Maxima and minima of homogeneous Gaussian random fields over continuous time and uniform grids

TL;DR

This paper investigates the joint limiting distributions of maxima and minima of centered homogeneous Gaussian random fields over continuous time and various grids, revealing dependence structures based on the field's dependence strength.

Contribution

It provides new theoretical results on the asymptotic dependence or independence of maxima and minima for different types of Gaussian fields and grid configurations.

Findings

Maxima and minima are asymptotically dependent for strongly dependent fields with sparse, Pickands', or dense grids.

Maxima and minima are asymptotically independent for weakly dependent fields.

The results extend understanding of extremal behavior in Gaussian random fields.

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.

Equations

$$r(t)=1-|t{|}^{\alpha }+o\left(|t{|}^{\alpha }\right),t\to 0.$$

$$r(t)\log t\to 0\text{mbox}ast\to \infty .$$

$$M_T=\max \{X(t),t\in [0,T]\}$$

$$P\{a_T\left(M_T-b_T\right)\le x\}\to \Lambda (x)$$

$$\displaystyle a_{T}=\sqrt{2\log T},\;\ b_{T}=\sqrt{2\log T}+\frac{\log\Big{[}(2\pi)^{-1/2}H_{\alpha}(2\log T)^{-1/2+1/\alpha}\Big{]}}{\sqrt{2\log T}}.$$

$$H_{\alpha }(\lambda )=E\exp \left(\underset{t\in [0,\lambda ]}{\max }\sqrt{2}{B}_{\alpha /2}(t)-t^{\alpha }\right)$$

$${\mathbf{M}}_T=\left(M_T,M_T^{\mathbf{\delta }}\right)=\left(\underset{t\in I_T}{\max }X(t),\underset{t\in I_T\bigcap {\prod }_{i=1}^d\Re ({\delta }_i)}{\max }X(t)\right),$$

$${\delta }_i{\left(2\log \prod _{i=1}^d{T}_i\right)}^{1/{\alpha }_i}\to D_i,i=1,2,\cdots ,d.$$

$${\mathbf{m}}_T=\left(m_T,m_T^{\mathbf{\delta }}\right)=\left(\underset{t\in I_T}{\min }X(t),\underset{t\in I_T\bigcap {\prod }_{i=1}^d\Re ({\delta }_i)}{\min }X(t)\right)$$

$$\{\begin{array}{l}{b}_T^{\mathbf{\delta }}=a_T+a_T^{-1}\log \left((2\pi {)}^{-1/2}({\prod }_{i=1}^d{\delta }_i^{-1})(a_T{)}^{-1}\right)\\ b_{a,T}=a_T+a_T^{-1}\log \left((2\pi {)}^{-1/2}({\prod }_{i=1}^d{H}_{a_i,{\alpha }_i})(a_T{)}^{{\sum }_{i=1}^d\frac{2}{{\alpha }_i}-1}\right)\\ b_T=a_T+a_T^{-1}\log \left((2\pi {)}^{-1/2}({\prod }_{i=1}^d{H}_{{\alpha }_i})(a_T{)}^{{\sum }_{i=1}^d\frac{2}{{\alpha }_i}-1}\right),\end{array}$$

$$H_{a_i,{\alpha }_i}({\lambda }_i)=E\exp \left(\underset{k_i{a}_i\in [0,{\lambda }_i]}{\max }\sqrt{2}{B}_{{\alpha }_i/2}^{(i)}(k_i{a}_i)-(k_i{a}_i{)}^{{\alpha }_i}\right),i=1,2,\cdots ,d,$$

$$\{\begin{array}{l}{\mathbf{u}}_T=\left(u_T(x_2),u_T^{\mathbf{\delta }}(y_2)\right)=\left(b_T+\frac{x_2}{a_T},b_T^{\ast }+\frac{y_2}{a_T}\right)\\ {\mathbf{v}}_T=\left(v_T(x_1),v_T^{\mathbf{\delta }}(y_1)\right)=\left(-b_T+\frac{x_1}{a_T},-b_T^{\ast }+\frac{y_1}{a_T}\right),\end{array}$$

$$\underset{T\to \infty }{\lim }P\left({\mathbf{M}}_T\le {\mathbf{u}}_T,{\mathbf{m}}_T\le {\mathbf{v}}_T\right)$$

$$\displaystyle H_{\operatorname*{\mathbf{a}},\boldsymbol{\alpha}}^{x,y}(\boldsymbol{\lambda})=\int_{-\infty}^{\infty}e^{s}\operatorname*{\mathbf{P}}\Big{(}\max_{\operatorname*{\mathbf{k}}\operatorname*{\mathbf{a}}\in\prod_{i=1}^{d}[0,\lambda_{i}]}\sqrt{2}\chi(\operatorname*{\mathbf{k}}\operatorname*{\mathbf{a}})>s+x,\max_{\operatorname*{\mathbf{t}}\in\prod_{i=1}^{d}[0,\lambda_{i}]}\sqrt{2}\chi(\operatorname*{\mathbf{t}})>s+y\Big{)}ds,$$

$$H_{a,\mathbf{\alpha }}^{x,y}=\underset{\mathbf{\lambda }\to \infty }{\lim }{H}_{a,\mathbf{\alpha }}^{x,y}(\mathbf{\lambda })/(\prod _{i=1}^d{\lambda }_i)\in (0,\infty )$$

$$\underset{T\to \infty }{\lim }P\left({\mathbf{M}}_T\le {\mathbf{u}}_T,{\mathbf{m}}_T\le {\mathbf{v}}_T\right)$$

$$P(u,x)=P\left(X(0)>u,\underset{t\in I_{(\mathbf{\delta })}}{\max }X(t)>u+\frac{v^2+x}{u}\right),$$

$$P_S(u,x)=P\left(\underset{t\in I_S\cap {\prod }_{i=1}^d\Re ({\delta }_i)}{\max }X(t)>u,\underset{t\in I_S}{\max }X(t)>u+\frac{v^2+x}{u}\right)$$

$$\delta (ϵ)=\underset{\max \{|t_i|,i=1,2,\cdots ,d\}>ϵ}{\inf }(1-r(t))>0.$$

$$P\left(\underset{t\in I_S}{\max }X(t)>u+\frac{v^2+x}{u}\right)\sim \left(\prod _{i=1}^d{S}_i{\delta }_i^{-1}{H}_{{\alpha }_i}\right)e^{-x}\Psi (u),$$

$$P\left(\underset{t\in I_S\cap {\prod }_{i=1}^d\Re ({\delta }_i)}{\max }X(t)>u\right)\sim \left(\prod _{i=1}^d{S}_i{\delta }_i^{-1}\right)\Psi (u)$$

$$P_S(u,x)=o\left(P\left(\underset{t\in I_S\cap {\prod }_{i=1}^d\Re ({\delta }_i)}{\max }X(t)>u\right)+P\left(\underset{t\in I_S}{\max }X(t)>u+\frac{v^2+x}{u}\right)\right)$$

$$1-P\left(\underset{t\in I_S\cap {\prod }_{i=1}^d\Re ({\delta }_i)}{\max }X(t)\le u,\underset{t\in I_S}{\max }X(t)\le u+\frac{v^2+x}{u}\right)$$

$$\sim P\left(\underset{t\in I_S\cap {\prod }_{i=1}^d\Re ({\delta }_i)}{\max }X(t)>u\right)+P\left(\underset{t\in I_S}{\max }X(t)>u+\frac{v^2+x}{u}\right)$$

$$P\left(\underset{t\in I_S\cap {\prod }_{i=1}^d\Re ({\delta }_i)}{\max }X(t)\le u\right)-P\left(\underset{t\in I_S}{\max }X(t)\le u\right)\le {\rho }_a\left(\prod _{i=1}^d{H}_{{\alpha }_i}{S}_i{u}^{\frac{2}{{\alpha }_i}}\right)\Psi (u),$$

$${\xi }_T(t)=(1-\rho (T){)}^{1/2}\eta (t)+{\rho }^{1/2}(T)U,t\in {\mathbf{I}}_T,$$

$$\varrho (s,t)$$

$${\mathbf{M}}_O=\left(M_O,M_O^{\mathbf{\delta }}\right)=\left(\underset{t\in O}{\max }X(t),\underset{t\in O\bigcap {\prod }_{i=1}^d\Re ({\delta }_i)}{\max }X(t))\right),$$

$${\mathbf{m}}_O=\left(m_O,m_O^{\mathbf{\delta }}\right)=\left(\underset{t\in O}{\min }X(t),\underset{t\in O\bigcap {\prod }_{i=1}^d\Re ({\delta }_i)}{\min }X(t))\right).$$

$$|P\left({\mathbf{v}}_T<{\mathbf{m}}_T\le {\mathbf{M}}_T\le {\mathbf{u}}_T\right)-P\left({\mathbf{v}}_T<{\mathbf{m}}_O\le {\mathbf{M}}_O\le {\mathbf{u}}_T\right)|\to 0$$

Full text

Maxima and minima of homogeneous Gaussian random fields over continuous time and uniform grids

Yingyin Lu     Zuoxiang Peng

School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China Corresponding author. Email: pzx@swu.edu.cn

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.

1 Introduction

Let $\{X(t),t\geq 0\}$ be a stationary Gaussian process with mean zero, variance one and the correlation function $r(t)$ which satisfies for some $\alpha\in(0,2]$,

[TABLE]

Assume that

[TABLE]

Under conditions (1.1) and (1.2), the limit distribution theory on the maximum of $\{X(t),t\geq 0\}$ up to time $T$,

[TABLE]

is well developed that

[TABLE]

as $T\to\infty$, where $\Lambda(x)=\exp\left(-e^{-x}\right)$ is the standard Gumbel distribution and

[TABLE]

Here $H_{\alpha}$ is the well-known Pickands’ constant, which is defined by $H_{\alpha}=\lim_{\lambda\to\infty}H_{\alpha}(\lambda)/\lambda$ with

[TABLE]

and $B_{H}$ is a fractional Brownian motion with $\operatorname*{\mathbf{E}}B_{H}^{2}(t)=|t|^{2H}$. It is well known that $0<H_{\alpha}<\infty$, see e.g. 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.

It is known that extreme value theory of Gaussian random fields may be applied to image analysis, quantum chaos, queuing theory, insurance mathematics, number theory and so on; see e.g. Adler (2000), Adler et al. (2014), and Hashorva and Ji (2016). Further, numerical simulation of trajectories of high extremes of continuous random processes may be performed through the discrete time random processes depending heavily on the sampling frequency, see, for instance Leadbetter et al. (1983), Piterbarg (2004) and recent work of Song et al. (2018, 2019) on flaw detection by using ultrasonic response signals.

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 (1.1) and (1.2), where the uniform grid 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 on 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 behavior 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 $\{X(\operatorname*{\mathbf{t}}),\operatorname*{\mathbf{t}}\geq\mathbf{0}\}$ be a homogeneous Gaussian field with mean zero, variance one and covariance function $r(\mathbf{t})=\operatorname*{\mathbf{Cov}}(X(\mathbf{t}),X(\mathbf{0}))$ satisfies the following conditions, for $d\geq 2$:

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

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

$\mathbf{A3}:\lim_{\operatorname*{\mathbf{T}}\to\infty}r(\operatorname*{\mathbf{T}})\log(\prod_{i=1}^{d}T_{i})=r\in[0,\infty)$, where $\operatorname*{\mathbf{T}}\to\infty$ 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

[TABLE]

where $\operatorname*{\mathbf{I}}_{\operatorname*{\mathbf{T}}}=\prod_{i=1}^{d}[0,T_{i}]$, and $\operatorname*{\mathbf{I}}_{\operatorname*{\mathbf{T}}}\bigcap\prod_{i=1}^{d}\Re(\delta_{i})$ means $\prod_{i=1}^{d}\{[0,T_{i}]\bigcap\Re(\delta_{i})\}$, and the uniform grid $\Re(\delta_{i})=\{k\delta_{i},k\in N\}$ is given by

[TABLE]

We say that the grid is dense if all $D_{i}=0$ 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 $\operatorname*{\mathbf{M}}_{\operatorname*{\mathbf{T}}}$ 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).

Similarly to the definition of maxima $\operatorname*{\mathbf{M}}_{\operatorname*{\mathbf{T}}}$, define the minima $\operatorname*{\mathbf{m}}_{\operatorname*{\mathbf{T}}}$ as follows:

[TABLE]

and let

[TABLE]

where $a_{\operatorname*{\mathbf{T}}}=\sqrt{2\log(\prod_{i=1}^{d}T_{i})}$ 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

[TABLE]

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 $\mathbf{u}_{\operatorname*{\mathbf{T}}}$ and $\mathbf{v}_{\operatorname*{\mathbf{T}}}$ be given by

[TABLE]

where $b_{\operatorname*{\mathbf{T}}}^{*}=b_{\operatorname*{\mathbf{T}}}^{\boldsymbol{\delta}}$ for sparse grids, $b_{\operatorname*{\mathbf{T}}}^{*}=b_{\operatorname*{\mathbf{a}},\operatorname*{\mathbf{T}}}$ for Pickands grids and $b_{\operatorname*{\mathbf{T}}}^{*}=b_{\operatorname*{\mathbf{T}}}$ 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 mean componentwise operating. For example, for vectors $\operatorname*{\mathbf{t}}=(t_{1},t_{2},\cdots,t_{d})$ and $\operatorname*{\mathbf{s}}=(s_{1},s_{2},\cdots,s_{d})$, operations of $\operatorname*{\mathbf{s}}\leq\operatorname*{\mathbf{t}}$, $\operatorname*{\mathbf{s}}-\operatorname*{\mathbf{t}}$, $\operatorname*{\mathbf{s}}\operatorname*{\mathbf{t}}$, and $\operatorname*{\mathbf{s}}^{\operatorname*{\mathbf{t}}}$ 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 2 presents the main results and Section 3 gives some auxiliary lemmas. The proofs of the main results will be given in Section 4.

2 Main results

Theorem 2.1**.**

Let $X(\operatorname*{\mathbf{t}})$ be a centered homogenous Gaussian field with unit variance and covariance function $r(\operatorname*{\mathbf{t}})$ satisfying $\mathbf{A1}-\mathbf{A3}$. Then for any sparse grids $\Re(\delta_{i})$, $i=1,2,\cdots,d$,

[TABLE]

where $\mathbf{u}_{\operatorname*{\mathbf{T}}}$, $\mathbf{v}_{\operatorname*{\mathbf{T}}}$ are given by (1.4).

Define,

[TABLE]

where $\chi(\operatorname*{\mathbf{t}})=\sum_{i=1}^{d}B_{\alpha_{i}/2}^{(i)}(t_{i})-\sum_{i=1}^{d}|t_{i}|^{\alpha_{i}}$.

Theorem 2.2**.**

Let $X(\operatorname*{\mathbf{t}})$ be a centered homogenous Gaussian field with unit variance and covariance function $r(\operatorname*{\mathbf{t}})$ satisfying $\mathbf{A1}-\mathbf{A3}$ and $\mathbf{u}_{\operatorname*{\mathbf{T}}}$, $\mathbf{v}_{\operatorname*{\mathbf{T}}}$ be given by (1.4). Then for any Pickands grids $\Re(a_{i}(2\log\prod_{i=1}^{d}T_{i})^{-1/\alpha_{i}})$ with $a_{i}>0$, $i=1,2,\cdots,d$, the following limit exists,

[TABLE]

and

[TABLE]

Theorem 2.3**.**

Let $X(\operatorname*{\mathbf{t}})$ be a centered homogenous Gaussian field with unit variance and covariance function $r(\operatorname*{\mathbf{t}})$ satisfying $\mathbf{A1}-\mathbf{A3}$. Then for any dense grids $\Re(\delta_{i})$, $i=1,2,\cdots,d$, we have

[TABLE]

where $\mathbf{u}_{\operatorname*{\mathbf{T}}}$, $\mathbf{v}_{\operatorname*{\mathbf{T}}}$ are given by (1.4).

Remark 2.1**.**

Similar to the weakly dependent stationary Gaussian sequences and processes, for homogeneous Gaussian random fields, Theorems 2.1-2.3 shows that $\mathbf{M}_{\operatorname*{\mathbf{T}}}$ and $\mathbf{m}_{\operatorname*{\mathbf{T}}}$ are asymptotically independent if $r=0$.

3 Auxiliary results

For simplicity, let $u=a_{\operatorname*{\mathbf{T}}}=\sqrt{2\log(\prod_{i=1}^{d}T_{i})}$, so for $i=1,2,\cdots,d$, if $\delta_{i}u^{2/\alpha_{i}}\to 0$, the grid is dense; 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 $\operatorname*{\mathbf{I}}_{(\boldsymbol{\delta})}=\prod_{i=1}^{d}[-\delta_{i},\delta_{i}]$ and

[TABLE]

where $v^{2}=(\sum_{i=1}^{d}{2}/{\alpha_{i}})\log u+\log(\prod_{i=1}^{d}\delta_{i}).$ The following Lemmas 3.1-3.3 extended Lemmas 1, 2 and 4 of Piterbarg (2004) from stationary Gaussian processes to homogenous Gaussian random fields.

Lemma 3.1**.**

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 (3.1) satisfies $P(u,x)=o(\Psi(u))$ as $u\to\infty$.

Proof. The proof is similar to Lemma 1 of Piterbarg (2004) and Lemma A1 of Tan and Wang (2015). ∎

Now, define

[TABLE]

and for small $\epsilon>0$

[TABLE]

Lemma 3.2**.**

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

[TABLE]

[TABLE]

as $u\to\infty$ and

[TABLE]

as $u\to\infty$, so that

[TABLE]

as $u\to\infty$.

Proof. The proof is similar to Lemma 2 of Piterbarg (2004) and Lemma A2 of Tan and Wang (2015). ∎

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

Lemma 3.3**.**

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

[TABLE]

where $\rho_{\operatorname*{\mathbf{a}}}\to 0$ as $\operatorname*{\mathbf{a}}\to 0$.

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). ∎

Following Tan and Wang (2015), define $\rho(\operatorname*{\mathbf{T}})=r/\log(\prod_{i=1}^{d}T_{i})$ 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 $n_{i}=\lfloor T_{i}/(T_{i}^{a}+T_{i}^{b})\rfloor$. Here $\lfloor\cdot\rfloor$ represents the integer part of the real number. Denote $\operatorname*{\mathbf{O}}_{\operatorname*{\mathbf{i}}}=\prod_{j=1}^{d}[(i_{j}-1)(T_{j}^{a}+T_{j}^{b}),(i_{j}-1)(T_{j}^{a}+T_{j}^{b})+T_{j}^{a}]$, $\operatorname*{\mathbf{E}}_{\operatorname*{\mathbf{i}}}=\prod_{j=1}^{d}[(i_{j}-1)(T_{j}^{a}+T_{j}^{b}),i_{j}(T_{j}^{a}+T_{j}^{b})]$, $\operatorname*{\mathbf{i}}=\mathbf{1},\cdots,\mathbf{n}$ and $\operatorname*{\mathbf{O}}=\bigcup_{\operatorname*{\mathbf{i}}}\operatorname*{\mathbf{O}}_{\operatorname*{\mathbf{i}}}$. Let $\{X_{\operatorname*{\mathbf{i}}}(\operatorname*{\mathbf{t}}),\operatorname*{\mathbf{t}}\geq\mathbf{0}\}$, $\operatorname*{\mathbf{i}}\geq 1$ be independent copies of $\{X(\operatorname*{\mathbf{t}}),\operatorname*{\mathbf{t}}\geq\mathbf{0}\}$ and $\{\eta(\operatorname*{\mathbf{t}}),\operatorname*{\mathbf{t}}\geq\mathbf{0}\}$ be such that $\eta(\operatorname*{\mathbf{t}})=X_{\operatorname*{\mathbf{i}}}(\operatorname*{\mathbf{t}})$ for $\operatorname*{\mathbf{t}}\in\operatorname*{\mathbf{E}}_{\operatorname*{\mathbf{i}}}$.

Define

[TABLE]

where $U$ is a standard normal variable independent of $\{\eta(\operatorname*{\mathbf{t}}),\operatorname*{\mathbf{t}}\geq 0\}$. Then $\varrho(\operatorname*{\mathbf{s}},\operatorname*{\mathbf{t}})$, covariance function of $\{\xi_{\operatorname*{\mathbf{T}}}(\operatorname*{\mathbf{t}}),\;\ \operatorname*{\mathbf{t}}\in\operatorname*{\mathbf{I}}_{\operatorname*{\mathbf{T}}}\}$, is

[TABLE]

Let

[TABLE]

[TABLE]

Lemma 3.4**.**

Suppose $\Re(\delta_{i}),i=1,2,\cdots,d$ are all sparse grids or all Pickands grids. Then for the $\mathbf{u}_{\operatorname*{\mathbf{T}}}$, $\mathbf{v}_{\operatorname*{\mathbf{T}}}$ given by (1.4) we have

[TABLE]

as $\operatorname*{\mathbf{T}}\to\infty$.

Proof. Note that the homogeneous Gaussian fields $\{X(\operatorname*{\mathbf{t}}),\operatorname*{\mathbf{t}}\geq\mathbf{0}\}$ and $\{-X(\operatorname*{\mathbf{t}}),\operatorname*{\mathbf{t}}\geq\mathbf{0}\}$ have the same distribution so that

[TABLE]

By arguments similar to Lemma 3.1 of Tan and Wang (2015) (denote by $mes(\cdot)$ the Lebesgue measure), we have

[TABLE]

as $\operatorname*{\mathbf{T}}\to\infty$. Combining (3.4), we can get the assertion of this lemma.∎

Lemma 3.5**.**

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 ${\operatorname*{\mathbf{u}}}_{\operatorname*{\mathbf{T}}}$ and $\operatorname*{\mathbf{v}}_{\operatorname*{\mathbf{T}}}$ given by (1.4) we have

[TABLE]

as $\operatorname*{\mathbf{T}}\to\infty$ and $\boldsymbol{\gamma}=(\gamma_{1},\gamma_{2},\cdots,\gamma_{d})\to 0$.

Proof. It follows from Lemma 3.3 and the fact $\{X(\operatorname*{\mathbf{t}}),\operatorname*{\mathbf{t}}\geq\mathbf{0}\}=^{d}\{-X(\operatorname*{\mathbf{t}}),\operatorname*{\mathbf{t}}\geq\mathbf{0}\}$ that the left hand side of (3.5) can be bounded by

[TABLE]

as $\operatorname*{\mathbf{T}}\to\infty$ and $\boldsymbol{\gamma}=(\gamma_{1},\gamma_{2},\cdots,\gamma_{d})\to 0$. The result follows. ∎

Lemma 3.6**.**

Suppose $\Re(\delta_{i}),i=1,2,\cdots,d$ are all sparse grids or all Pickands grids, then for ${\operatorname*{\mathbf{u}}}_{\operatorname*{\mathbf{T}}}$ and ${\operatorname*{\mathbf{v}}}_{\operatorname*{\mathbf{T}}}$ given by (1.4) we have

[TABLE]

as $\operatorname*{\mathbf{T}}\to\infty$.

Proof. It follows from the Normal Comparison Lemma (see e.g. Leadbetter et al. (1983) and Li and Shao (2002)) that the left hand side of (3.6) can be bounded by

[TABLE]

where $r^{h}(\mathbf{x},\mathbf{y})=hr(\mathbf{x},\mathbf{y})+(1-h)\varrho(\mathbf{x},\mathbf{y})$ with $h\in[0,1]$.

Next we will show that $F_{1}\to 0$, $F_{2}\to 0$ and $F_{3}\to\ 0$ as $\operatorname*{\mathbf{T}}\to\infty$, respectively. For $F_{1}$, we first consider the case that $\operatorname*{\mathbf{k}}\operatorname*{\mathbf{q}}$, $\operatorname*{\mathbf{l}}\operatorname*{\mathbf{q}}$ in the same interval firstly, and split $F_{1}$ into the following two parts:

[TABLE]

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 $|t_{i}|\leq\epsilon<2^{-1/\alpha_{i}}$,

[TABLE]

then, by definition of $\xi_{\operatorname*{\mathbf{T}}}(\operatorname*{\mathbf{t}})$, we have $\varrho(\operatorname*{\mathbf{k}}\operatorname*{\mathbf{q}},\operatorname*{\mathbf{l}}\operatorname*{\mathbf{q}})-r(\operatorname*{\mathbf{k}}\operatorname*{\mathbf{q}},\operatorname*{\mathbf{l}}\operatorname*{\mathbf{q}})=\rho(\operatorname*{\mathbf{T}})\left(1-r(\operatorname*{\mathbf{k}}\operatorname*{\mathbf{q}},\operatorname*{\mathbf{l}}\operatorname*{\mathbf{q}})\right)$ and $\varrho(\operatorname*{\mathbf{k}}\operatorname*{\mathbf{q}},\operatorname*{\mathbf{l}}\operatorname*{\mathbf{q}})\sim r(\operatorname*{\mathbf{k}}\operatorname*{\mathbf{q}},\operatorname*{\mathbf{l}}\operatorname*{\mathbf{q}})$ for sufficiently large $\operatorname*{\mathbf{T}}$. With $v_{\operatorname*{\mathbf{T}}}$ and $u_{\operatorname*{\mathbf{T}}}$ given by (1.4) we have

[TABLE]

Hence,

[TABLE]

as $u\to\infty$.

Let $\varpi(\operatorname*{\mathbf{t}},\operatorname*{\mathbf{s}})=\max\{|r(\operatorname*{\mathbf{t}},\operatorname*{\mathbf{s}})|,|\varrho(\operatorname*{\mathbf{t}},\operatorname*{\mathbf{s}})|\}$ and

[TABLE]

For $J_{\operatorname*{\mathbf{T}},2}$, by the fact that $u_{\operatorname*{\mathbf{T}}}^{2}(x_{2})\sim 2\log\prod_{i=1}^{d}T_{i}=u^{2}$ we have

[TABLE]

as $u\to\infty$ since $a<\frac{1-\theta(\epsilon)}{1+\theta(\epsilon)}$. Combining with (3.17), (3) and (3.26), we show that $F_{1}\to 0$ for the first case that $\operatorname*{\mathbf{k}}\operatorname*{\mathbf{q}}$, $\operatorname*{\mathbf{l}}\operatorname*{\mathbf{q}}$ in the same interval.

Second, we consider the case that $\operatorname*{\mathbf{k}}\operatorname*{\mathbf{q}}\in\operatorname*{\mathbf{O}}_{\operatorname*{\mathbf{i}}}$, $\operatorname*{\mathbf{l}}\operatorname*{\mathbf{q}}\in\operatorname*{\mathbf{O}}_{\operatorname*{\mathbf{j}}}$, $\operatorname*{\mathbf{i}}\neq\operatorname*{\mathbf{j}}$. Note that the distance between the points in any two rectangles $\operatorname*{\mathbf{O}}_{\operatorname*{\mathbf{i}}}$ and $\operatorname*{\mathbf{O}}_{\operatorname*{\mathbf{j}}}$ are larger than $\min\{T_{i}^{b},i=1,2,\cdots,d\}$ and $\varrho(\operatorname*{\mathbf{k}}\operatorname*{\mathbf{q}},\operatorname*{\mathbf{l}}\operatorname*{\mathbf{q}})=\rho(\operatorname*{\mathbf{T}})$ for $\operatorname*{\mathbf{k}}\operatorname*{\mathbf{q}}\in\operatorname*{\mathbf{O}}_{\operatorname*{\mathbf{i}}}$, $\operatorname*{\mathbf{l}}\operatorname*{\mathbf{q}}\in\operatorname*{\mathbf{O}}_{\operatorname*{\mathbf{j}}}$, $\operatorname*{\mathbf{i}}\neq\operatorname*{\mathbf{j}}$. Then, $F_{1}$ can be bounded by

[TABLE]

Split (3.33) 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 $S_{\operatorname*{\mathbf{T}},1}$, $S_{\operatorname*{\mathbf{T}},2}$, respectively. Let $\beta$ be such that $0<b<a<\beta<\frac{1-\theta(\epsilon)}{1+\theta(\epsilon)}$ for all sufficiently large $\operatorname*{\mathbf{T}}$.

For $S_{\operatorname*{\mathbf{T}},1}$, we can also divide it into the following two parts:

[TABLE]

then, by the same arguments as used in (3.26), we have

[TABLE]

as $u\to\infty$ since $\beta<\frac{1-\theta(\epsilon)}{1+\theta(\epsilon)}$.

To deal with $S_{\operatorname*{\mathbf{T}},12}$, we can define $w_{1}(\operatorname*{\mathbf{t}})=\max\{|r(\operatorname*{\mathbf{t}})|,|\rho(\operatorname*{\mathbf{T}})|\}$ and

[TABLE]

Denote $\operatorname*{\mathbf{T}}^{\beta}=(T_{1}^{\beta},T_{1}^{\beta},\cdots,T_{d}^{\beta})$ and by the condition $\mathbf{A3}$, we have $\theta_{1}(\operatorname*{\mathbf{T}}^{\beta})\leq Cu^{-2}$ for sufficiently large $\operatorname*{\mathbf{T}}$ and $\prod_{j=1}^{d}k_{j}q_{j}>{\prod_{i=1}^{d}T_{i}}^{\beta}=e^{\beta u^{2}/2}$, then, using (3.19), we have

[TABLE]

as $u\to\infty$. Therefore, we have

[TABLE]

Note that

[TABLE]

as $u\to\infty$. Combining condition $\mathbf{A3}$, and (3.38)-(3), we can get $S_{\operatorname*{\mathbf{T}},1}\to 0$ as $\operatorname*{\mathbf{T}}\to\infty$.

Now, for $S_{\operatorname*{\mathbf{T}},2}$, 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 the same arguments as used in (3) that

[TABLE]

as $u\to\infty$. If $\prod_{i=2}^{d}T_{i}>\prod_{i=1}^{d}T_{i}^{\beta}=e^{\beta u^{2}/2}$, split $S_{\operatorname*{\mathbf{T}},2}$ as follows:

[TABLE]

For $S_{\operatorname*{\mathbf{T}},21}$, similar to the arguments as used in (3), we have

[TABLE]

as $u\to\infty$. To deal with $S_{\operatorname*{\mathbf{T}},22}$, let $w_{2}(\operatorname*{\mathbf{t}})=\max\{|r(0,t_{2},\cdots,t_{d})|,|\rho(\operatorname*{\mathbf{T}})|\}$ and

[TABLE]

By condition $\mathbf{A3}$, we have $\theta_{2}(\operatorname*{\mathbf{T}}^{\beta})\leq Cu^{-2}$ for sufficiently large $\operatorname*{\mathbf{T}}$ and $\prod_{j=2}^{d}k_{j}q_{j}>\prod_{i=1}^{d}T_{i}^{\beta}=e^{\beta u^{2}/2}$, then, by the same arguments as used in (3.38),

[TABLE]

thus we have

[TABLE]

which implies $S_{\operatorname*{\mathbf{T}},22}\to 0$ as $\operatorname*{\mathbf{T}}\to\infty$. Therefore, we showed that $S_{\operatorname*{\mathbf{T}},2}\to 0$ as $\operatorname*{\mathbf{T}}\to\infty$. Combining with $S_{\operatorname*{\mathbf{T}},1}\to 0$, we showed that $F_{1}\to 0$ as $\operatorname*{\mathbf{T}}\to\infty$ for the second case.

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

If $\operatorname*{\mathbf{k}}\operatorname*{\mathbf{q}}$, $\operatorname*{\mathbf{l}}\boldsymbol{\delta}$ in the same interval $\operatorname*{\mathbf{O}}_{\operatorname*{\mathbf{i}}}$, split $F_{3}$ into two parts as

[TABLE]

Following condition $\mathbf{A1}$, we can choose small enough $\epsilon>0$ so (3.18) is satisfied and $\max\{|l_{j}\delta_{j}-k_{j}q_{j}|,j=1,2,\cdots,d\}\leq\epsilon$. By definition of $\xi_{\operatorname*{\mathbf{T}}}(\operatorname*{\mathbf{t}})$, we have $\varrho(\operatorname*{\mathbf{k}}\operatorname*{\mathbf{q}},\operatorname*{\mathbf{l}}\boldsymbol{\delta})=r(\operatorname*{\mathbf{k}}\operatorname*{\mathbf{q}},\operatorname*{\mathbf{l}}\boldsymbol{\delta})+\rho(\operatorname*{\mathbf{T}})\left(1-r(\operatorname*{\mathbf{k}}\operatorname*{\mathbf{q}},\operatorname*{\mathbf{l}}\boldsymbol{\delta})\right)\sim r(\operatorname*{\mathbf{k}}\operatorname*{\mathbf{q}},\operatorname*{\mathbf{l}}\boldsymbol{\delta})$ for sufficient large $\operatorname*{\mathbf{T}}$. 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 $F_{3}\to 0$. If $\Re(\delta_{i}),i=1,2,\cdots,d$ are all sparse grids, it follows from (1.4) that

[TABLE]

Similarly for $\frac{1}{2}\left(v_{\operatorname*{\mathbf{T}}}^{2}(x_{1})+(v_{\operatorname*{\mathbf{T}}}^{\boldsymbol{\delta}}(y_{1}))^{2}\right)$, $\frac{1}{2}\left(u_{\operatorname*{\mathbf{T}}}^{2}(x_{2})+(v_{\operatorname*{\mathbf{T}}}^{\boldsymbol{\delta}}(y_{1}))^{2}\right)$ and $\frac{1}{2}\left(v_{\operatorname*{\mathbf{T}}}^{2}(x_{1})+(u_{\operatorname*{\mathbf{T}}}^{\boldsymbol{\delta}}(y_{2}))^{2}\right)$. In view of (3.18), we have

[TABLE]

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, (3.68) can be bounded by

[TABLE]

as $u\to\infty$.

Noting that $u_{\operatorname*{\mathbf{T}}}\sim\left(2\log\prod_{i=1}^{d}T_{i}\right)^{1/2}=u$, we have

[TABLE]

as $u\to\infty$.

Next, we consider the case that $\operatorname*{\mathbf{k}}\operatorname*{\mathbf{q}}\in\operatorname*{\mathbf{O}}_{\operatorname*{\mathbf{i}}}$, $\operatorname*{\mathbf{l}}\boldsymbol{\delta}\in\operatorname*{\mathbf{O}}_{\operatorname*{\mathbf{j}}}$, $\operatorname*{\mathbf{i}}\neq\operatorname*{\mathbf{j}}$. Note that the distance between the points in any two rectangles $\operatorname*{\mathbf{O}}_{\operatorname*{\mathbf{i}}}$ and $\operatorname*{\mathbf{O}}_{\operatorname*{\mathbf{j}}}$ is larger than $\min\{T_{i}^{b},i=1,2,\cdots,d\}$ and $\varrho(\operatorname*{\mathbf{k}}\operatorname*{\mathbf{q}},\operatorname*{\mathbf{l}}\boldsymbol{\delta})=\rho(\operatorname*{\mathbf{T}})$. Then, $F_{3}$ is at most

[TABLE]

Split (3.85) 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 $H_{\operatorname*{\mathbf{T}},1}$, $H_{\operatorname*{\mathbf{T}},2}$, respectively. Then,

[TABLE]

By the arguments as used in (3.78), we have

[TABLE]

as $u\to\infty$.

For $H_{\operatorname*{\mathbf{T}},12}$, by the same arguments as used in (3.38), we have

[TABLE]

as $u\to\infty$. Then,

[TABLE]

By condition $\mathbf{A3}$, the first term on the right hand side of (3.92) tends to [math] as $\operatorname*{\mathbf{T}}\to\infty$, and the second term also tends to [math] by the same arguments of (3), then $H_{\operatorname*{\mathbf{T}},12}\to 0$ as $\operatorname*{\mathbf{T}}\to\infty$. Combining (3), we can get that $H_{\operatorname*{\mathbf{T}},1}\to 0$ as $\operatorname*{\mathbf{T}}\to\infty$.

Now, we consider $H_{\operatorname*{\mathbf{T}},2}$, 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 the same arguments as used in (3) that

[TABLE]

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

[TABLE]

For $H_{\operatorname*{\mathbf{T}},21}$, it follows from (3.101) that

[TABLE]

For $H_{\operatorname*{\mathbf{T}},22}$, by the same arguments as used in (3.38), we have

[TABLE]

hence,

[TABLE]

which implies $H_{\operatorname*{\mathbf{T}},22}\to 0$ as $\operatorname*{\mathbf{T}}\to\infty$. Furthermore, $H_{\operatorname*{\mathbf{T}},2}\to 0$, as $\operatorname*{\mathbf{T}}\to\infty$. Combining (3.75), (3.78), (3) and (3.92), we can get $F_{3}\to 0$ as $\operatorname*{\mathbf{T}}\to\infty$. The proof is complete. ∎

Lemma 3.7**.**

Suppose $\Re(\delta_{i}),i=1,2,\cdots,d$ are all sparse grids or all Pickands grids and ${\operatorname*{\mathbf{u}}}_{\operatorname*{\mathbf{T}}}$, $\operatorname*{\mathbf{v}}_{\operatorname*{\mathbf{T}}}$ are given by (1.4). 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

[TABLE]

as $\operatorname*{\mathbf{T}}\to\infty$ and all $\gamma_{i}\downarrow 0$, where

[TABLE]

Proof. By the definition of $\{\xi_{\operatorname*{\mathbf{T}}}(\operatorname*{\mathbf{t}}),\operatorname*{\mathbf{t}}\geq\mathbf{0}\}$ and $\{\eta(\operatorname*{\mathbf{t}}),\operatorname*{\mathbf{t}}\geq\mathbf{0}\}$, we have

[TABLE]

By Lemma 3.3 and the dominated convergence theorem, we have

[TABLE]

as $\operatorname*{\mathbf{T}}\to\infty$ and all $\gamma_{i}\downarrow 0$. Combining (3.109), we finish the proof. ∎

Lemma 3.8**.**

Let $X(\operatorname*{\mathbf{t}})$ be a centered homogeneous Gaussian field with unit variance and covariance function $r(\operatorname*{\mathbf{t}})$ satisfying $\mathbf{A1}-\mathbf{A3}$. Then for any sparse grids,

[TABLE]

where $\mathbf{u}_{\operatorname*{\mathbf{T}}}$, $\mathbf{v}_{\operatorname*{\mathbf{T}}}$ are given by (1.4).

Proof. Noting that $\{\eta(\operatorname*{\mathbf{t}}),\operatorname*{\mathbf{t}}\geq\mathbf{0}\}=^{d}\{-\eta(\operatorname*{\mathbf{t}}),\operatorname*{\mathbf{t}}\geq\mathbf{0}\}$, we have

[TABLE]

Since

[TABLE]

by Borell theorem and (3.108) that for some constant $c$, with $n_{i}=\lfloor T_{i}/(T_{i}^{a}+T_{i}^{b})\rfloor$ defined as before we have

[TABLE]

as $\operatorname*{\mathbf{T}}\to\infty.$ Similarly,

[TABLE]

as $\operatorname*{\mathbf{T}}\to\infty$. Using Lemma 3.2 and (3.7), we have

[TABLE]

as $\operatorname*{\mathbf{T}}\to\infty$, and

[TABLE]

as $\operatorname*{\mathbf{T}}\to\infty$. Since

[TABLE]

where

[TABLE]

as $\operatorname*{\mathbf{T}}\to\infty$, and

[TABLE]

as $\operatorname*{\mathbf{T}}\to\infty$. Then, combining (3.110)-(3.115) and Lemmas 3.4-3.7, the assertion of this lemma follows.

Lemma 3.9**.**

Let $X(\operatorname*{\mathbf{t}})$ be a centered homogeneous Gaussian field with unit variance and covariance function $r(\operatorname*{\mathbf{t}})$ satisfying $\mathbf{A1}-\mathbf{A3}$ and the $\mathbf{u}_{\operatorname*{\mathbf{T}}}$, $\mathbf{v}_{\operatorname*{\mathbf{T}}}$ be given by (1.4). Then for $H_{\operatorname*{\mathbf{a}},\boldsymbol{\alpha}}^{x,y}$ given in Theorem 2.2 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

[TABLE]

Proof. Suppose $\Re(\delta_{i})$ are all Pickands grids and $S_{i}$ satisfies the conditions in Lemma 3.3. Denote

[TABLE]

then by using Lemma 6.1 of Piterbarg (1996), we have

[TABLE]

as $u\to\infty$. Note that for Pickands grids,

[TABLE]

and

[TABLE]

Hence, from (3.116), we have

[TABLE]

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 $H_{\operatorname*{\mathbf{a}},\boldsymbol{\alpha}}^{x,y}$, we get $H_{\operatorname*{\mathbf{a}},\boldsymbol{\alpha}}^{0,Z_{x_{2},y_{2}}}\left(\prod_{i=1}^{d}H_{a_{i},\alpha_{i}}\right)^{-1}e^{-y_{2}}=H_{\operatorname*{\mathbf{a}},\boldsymbol{\alpha}}^{x_{2}+\log(\prod_{i=1}^{d}H_{\alpha_{i}}),y_{2}+\log(\prod_{i=1}^{d}H_{a_{i},\alpha_{i}})}$.

Furthermore, from (3.3) and (3), we have

[TABLE]

Similarly,

[TABLE]

Following the same arguments as used in (3.111), we have

[TABLE]

as $\operatorname*{\mathbf{T}}\to\infty$, and

[TABLE]

as $\operatorname*{\mathbf{T}}\to\infty$. Combining (3.110), (3.115), (3)-(3.121) and Lemmas 3.4-3.7, the assertion of this lemma follows.∎

4 Proofs of the main results

Proof of Theorem 2.1. Using Lemma 3.8 and letting $x_{1}\to-\infty$, $y_{1}\to-\infty$, respectively, we have

[TABLE]

and

[TABLE]

Similarly, letting $x_{1}\to-\infty$, $y_{1}\to-\infty$ in the same time, we can get

[TABLE]

Therefore by (4.1)-(4.3) and Lemma 3.8, we can get the desired result since

[TABLE]

∎

Proof of Theorem 2.2. It follows from Lemma 3.9 and the definition of $H_{\operatorname*{\mathbf{a}},\boldsymbol{\alpha}}^{x,y}$ that

[TABLE]

Similarly,

[TABLE]

and

[TABLE]

By the same arguments as used in the proof of Theorem 2.1, we can get the assertion of this theorem. ∎

Proof of Theorem 2.3. It follows from Lemma 3.3 that

[TABLE]

as $\operatorname*{\mathbf{T}}\to\infty$. Then from (3.113) and (3.114), we have

[TABLE]

as $\operatorname*{\mathbf{T}}\to\infty$, and

[TABLE]

as $\operatorname*{\mathbf{T}}\to\infty$. Using Lemma 3.3 again, we can get

[TABLE]

as $\operatorname*{\mathbf{T}}\to\infty$. Then by Borell theorem and (3.111), we have

[TABLE]

as $\operatorname*{\mathbf{T}}\to\infty$, and similarly,

[TABLE]

as $\operatorname*{\mathbf{T}}\to\infty$.

Hence, combining (3.110), (3.115) and (4.5)-(4.8), we have

[TABLE]

By the same arguments as used in (4.1)-(4.3), we have

[TABLE]

[TABLE]

and

[TABLE]

Therefore, from (4.4), we complete the proof. ∎