max_{\t\in \I_{\T^{a}} \cap\prod_{i=1}^{d} \Re(\delta_{i})} -\eta(\t)\geq -v_{\T}^{*}(y_{1}) \right)\nonumber\\
&&-\P\left(\max_{\t\in\I_{\T^{a}}} -\eta(\t)\geq-v_{\T}^{'}(x_{1}), \max_{\t\in \I_{\T^{a}}\cap\prod_{i=1}^{d} \Re(\delta_{i})} \eta(\t)>  u_{\T}^{*}(y_{2})  \right).\end{aligned}$$ Since $$\begin{aligned}
&&\P\left(\max_{\t\in\I_{\T^{a}}} \eta(\t)> u_{\T}^{'}(x_{2}),\max_{\t\in \I_{\T^{a}} \cap\prod_{i=1}^{d} \Re(\delta_{i})} -\eta(\t)\geq -v_{\T}^{*}(y_{1}) \right)\\
&&\leq \P\left(\max_{(\t,\s)\in\I_{\T^{a}}\times \I_{\T^{a}}} \eta(\t)-\eta(\s)> u_{\T}^{'}(x_{2})-v_{\T}^{*}(y_{1}) \right)\\
&&\leq 2\P\left( \max_{\t\in\I_{\T^{a}}} \eta(\t)>\frac{u_{\T}^{'}(x_{2})-v_{\T}^{*}(y_{1})}{2} \right),\end{aligned}$$ by Borell theorem and that for some constant $ c $, with defined as before we have[$$\begin{aligned}
\label{eq8.2}
&& \prod_{i=1}^{d} n_{i} \P\left(\max_{\t\in\I_{\T^{a}}} \eta(\t)> u_{\T}^{'}(x_{2}),\max_{\t\in \I_{\T^{a}} \cap\prod_{i=1}^{d} \Re(\delta_{i})} -\eta(\t)\geq -v_{\T}^{*}(y_{1}) \right)\nonumber\\
& \leq &C \prod_{i=1}^{d} n_{i} \Psi\left(\frac{u_{\T}^{'}(x_{2})-v_{\T}^{*}(y_{1})}{2} -c\right)\nonumber\\
& \leq &C \prod_{i=1}^{d} n_{i} \left(2\log \prod_{i=1}^{d} T_{i}\right)^{-\frac{1}{2}}\Big(1+\frac{-2c\sqrt{2\log \prod_{i=1}^{d} T_{i}}+x_{2}-y_{1}+2r}{4\log\left(\prod_{i=1}^{d} T_{i}\right)}+\nonumber\\
&&\frac{\log\left[(2\pi)^{-1/2}\left(\prod_{i=1}^{d} H_{\alpha_{i}}\right)\left(2\log \prod_{i=1}^{d} T_{i}\right)^{-1/2+\sum_{i=1}^{d}\frac{1}{\alpha_{i}}}\right]+\log\left[(2\pi)^{-1/2}\left(\prod_{i=1}^{d} \delta_{i}^{-1}\right)\left(2\log \prod_{i=1}^{d} T_{i}\right)^{-1/2}\right]}{4\log\left(\prod_{i=1}^{d} T_{i}\right)}\Big)^{-1}\nonumber\\
&&\times \exp\Big( -\log\left(\prod_{i=1}^{d} T_{i}\right)-\frac{-2c\sqrt{2\log \prod_{i=1}^{d} T_{i}}+x_{2}-y_{1}+2r}{2} -\nonumber\\
&&\frac{\log\left[(2\pi)^{-1/2}\left(\prod_{i=1}^{d} H_{\alpha_{i}}\right)\left(2\log \prod_{i=1}^{d} T_{i}\right)^{-1/2+\sum_{i=1}^{d}\frac{1}{\alpha_{i}}}\right]+\log\left[(2\pi)^{-1/2}\left(\prod_{i=1}^{d} \delta_{i}^{-1}\right)\left(2\log \prod_{i=1}^{d} T_{i}\right)^{-1/2}\right]}{2} \Big)\nonumber\\
& \leq &C \exp\Big(-\sum_{i=1}^{d} \log(T_{i}^{a}+T_{i}^{b})-\sum_{i=1}^{d}\frac{1}{2\alpha_{i}}\log(2\log \prod_{i=1}^{d} T_{i})+c\sqrt{2\log (\prod_{i=1}^{d} T_{i})}\Big)
\to 0\end{aligned}$$ ]{} as $\T\to\infty.$ Similarly, $$\begin{aligned}
\label{eq8.3}
\prod_{i=1}^{d} n_{i} \P\left(\max_{\t\in\T_{\T^{a}}} -\eta(\t)\geq-v_{\T}^{'}(x_{1}), \max_{\t\in \I_{\T^{a}}\cap\prod_{i=1}^{d} \Re(\delta_{i})} \eta(\t)>  u_{\T}^{*}(y_{2})\right) \to 0\end{aligned}$$ as $ \T\to\infty $. Using Lemma \[lem2\] and , we have $$\begin{aligned}
\label{eq8.4}
&&\prod_{i=1}^{d} n_{i} \left(1-\P\left(\max_{\t\in\I_{\T^{a}}} \eta(\t)\leq u_{\T}^{'}(x_{2}), \max_{\t\in \I_{\T^{a}}\cap\prod_{i=1}^{d} \Re(\delta_{i})} \eta(\t)\leq  u_{\T}^{*}(y_{2}) \right)\right) \nonumber\\
&\to& e^{-x_{2}-r+\sqrt{2r}z}+e^{-y_{2}-r+\sqrt{2r}z}\end{aligned}$$ as $ \T\to\infty $, and $$\begin{aligned}
\label{eq8.5}
&&\prod_{i=1}^{d} n_{i} \left(1-\P\left(\max_{\t\in\T_{\T^{a}}} -\eta(\t)<-v_{\T}^{'}(x_{1}),\max_{\t\in \I_{\T^{a}} \cap\prod_{i=1}^{d} \Re(\delta_{i})} -\eta(\t)< -v_{\T}^{*}(y_{1}) \right)\right)\nonumber\\
&\to& e^{x_{1}+r+\sqrt{2r}z}+e^{y_{1}+r+\sqrt{2r}z}\end{aligned}$$ as $ \T\to\infty $. Since $$\begin{aligned}
\label{eq8.6}
&&\prod_{j=1}^{d} \prod_{i_{j}=1}^{n_{j}}\P\Big( v_{\T}^{'}(x_{1})<\min_{\t\in\O_{\i}} \eta(\t)\leq \max_{\t\in\O_{\i}} \eta(\t)\leq u_{\T}^{'}(x_{2}),\nonumber\\
&&\qquad v_{\T}^{*}(y_{1}) < \min_{\t\in \O_{\i} \cap\prod_{i=1}^{d} \Re(\delta_{i})} \eta(\t) \leq \max_{\t\in \O_{\i}\cap\prod_{i=1}^{d} \Re(\delta_{i})} \eta(\t)\leq  u_{\T}^{*}(y_{2}) \Big)\nonumber\\
&=&\Big( \P\Big(v_{\T}^{'}(x_{1})<\min_{\t\in\I_{\T^{a}}} \eta(\t)\leq \max_{\t\in\I_{\T^{a}}} \eta(\t)\leq u_{\T}^{'}(x_{2}),\nonumber\\
&& v_{\T}^{*}(y_{1}) < \min_{\t\in \I_{\T^{a}} \cap\prod_{i=1}^{d} \Re(\delta_{i})} \eta(\t) \leq \max_{\t\in \I_{\T^{a}}\cap\prod_{i=1}^{d} \Re(\delta_{i})} \eta(\t)\leq  u_{\T}^{*}(y_{2}) \Big)\Big)^{\prod_{i=1}^{d} n_{i}}\nonumber\\
&&=\exp\Big(\prod_{i=1}^{d} n_{i}\log\Big(\P\Big(v_{\T}^{'}(x_{1})<\min_{\t\in\I_{\T^{a}}} \eta(\t)\leq \max_{\t\in\I_{\T^{a}}} \eta(\t)\leq u_{\T}^{'}(x_{2}),\nonumber\\
&&\qquad v_{\T}^{*}(y_{1}) < \min_{\t\in \I_{\T^{a}} \cap\prod_{i=1}^{d} \Re(\delta_{i})} \eta(\t) \leq \max_{\t\in \I_{\T^{a}}\cap\prod_{i=1}^{d} \Re(\delta_{i})} \eta(\t)\leq  u_{\T}^{*}(y_{2}) \Big)\Big)\Big)\nonumber\\
&=&\exp\left(-\prod_{i=1}^{d} n_{i}(1-P_{\T})+R_{\T}\right),\end{aligned}$$ where $$\begin{aligned}
&&P_{\T}=\P\Big(v_{\T}^{'}(x_{1})<\min_{\t\in\I_{\T^{a}}} \eta(\t)\leq \max_{\t\in\I_{\T^{a}}} \eta(\t)\leq u_{\T}^{'}(x_{2}),\nonumber\\
&&\qquad v_{\T}^{*}(y_{1}) < \min_{\t\in \I_{\T^{a}} \cap\prod_{i=1}^{d} \Re(\delta_{i})} \eta(\t) \leq \max_{\t\in \I_{\T^{a}}\cap\prod_{i=1}^{d} \Re(\delta_{i})} \eta(\t)\leq  u_{\T}^{*}(y_{2}) \Big)\\
&&\to 1\end{aligned}$$ as $ \T\to\infty $, and $$\begin{aligned}
R_{\T}=o\left(\prod_{i=1}^{d} n_{i}(1-P_{\T})\right)\end{aligned}$$ as $ \T\to\infty $. Then, combining - and Lemmas \[lem4\]-\[lem7\], the assertion of this lemma follows.

\[lem9\] Let $ X(\t) $ be a centered homogeneous Gaussian field with and covariance function $ r(\t) $ satisfying $ \mathbf{A1}-\mathbf{A3} $ and the $\mathbf{u}_{\T}$, $\mathbf{v}_{\T}$ be given by . Then $H_{\a,\boldsymbol{\alpha}}^{x,y}$ given Theorem \[th2\] and any Pickands grids $\Re(a_{i}(2\log \prod_{i=1}^{d} T_{i})^{-1/\alpha_{i}})$ with $a_{i}>0$, $ i=1,2,\cdots,d $, we have $$\begin{aligned}
&&\lim_{\T\to\infty}\P\left(\mathbf{v}_{\T}<\mathbf{m}_{\T}\leq \mathbf{M}_{\T}\leq \mathbf{u}_{\T}\right)\\
&&=\int_{-\infty}^{+\infty}\exp\Big( -\Big(e^{-x_{2}-r+\sqrt{2r}z}+e^{-y_{2}-r+\sqrt{2r}z}
-H_{\a,\boldsymbol{\alpha}}^{x_{2}+\log( \prod_{i=1}^{d}H_{\alpha_{i}}),y_{2}+\log(\prod_{i=1}^{d} H_{a_{i},\alpha_{i}})}e^{-r+\sqrt{2r}z}\\&&+e^{x_{1}+r+\sqrt{2r}z}
+e^{y_{1}+r+\sqrt{2r}z}
-H_{\a,\boldsymbol{\alpha}}^{-x_{1}+\log( \prod_{i=1}^{d}H_{\alpha_{i}}),-y_{1}+\log(\prod_{i=1}^{d} H_{a_{i},\alpha_{i}})}e^{r+\sqrt{2r}z}\Big) \Big)\phi(z)dz.\end{aligned}$$

[**Proof.**]{} Suppose $ \Re(\delta_{i})$ are all Pickands grids and $S_{i}$ satisfies the conditions in Lemma \[lem3\]. Denote $$P_{\S}^{'}(u,x)=\P\left( \max_{\t\in\I_{\S}\cap\prod_{i=1}^{d} \Re(\delta_{i})} X(\t)> u, \max_{\t\in\I_{\S}} X(\t)> u+\frac{x}{u}\right),$$ then by using Lemma 6.1 of Piterbarg (1996), we have $$\begin{aligned}
\label{eq2}
P_{\S}^{'}(u,x)\sim \left(\prod_{i=1}^{d} S_{i}u^{2/\alpha_{i}}\right) H_{\a,\boldsymbol{\alpha}}^{0,x} \Psi(u)\end{aligned}$$ as $ u\to\infty $. Note that for Pickands grids, $$\begin{aligned}
u_{\T}^{'}(x_{2})
&=&  u_{\T}^{*}(y_{2})+\frac{\log( \prod_{i=1}^{d}H_{\alpha_{i}})-\log(\prod_{i=1}^{d} H_{a_{i},\alpha_{i}})+x_{2}-y_{2}}{u_{\T}^{*}(y_{2})}
+O\left(\frac{\log u}{u^{3}}\right)\end{aligned}$$ and $$\begin{aligned}
v_{\T}^{'}(x_{1})
&=&  v_{\T}^{*}(y_{1})+\frac{\log( \prod_{i=1}^{d}H_{\alpha_{i}})-\log(\prod_{i=1}^{d} H_{a_{i},\alpha_{i}})-x_{1}+y_{1}}{v_{\T}^{*}(y_{1})}
+O\left(\frac{\log u}{u^{3}}\right).\end{aligned}$$

Hence, from , we have $$\begin{aligned}
\label{eq9.2}
&&\prod_{i=1}^{d} n_{i} \left(\P\left(\max_{\t\in\I_{\T^{a}}} \eta(\t)> u_{\T}^{'}(x_{2}), \max_{\t\in \I_{\T^{a}}\cap\prod_{i=1}^{d} \Re(\delta_{i})} \eta(\t)>  u_{\T}^{*}(y_{2}) \right)\right) \nonumber\\
&&\sim \left(\prod_{i=1}^{d} n_{i} T_{i}^{a}\right) H_{\a,\boldsymbol{\alpha}}^{0,Z_{x_{2},y_{2}}}(u_{\T}^{*}(y_{2}))^{\sum_{i=1}^{d} \frac{2}{\alpha_{i}}}\Psi(u_{\T}^{*}(y_{2}))\nonumber\\
&&\sim H_{\a,\boldsymbol{\alpha}}^{0,Z_{x_{2},y_{2}}} \left(\prod_{i=1}^{d}H_{a_{i},\alpha_{i}} \right)^{-1}e^{-y_{2}-r+\sqrt{2r}z},\end{aligned}$$ where $Z_{x_{2},y_{2}}=\log( \prod_{i=1}^{d}H_{\alpha_{i}})-\log(\prod_{i=1}^{d} H_{a_{i},\alpha_{i}})+x_{2}-y_{2} $. By the definition of $ H_{\a,\boldsymbol{\alpha}}^{x,y} $, we get $ H_{\a,\boldsymbol{\alpha}}^{0,Z_{x_{2},y_{2}}}\left(\prod_{i=1}^{d}H_{a_{i},\alpha_{i}} \right)^{-1}e^{-y_{2}}=H_{\a,\boldsymbol{\alpha}}^{x_{2}+\log( \prod_{i=1}^{d}H_{\alpha_{i}}),y_{2}+\log(\prod_{i=1}^{d} H_{a_{i},\alpha_{i}})} $.

Furthermore, from and , we have $$\begin{aligned}
\label{eq9.3}
&&\prod_{i=1}^{d} n_{i} \left(1- \P\left(\max_{\t\in\I_{\T^{a}}} \eta(\t)\leq u_{\T}^{'}(x_{2}), \max_{\t\in \I_{\T^{a}}\cap\prod_{i=1}^{d} \Re(\delta_{i})} \eta(\t)\leq  u_{\T}^{*}(y_{2}) \right) \right) \nonumber\\
&&=\prod_{i=1}^{d} n_{i}\Big[\P\left( \max_{\t\in\I_{\T^{a}}} \eta(\t)> u_{\T}^{'}(x_{2})\right)+\P\left( \max_{\t\in\I_{\T^{a}}\cap\prod_{i=1}^{d} \Re(\delta_{i})} \eta(\t)> u_{\T}^{*}(y_{2})\right)\nonumber\\
&&-\P\left( \max_{\t\in\I_{\T^{a}}\cap\prod_{i=1}^{d} \Re(\delta_{i})} \eta(\t)>u_{\T}^{*}(y_{2}) , \max_{\t\in\I_{\T^{a}}} \eta(\t)> u_{\T}^{'}(x_{2})\right)\Big]\nonumber\\
&&\sim e^{-x_{2}-r+\sqrt{2r}z}+e^{-y_{2}-r+\sqrt{2r}z}-H_{\a,\boldsymbol{\alpha}}^{x_{2}+\log( \prod_{i=1}^{d}H_{\alpha_{i}}),y_{2}+\log(\prod_{i=1}^{d} H_{a_{i},\alpha_{i}})}e^{-r+\sqrt{2r}z}.\end{aligned}$$ Similarly, $$\begin{aligned}
\label{eq9.4}
&&\prod_{i=1}^{d} n_{i} \left(1- \P\left(\max_{\t\in\I_{\T^{a}}} -\eta(\t)\leq -v_{\T}^{'}(x_{1}), \max_{\t\in \I_{\T^{a}}\cap\prod_{i=1}^{d} \Re(\delta_{i})} \eta(\t)\leq  v_{\T}^{*}(y_{1}) \right) \right) \nonumber\\
&&\sim e^{x_{1}+r+\sqrt{2r}z}+e^{y_{1}+r+\sqrt{2r}z}-H_{\a,\boldsymbol{\alpha}}^{-x_{1}+\log( \prod_{i=1}^{d}H_{\alpha_{i}}),-y_{1}+\log(\prod_{i=1}^{d} H_{a_{i},\alpha_{i}})}e^{r+\sqrt{2r}z}.\end{aligned}$$ Following the same arguments as used in , we have $$\begin{aligned}
\label{eq9.5}
&&\prod_{i=1}^{d} n_{i} \left(\P\left(\max_{\t\in\I_{\T^{a}}} -\eta(\t)> -v_{\T}^{'}(x_{1}), \max_{\t\in \I_{\T^{a}}\cap\prod_{i=1}^{d} \Re(\delta_{i})} \eta(\t)>  u_{\T}^{*}(y_{2}) \right)\right) \to 0\end{aligned}$$ as $ \T\to\infty $, and $$\begin{aligned}
\label{eq9.6}
&&\prod_{i=1}^{d} n_{i} \left(\P\left(\max_{\t\in\I_{\T^{a}}} \eta(\t)> u_{\T}^{'}(x_{2}), \max_{\t\in \I_{\T^{a}}\cap\prod_{i=1}^{d} \Re(\delta_{i})} \eta(\t)\geq  -v_{\T}^{*}(y_{1}) \right)\right) \to 0\end{aligned}$$ as $ \T\to\infty $. Combining , , - and Lemmas \[lem4\]-\[lem7\], the assertion of this lemma follows.

Proofs of the main results {#sec4}
==========================

[**Proof of Theorem \[th1\].**]{} we have $$\begin{aligned}
\label{eq10.3}
&&\lim_{\T\to\infty}\P\left(\mathbf{M}_{\T}\leq \mathbf{u}_{\T},m_{\T}^{\boldsymbol{\delta}}>v_{\T}^{\boldsymbol{\delta}}(y_{1})\right)\nonumber\\
&=&\int_{-\infty}^{+\infty} \exp\Big( -(e^{-x_{2}-r+\sqrt{2r}z}
+e^{-y_{2}-r+\sqrt{2r}z}+e^{y_{1}+r+\sqrt{2r}z})\Big) \phi(z) dz,\end{aligned}$$ and $$\begin{aligned}
\label{eq10.4}
&&\lim_{\T\to\infty}\P\left(\mathbf{M}_{\T}\leq \mathbf{u}_{\T},m_{\T}>v_{\T}(x_{1})\right)\nonumber\\
&=&\int_{-\infty}^{+\infty} \exp\Big( -(e^{-x_{2}-r+\sqrt{2r}z}
+e^{-y_{2}-r+\sqrt{2r}z}+e^{x_{1}+r+\sqrt{2r}z})\Big) \phi(z) dz.\end{aligned}$$

Therefore by - and Lemma \[lem8\], we can get the desired result since $$\begin{aligned}
\label{eq10.5}
\P\left(\mathbf M_{\T}\leq\mathbf u_{\T},\mathbf m_{\T}\leq\mathbf v_{\T}\right) &=& \P\left(\mathbf{v}_{\T}<\mathbf{m}_{\T}\leq \mathbf{M}_{\T}\leq \mathbf{u}_{\T}\right)-\P\left(\mathbf M_{\T}\leq\mathbf u_{\T}, m_{\T}> v_{\T}(x_{1})\right)\nonumber\\
&-&\P\left(\mathbf M_{\T}\leq\mathbf u_{\T}, m_{\T}^{\boldsymbol{\delta}}> v_{\T}^{\boldsymbol{\delta}}(y_{1})\right)+\P\left(\mathbf M_{\T}\leq\mathbf u_{\T}\right).\end{aligned}$$

[**Proof of Theorem \[th2\].**]{} It follows from Lemma \[lem9\] and the definition of $ H_{\a,\boldsymbol{\alpha}}^{x,y} $ that $$\begin{aligned}
\lim_{\T\to\infty}\P\left(\mathbf{M}_{\T}\leq \mathbf{u}_{\T},m_{\T}>v_{\T}(x_{1})\right)&=&\int_{-\infty}^{+\infty} \exp\Big( -(e^{-x_{2}-r+\sqrt{2r}z}
+e^{-y_{2}-r+\sqrt{2r}z}\nonumber\\&-&H_{\a,\boldsymbol{\alpha}}^{x_{2}+\log( \prod_{i=1}^{d}H_{\alpha_{i}}),y_{2}+\log(\prod_{i=1}^{d} H_{a_{i},\alpha_{i}})}e^{-r+\sqrt{2r}z}+e^{x_{1}+r+\sqrt{2r}z})\Big) \phi(z) dz.\end{aligned}$$ Similarly, $$\begin{aligned}
\lim_{\T\to\infty}\P\left(\mathbf{M}_{\T}\leq \mathbf{u}_{\T},m_{\T}^{\boldsymbol{\delta}}>v_{\T}^{\boldsymbol{\delta}}(y_{1})\right)&=&\int_{-\infty}^{+\infty} \exp\Big( -(e^{-x_{2}-r+\sqrt{2r}z}
+e^{-y_{2}-r+\sqrt{2r}z}\nonumber\\&-&H_{\a,\boldsymbol{\alpha}}^{x_{2}+\log( \prod_{i=1}^{d}H_{\alpha_{i}}),y_{2}+\log(\prod_{i=1}^{d} H_{a_{i},\alpha_{i}})}e^{-r+\sqrt{2r}z}+e^{y_{1}+r+\sqrt{2r}z})\Big) \phi(z) dz,\end{aligned}$$ and By the same arguments as used in the proof of Theorem \[th1\], we can get the assertion of this theorem.

[**Proof of Theorem \[th3\].**]{} It follows from Lemma \[lem3\] that $$\begin{aligned}
&&\Big|\prod_{i=1}^{d} n_{i} \left(1- \P\left(\max_{\t\in\I_{\T^{a}}} \eta(\t)\leq u_{\T}^{'}(x_{2}), \max_{\t\in \I_{\T^{a}}\cap\prod_{i=1}^{d} \Re(\delta_{i})} \eta(\t)\leq  u_{\T}^{*}(y_{2}) \right) \right) \nonumber\\
&&\qquad-\prod_{i=1}^{d} n_{i} \left(1- \P\left(\max_{\t\in\I_{\T^{a}}} \eta(\t)\leq u_{\T}^{'}(x_{2}), \max_{\t\in \I_{\T^{a}}} \eta(\t)\leq  u_{\T}^{*}(y_{2}) \right) \right)\Big| \nonumber\\
&\leq& \prod_{i=1}^{d} n_{i} \left(\P\left(\max_{\t\in \I_{\T^{a}}\cap\prod_{i=1}^{d} \Re(\delta_{i})} \eta(\t)\leq  u_{\T}^{*}(y_{2}) \right)-\P\left(\max_{\t\in\I_{\T^{a}}} \eta(\t)\leq u_{\T}^{*}(y_{2})\right)\right)
\to 0\end{aligned}$$ as $ \T\to\infty $. Then from and , we have

Hence, combining , and -, we have $$\begin{aligned}
\lim_{\T\to\infty}\P\left(\mathbf{v}_{\T}<\mathbf{m}_{\T}\leq \mathbf{M}_{\T}\leq \mathbf{u}_{\T}\right)=
\int_{-\infty}^{+\infty}\exp\Big(-e^{-\min(x_{2},y_{2})-r+\sqrt{2r}z}-e^{\max(x_{1},y_{1})+r+\sqrt{2r}z}\Big)\phi(z)dz.\end{aligned}$$ , we have $$\begin{aligned}
\lim_{\T\to\infty}\P\left(\mathbf{M}_{\T}\leq \mathbf{u}_{\T},m_{\T}>v_{\T}(x_{1})\right)=\int_{-\infty}^{+\infty} \exp\Big( -e^{-\min(x_{2},y_{2})-r+\sqrt{2r}z}
-e^{x_{1}+r+\sqrt{2r}z}\Big) \phi(z) dz,\end{aligned}$$

$$\begin{aligned}
\lim_{\T\to\infty}\P\left(\mathbf{M}_{\T}\leq \mathbf{u}_{\T},m_{\T}^{\boldsymbol{\delta}}>v_{\T}(y_{1})\right)=\int_{-\infty}^{+\infty} \exp\Big( -e^{-\min(x_{2},y_{2})-r+\sqrt{2r}z}
-e^{y_{1}+r+\sqrt{2r}z}\Big) \phi(z) dz.\end{aligned}$$

and $$\begin{aligned}
\lim_{\T\to\infty}\P\left(\mathbf{M}_{\T}\leq \mathbf{u}_{\T}\right)=\int_{-\infty}^{+\infty} \exp\Big( -e^{-\min(x_{2},y_{2})-r+\sqrt{2r}z}\Big) \phi(z) dz,\end{aligned}$$ Therefore, from , we complete the proof.

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[^1]: Corresponding author. Email: pzx@swu.edu.cn
