# On Piterbarg’s max-discretisation theorem for homogeneous Gaussian random
fields††thanks: Research supported by National Science Foundation of China
(No. 11326175), Natural Science Foundation of Zhejiang Province of China (No.
LQ14A010012) and Research Start-up Foundation of Jiaxing University (No.
70512021).

Zhongquan Tan111 E-mail address<EMAIL_ADDRESS>
College of Mathematics, Physics and Information Engineering, Jiaxing
University, Jiaxing 314001, PR China
Kaiyong Wang
School of Mathematics and Physics, Suzhou University of Science and
Technology, Suzhou 215009, PR China

> Abstract: Motivated by the papers of Piterbarg (2004) and Hüsler (2004), in
> this paper the asymptotic relation between the maximum of a continuous
> dependent homogeneous Gaussian random field and the maximum of this field
> sampled at discrete time points is studied. It is shown that, for the weakly
> dependent case, these two maxima are asymptotically independent, dependent
> and coincide when the grid of the discrete time points is a sparse grid,
> Pickands grid and dense grid, respectively, while for the strongly dependent
> case, these two maxima are asymptotically totally dependent if the grid of
> the discrete time points is sufficiently dense, and asymptotically dependent
> if the the grid points are sparse or Pickands grids.
>
> Key Words: continuous time process, dependence, discrete time process,
> extreme values, homogeneous Gaussian random fields.
>
> AMS Classification: Primary 60F05; secondary 60G15

##  1 Introduction

Let $\\{X(t),t\geq 0\\}$ be a stationary Gaussian process with mean 0,
variance 1, correlation function $r(t)$ and continuous sample functions. The
study on the limit distribution theory on the maximum of $\\{X(t),t\geq 0\\}$
up to time $T$: $M_{T}=\max\\{X(t),0\leq t\leq T\\}$ can be dated back to
Pickands (1969). Assume that the correlation function $r(t)$ satisfies for
some $\alpha\in(0,2]$,

$\displaystyle r(t)=1-|t|^{\alpha}+o(|t|^{\alpha})\ \ \mbox{as}\ \
t\rightarrow 0\ \ \mbox{and}\ \ r(t)<1\ \ \mbox{for}\ \ t>0$ (1)

and

$\displaystyle r(t)\log t\rightarrow 0,\ \ \mbox{as}\ \ t\rightarrow\infty.$
(2)

It is well known (see e.g. Pickands (1969), Leadbetter et al. 1983) that (1)
and (2) imply the following classic limit relation

$\displaystyle P\\{a_{T}(M_{T}-b_{T})\leq x\\}\rightarrow\exp(-e^{-x})$ (3)

as $T\rightarrow\infty$, where

$a_{T}=\sqrt{2\log T},\ \ b_{T}=\sqrt{2\log
T}+\frac{\log[(2\pi)^{-1/2}\mathcal{H}_{\alpha}(2\log
T)^{-1/2+1/\alpha}]}{\sqrt{2\log T}}.$

Here $\mathcal{H}_{\alpha}$ denotes Pickands constant, which is defined by
$\mathcal{H}_{\alpha}=\lim_{\lambda\rightarrow\infty}\mathcal{H}_{\alpha}(\lambda)/\lambda$,
with

$\mathcal{H}_{\alpha}(\lambda)=\mathbb{E}\exp\left(\max_{t\in[0,\lambda]}\sqrt{2}B_{\alpha/2}(t)-t^{\alpha}\right)$

and $B_{H}$ is a fractional Brownian motion, that is a Gaussian zero mean
process with stationary increments such that
$\mathbb{E}B_{H}^{2}(t)=|t|^{2H}$. It is also well known that
$0<\mathcal{H}_{\alpha}<\infty$, see e.g. Pickands (1969), Leadbetter et al.
(1983), Piterbarg (1996).

The extensions of the classic result (3) to more general cases, such as for
non-stationary case, strongly dependent case, can be found in Mittal and
Ylvisaker (1975), McCormick (1980), McCormick and Qi (2000), Hülser (1990),
Konstant and Piterbarg (1993), Seleznjev (1991, 1996), Hülser (1999), Hülser
et al. (2003), Tan et al. (2012) and among others.

In applied fields, however, the classic result (3) can not be used directly,
since the available samples are discrete. Usually, simulation techniques are
applied to derive results for continuous process when they can not be derived
with mathematical analytic tools. Simulations of such processes are performed
for discrete time-grids, while the results should be interpreted in the
context of continuous time. Therefore, it is crucial to investigate the
relation between the extremes of the continuous process and the extremes of
the discrete process.

Piterbarg (2004) first studied the asymptotic relation between $M_{T}$ and the
maximum of the discrete version $M_{T}^{\delta}=\max\\{X(k\delta),0\leq
k\delta\leq T\\}$ for some $\delta=\delta(T)>0,k\in\mathbb{N}$, where
$\mathbb{N}$ denotes the set of all natural numbers. Following Piterbarg
(2004), we consider uniform grids
$\mathfrak{R}=\mathfrak{R}(\delta)=\\{k\delta:k\in\mathbb{N}\\}$, $\delta>0$.
A grid is called sparse if $\delta$ is such that

$\delta(2\log T)^{1/\alpha}\rightarrow D$

with $D=\infty$. If $D\in(0,\infty)$, the grid is a Pickands grid, and if
$D=0$, the grid is dense.

For the stationary Gaussian processes, Piterbarg (2004) first showed that the
maximum $M_{T}^{\delta}$ of discrete time points and the maximum $M_{T}$ of
the continuous time points can be asymptotically independent, dependent or
totally dependent if the grid is a sparse, a Pickands or a dense grid,
respectively. This type of results are called Piterbarg’s max-discretisation
theorems in the literature, see eg. Tan and Hashorva (2014a).

Based on the results of Hüsler (1990), Piterbarg’s max-discretisation theorems
were extended by Hüsler (2004) to a class of locally stationary Gaussian
processes which was introduced by Berman (1974). Other related results such as
for the storage process with fractional Brownian motion as input and
stationary non-Gaussian case can be found in Hüsler and Piterbarg (2004) and
Turkman (2012), respectively. The recent contributions Tan and Wang (2013) and
Tan and Tang (2014) present Piterbarg’s max-discretisation theorem for
strongly dependent stationary Gaussian processes. The Piterbarg’s max-
discretisation theorems for multivariate Gaussian processes can be found in
Tan and Hashorva (2014b) and their improvement to different grids can be found
in Tan and Hashorva (2015).

The Piterbarg’s max-discretisation theorems for Gaussian processes have been
studied extensively under different conditions in the past, but it is far from
complete. In this paper, we are interested in the similar problems for the
Gaussian random fields. It is well known that Gaussian random fields play a
very important role in many applied sciences, such as in image analysis,
atmospheric sciences, geostatistics, neuroimaging, astrophysics, oceanography,
hydrology and agriculture, among others, see eg. Adler and Taylor (2007) for
details. Extremes and their limit properties are particularly important in
these applications, see, for instance, Azaïs and Wschebor (2009).

The paper is organized as follows: In Section 2, we present the main results
for weakly and strongly dependent Gaussian fields. Section 3 gives the proofs.
Some technical auxiliary results are presented in Sections 4 and 5. Let $\phi$
and $\Psi$ denote the density function and tail distribution function of a
standard normal variable.

##  2 Main results

Denote the set of all real numbers by $\mathbb{R}$ and let $\mathbb{R}^{d}$ be
d-dimensions product space of $\mathbb{R}$, where $d\geq 2$. In this paper, we
only consider the case of $d=2$ since it is notationally simplest and the
results for higher dimensions follow analogouss arguments. Here the operations
with the vectors are meant component-wise. For instance for two vectors
$\mathbf{t}=(t_{1},t_{2})$ and $\mathbf{s}=(s_{1},s_{2})$,
$\mathbf{s}\leq\mathbf{t}$, $\mathbf{t-s}$ and $\mathbf{st}$ mean $s_{i}\leq
t_{i}$, $i=1,2$, $(t_{1}-s_{1},t_{2}-s_{2})$ and $(s_{1}t_{1},s_{2}t_{2})$,
respectively. $\mathbf{T}\rightarrow\infty$ means $T_{i}\rightarrow\infty$,
$i=1,2$. Let $\mathbf{I_{T}}=\\{\mathbf{t}\in\mathbb{R}^{2}:0\leq t_{i}\leq
T_{i},i=1,2\\}$. Let $\\{X(\mathbf{t}):\mathbf{t}\geq\mathbf{0}\\}$ denote a
homogeneous Gaussian field with covariance function

$r(\mathbf{t})=\mathbb{C}ov(X(\mathbf{t}),X(\mathbf{0})).$

In this paper we assume that the covariance function satisfies the following
conditions:

  * A1:

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

  * A2:

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

  * A3:

$\lim_{\mathbf{t}\rightarrow\infty}r(\mathbf{t})\log(t_{1}t_{2})=r\in[0,\infty)$
and both $r(0,t_{2})\log t_{2}$ and $r(t_{1},0)\log t_{1}$ are bounded.

Throughout the paper, for any set $\mathbf{E}\subset\mathbb{R}^{2}$ and
$\mathbf{k}\in\mathbb{N}^{2}$, define

$M_{\mathbf{E}}=\max_{\mathbf{t}\in\mathbf{E}}X(\mathbf{t})=\max\\{X(\mathbf{t}),\mathbf{t}\in\mathbf{E}\\},\
\ \
M_{\mathbf{E}}^{\mathbf{p}}=\max_{\mathbf{t}\in\mathbf{E}\cap\mathfrak{R}(p_{1})\times\mathfrak{R}(p_{2})}X(\mathbf{t})=\max\\{X(\mathbf{kp}),\mathbf{kp}\in\mathbf{E}\\},$

where $\mathfrak{R}(p_{i})=\\{kp_{i},k\in\mathbb{N}\\}$, $i=1,2$, are uniform
grids. If $\mathbf{E}=\mathbf{I_{T}}$, we write the above two maxima for
simplicity as $M_{\mathbf{T}}$ and $M_{\mathbf{T}}^{\mathbf{p}}$,
respectively. For dealing with the multivariate case, we redefine the uniform
grids $\mathfrak{R}(p_{i})=\\{kp_{i},k\in\mathbb{N}\\}$, $i=1,2$ as following.
The grid $\mathfrak{R}(p_{i})$ is called sparse if $p_{i}=p_{i}(\mathbf{T})$
is such that

$p_{i}(2\log T_{1}T_{2})^{1/\alpha_{i}}\rightarrow D_{i},\ \ i=1,2$

with $D_{i}=\infty$. If $D_{i}\in(0,\infty)$, the grid is a Pickands grid, and
if $D_{i}=0$, the grid is dense.

Under conditions A1 and A2, Theorem 7.1 of Piterbarg (1996) showed that for
any fixed $\mathbf{h}>\mathbf{0}$

$\displaystyle
P\left(\max_{\mathbf{t}\in\mathbf{I_{h}}}X(\mathbf{t})>u\right)=\mathcal{H}_{\alpha_{1}}\mathcal{H}_{\alpha_{2}}h_{1}h_{2}u^{2/\alpha_{1}+2/\alpha_{2}}\Psi(u)(1+o(1)),$
(4)

as $u\rightarrow\infty$, where $\mathcal{H}_{\alpha_{i}}$, $i=1,2$ are the
Pickands constants. This exact asymptotic plays crucial role in deriving the
Gumbel law and also will be used in the proofs of our main results.

Now, we state our main results which extend the existing results (including
Piterbarg (2004) and Tan and Wang (2013)) to Gaussian random fields. The
extensions are, however, nontrivial in that asymptotic relation between two
Gaussian fields is more complicated than that of Gaussian processes. This can
be seen from the proof that follows.

Theorem 2.1. Let $\\{X(\mathbf{t}):\mathbf{t}\geq\mathbf{0}\\}$ be a
homogeneous Gaussian field with covariance function $r(\mathbf{t})$ satisfying
A1, A2 and A3. Then for any sparse grids $\mathfrak{R}(p_{i})$, $i=1,2$,

$\displaystyle
P\left\\{a_{\mathbf{T}}\big{(}M_{\mathbf{T}}-b_{\mathbf{T}}\big{)}\leq
x,a_{\mathbf{T}}\big{(}M_{\mathbf{T}}^{\mathbf{p}}-b_{\mathbf{T}}^{\mathbf{p}}\big{)}\leq
y\right\\}$ $\displaystyle\ \ \ \ \ \
\longrightarrow\int_{-\infty}^{+\infty}\exp\left(-\big{(}e^{-x-r+\sqrt{2r}z}+e^{-y-r+\sqrt{2r}z}\big{)}\right)\phi(z)dz$
(5)

as $\mathbf{T}\rightarrow\infty,$ where

$a_{\mathbf{T}}=\sqrt{2\log T_{1}T_{2}},\ \
b_{\mathbf{T}}=a_{\mathbf{T}}+a_{\mathbf{T}}^{-1}\log\left((2\pi)^{-1/2}\mathcal{H}_{\alpha_{1}}\mathcal{H}_{\alpha_{2}}(a_{\mathbf{T}})^{1/\alpha_{1}+1/\alpha_{2}-1/2}\right)$

and

$b_{\mathbf{T}}^{\mathbf{p}}=a_{\mathbf{T}}+a_{\mathbf{T}}^{-1}\log\left((2\pi)^{-1/2}p_{1}^{-1}p_{2}^{-1}(a_{\mathbf{T}})^{-1/2}\right).$

As the special case, we can obtain the limit distribution of the maximum for a
homogeneous Gaussian random field, which will be used in the proof of Theorem
2.3.

Corollary 2.1. Let $\\{X(\mathbf{t}):\mathbf{t}\geq\mathbf{0}\\}$ be a
homogeneous Gaussian field with covariance function $r(\mathbf{t})$ satisfying
A1, A2 and A3. Then for any $x\in\mathbb{R}$,

$\displaystyle
P\bigg{\\{}a_{\mathbf{T}}\big{(}M_{\mathbf{T}}-b_{\mathbf{T}}\big{)}\leq
x\bigg{\\}}\longrightarrow\int_{-\infty}^{+\infty}\exp\left(-e^{-x-r+\sqrt{2r}z}\right)\phi(z)dz$
(6)

as $\mathbf{T}\rightarrow\infty.$

Remark 2.2. A similar result as Corollary 2.1 can also be derived from Theorem
15.2 in Chapter 4 of Piterbarg (1996), where the author dealt with the limit
properties of uncrossing point processes under some slight different
conditions.

Before presenting the result for Pickands grids, we introduce the following
Pickands type constants. For $a>0$, define,

$\mathcal{H}_{a,\alpha}(\lambda)=\mathbb{E}\exp\left(\max_{ka\in[0,\lambda]}\sqrt{2}B_{\alpha/2}(ka)-(ka)^{\alpha}\right),$

we have (see Leadbetter et al. 1983),

$\mathcal{H}_{a,\alpha}=\lim_{\lambda\rightarrow\infty}\frac{\mathcal{H}_{a,\alpha}(\lambda)}{\lambda}\in(0,\infty).$

For any $\mathbf{d}>\mathbf{0}$, define

$\displaystyle\mathcal{H}_{\mathbf{d},\alpha_{1},\alpha_{2}}^{x,y}(\lambda_{1},\lambda_{2})=\int_{-\infty}^{+\infty}e^{s}P\left(\max_{(t_{1},t_{2})\in[0,\lambda_{1}]\times[0,\lambda_{2}]}\sqrt{2}\chi(t_{1},t_{2})>s+x,\right.$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \
\left.\max_{(k_{1}d_{1},k_{2}d_{2})\in[0,\lambda_{1}]\times[0,\lambda_{2}]}\sqrt{2}\chi(k_{1}d_{1},k_{2}d_{2})>s+y\right)ds,$

where

$\chi(t_{1},t_{2})=B^{(1)}_{\alpha_{1}/2}(t_{1})+B^{(2)}_{\alpha_{2}/2}(t_{2})-|t_{1}|^{\alpha_{1}}-|t_{2}|^{\alpha_{2}}$

and $B^{(1)}_{\alpha_{1}/2}(\cdot)$, $B^{(2)}_{\alpha_{2}/2}(\cdot)$ are two
independent fractional Brownian motions.

Theorem 2.2. Let $\\{X(\mathbf{t}):\mathbf{t}\geq\mathbf{0}\\}$ be a
homogeneous Gaussian field with covariance function $r(\mathbf{t})$ satisfying
A1, A2 and A3. Then for any Pickands grids
$\mathfrak{R}(p_{i})=\mathfrak{R}(a_{i}(2\log T_{1}T_{2})^{-1/\alpha_{i}})$
with $a_{i}>0$, $i=1,2$, the following limit exists,

$\mathcal{H}_{\mathbf{a},\alpha_{1},\alpha_{2}}^{x,y}:=\lim_{\lambda_{1}\rightarrow\infty\atop\lambda_{2}\rightarrow\infty}\mathcal{H}_{\mathbf{a},\alpha_{1},\alpha_{2}}^{x,y}(\lambda_{1},\lambda_{2})/\lambda_{1}\lambda_{2}\in(0,\infty)$

and

$\displaystyle
P\left\\{a_{\mathbf{T}}\big{(}M_{\mathbf{T}}-b_{\mathbf{T}}\big{)}\leq
x,a_{\mathbf{T}}\big{(}M_{\mathbf{T}}^{\mathbf{p}}-b_{\mathbf{a},\mathbf{T}}\big{)}\leq
y\right\\}$ $\displaystyle\ \ \ \ \
\longrightarrow\int_{-\infty}^{+\infty}\exp\left(-\big{(}e^{-x-r+\sqrt{2r}z}+e^{-y-r+\sqrt{2r}z}-\mathcal{H}_{\mathbf{a},\alpha_{1},\alpha_{2}}^{\log(\mathcal{H}_{\alpha_{1}}\mathcal{H}_{\alpha_{2}})+x,\log(\mathcal{H}_{a_{1},\alpha_{1}}\mathcal{H}_{a_{2},\alpha_{2}})+y}e^{-r+\sqrt{2r}z}\big{)}\right)\phi(z)dz$
(7)

as $\mathbf{T}\rightarrow\infty,$ where

$b_{\mathbf{a},\mathbf{T}}=a_{\mathbf{T}}+a_{\mathbf{T}}^{-1}\log\left((2\pi)^{-1/2}\mathcal{H}_{a_{1},\alpha_{1}}\mathcal{H}_{a_{2},\alpha_{2}}(a_{\mathbf{T}})^{1/\alpha_{1}+1/\alpha_{2}-1/2}\right).$

Theorem 2.3. Let $\\{X(\mathbf{t}):\mathbf{t}\geq\mathbf{0}\\}$ be a
homogeneous Gaussian field with covariance function $r(\mathbf{t})$ satisfying
A1, A2 and A3. Then for any dense grids $\mathfrak{R}(p_{i})$, $i=1,2$,

$\displaystyle
P\left\\{a_{\mathbf{T}}\big{(}M_{\mathbf{T}}-b_{\mathbf{T}}\big{)}\leq
x,a_{\mathbf{T}}\big{(}M_{\mathbf{T}}^{\mathbf{p}}-b_{\mathbf{T}}\big{)}\leq
y\right\\}\longrightarrow\int_{-\infty}^{+\infty}\exp\left(-e^{-\min(x,y)-r+\sqrt{2r}z}\right)\phi(z)dz$
(8)

as $\mathbf{T}\rightarrow\infty.$

Remark 2.2. i). In the literature, the Gaussian field $X(\mathbf{t})$ with
correlation satisfying
$\lim_{\mathbf{t}\rightarrow\infty}r(\mathbf{t})\log(t_{1}t_{2})=r\in[0,\infty)$
is called weakly and strongly dependent for $r=0$ and $r>0$, respectively, see
eg., Mittal and Ylvisaker (1975). Theorems 2.1-2.3 show that for the weakly
dependent case the two maxima are asymptotically independent, dependent and
coincide when the grid of the discrete time points is a sparse grid, Pickands
grid and dense grid, respectively. For the strongly dependent case, the
asymptotic independence between the two maxima does not hold anymore because
of the strong dependence of $X(\mathbf{t})$. However, in this case these two
maxima are asymptotically totally dependent if the grid of the discrete time
points is sufficiently dense, and asymptotically dependent if the the grid
points are sparse or Pickands grids.
ii). If $T_{1}=O(T_{2})$, as $\mathbf{T}\rightarrow\infty$, then the condition
that $r(0,t_{2})\log t_{2}$ and $r(t_{1},0)\log t_{1}$ are bounded in
Assumption ${\bf A3}$ can be omitted. Noting that Assumption ${\bf A3}$ is
only used in the proof of Lemma 3.3, it is easy to check this point from the
bounds of $S_{\mathbf{T},22}$ and $M_{\mathbf{T},22}$ in the proofs of Lemma
B1 and B3, respectively. However, if $T_{1}=o(T_{2})$, as
$\mathbf{T}\rightarrow\infty$, then Assumption ${\bf A3}$ can be weakened as:
$\lim_{\mathbf{t}\rightarrow\infty}r(\mathbf{t})\log(t_{1}t_{2})=r\in[0,\infty)$
and $r(0,t_{2})\log t_{2}$ is bounded. A similar statement holds also for the
case $T_{2}=o(T_{1})$.

##  3 Proofs

First, define $\rho(\mathbf{T})=r/\log(T_{1}T_{2})$ and let $a>b$ be constants
which will be determined in the proof of Lemma 3.3. Following Piterbarg
(2004), divide $[0,T_{i}]$ into intervals with length $T_{i}^{a}$ alternating
with shorter intervals with length $T_{i}^{b}$, $i=1,2$. Note that the numbers
of the long intervals is at most $n_{i}=\lfloor
T_{i}/(T_{i}^{a}+T_{i}^{b})\rfloor$, where $\lfloor x\rfloor$ denote the
integral parts of $x$. Let
$\mathbf{O_{i}}=[(i_{1}-1)(T_{1}^{a}+T_{1}^{b}),(i_{1}-1)(T_{1}^{a}+T_{1}^{b})+T_{1}^{a}]\times[(i_{2}-1)(T_{2}^{a}+T_{2}^{b}),(i_{2}-1)(T_{2}^{a}+T_{2}^{b})+T_{2}^{a}]$,
$\mathbf{i}=\mathbf{1},\cdots,\mathbf{n}$ and
$\mathbf{O}=\cup_{\mathbf{i}}\mathbf{O}_{\mathbf{i}}$. We will show blow that
the remaining area $\mathbf{I_{T}}\backslash\mathbf{O}$ plays no role in our
consideration.

Let $\\{X_{\mathbf{i}}(\mathbf{t}),\mathbf{t}\geq\mathbf{0}\\}$,
$\mathbf{i}\geq\mathbf{1}$ be independent copies of
$\\{X(\mathbf{t}),\mathbf{t}\geq\mathbf{0}\\}$ and
$\\{\eta(\mathbf{t}),\mathbf{t}\geq\mathbf{0}\\}$ be such that
$\eta(\mathbf{t})=X_{\mathbf{i}}(\mathbf{t})$ for
$\mathbf{t}\in\mathbf{E_{i}}:=[(i_{1}-1)(T_{1}^{a}+T_{1}^{b}),i_{1}(T_{1}^{a}+T_{1}^{b}))\times[(i_{2}-1)(T_{2}^{a}+T_{2}^{b}),i_{2}(T_{2}^{a}+T_{2}^{b}))$,
$\mathbf{i}=\mathbf{1},\cdots,\mathbf{n}$. Define

$\xi_{\mathbf{T}}(\mathbf{t})=\big{(}1-\rho(\mathbf{T})\big{)}^{1/2}\eta(\mathbf{t})+\rho^{1/2}(\mathbf{T})U,\
\ \mathbf{t}\in\mathbf{I_{T}},$

where $U$ is a standard normal variable independent of
$\\{\eta(\mathbf{t}),\mathbf{t}\geq\mathbf{0}\\}$. Denote by
$\varrho(\mathbf{s},\mathbf{t})$ the covariance function of
$\\{\xi_{\mathbf{T}}(\mathbf{t}),\mathbf{t}\in\mathbf{I_{T}}\\}$. It is easy
to check that

$\varrho(\mathbf{s},\mathbf{t})=\left\\{\begin{array}[]{cc}{r(\mathbf{t},\mathbf{s})+(1-r(\mathbf{t},\mathbf{s}))\rho(\mathbf{T})},&\mathbf{s}\in\mathbf{E_{i}},\mathbf{t}\in\mathbf{E_{j}},\mathbf{i}=\mathbf{j};\\\
{\rho(\mathbf{T})},&\mathbf{s}\in\mathbf{E_{i}},\mathbf{t}\in\mathbf{E_{j}},\mathbf{i}\neq\mathbf{j}.\end{array}\right.$

The proofs of our main results rely on the following Lemmas. In the sequel,
$C$ shall denote positive constant whose values may vary from place to place.

Lemma 3.1. Suppose that the grids $\mathfrak{R}(p_{i})$, $i=1,2$ are sparse
grids or Pickands grids. For any $B>0$, we have for all $x,y\in[-B,B]$,

$\displaystyle\bigg{|}P\left\\{a_{\mathbf{T}}\big{(}M_{\mathbf{T}}-b_{\mathbf{T}}\big{)}\leq
x,a_{\mathbf{T}}\big{(}M_{\mathbf{T}}^{\mathbf{p}}-b_{\mathbf{T}}^{{}^{\prime}}\big{)}\leq
y\right\\}$ $\displaystyle\ \ \ \ \
-P\left\\{a_{\mathbf{T}}\big{(}\max_{\mathbf{t}\in\mathbf{O}}X(\mathbf{t})-b_{\mathbf{T}}\big{)}\leq
x,a_{\mathbf{T}}\big{(}\max_{\mathbf{t}\in\mathfrak{R}(\delta_{1})\times\mathfrak{R}(\delta_{2})\cap\mathbf{O}}X(\mathbf{t})-b_{\mathbf{T}}^{{}^{\prime}}\big{)}\leq
y\right\\}\bigg{|}\rightarrow 0$

as $\mathbf{T}\rightarrow\infty$, where
$b_{\mathbf{T}}^{{}^{\prime}}=b_{\mathbf{T}}^{\mathbf{p}}$ for sparse grids
and $b_{\mathbf{T}}^{{}^{\prime}}=b_{\mathbf{a},\mathbf{T}}$ for Pickands
grids.

Proof: The proof is similar to that of Lemma 6 of Piterbarg (2004). Clearly,
we have

$\displaystyle\bigg{|}P\left\\{a_{\mathbf{T}}\big{(}M_{\mathbf{T}}-b_{\mathbf{T}}\big{)}\leq
x,a_{\mathbf{T}}\big{(}M_{\mathbf{T}}^{\mathbf{p}}-b_{\mathbf{T}}^{{}^{\prime}}\big{)}\leq
y\right\\}$ $\displaystyle\ \ \ \ \
-P\left\\{a_{\mathbf{T}}\big{(}\max_{\mathbf{t}\in\mathbf{O}}X(\mathbf{t})-b_{\mathbf{T}}\big{)}\leq
x,a_{\mathbf{T}}\big{(}\max_{\mathbf{t}\in\mathfrak{R}(\delta_{1})\times\mathfrak{R}(\delta_{2})\cap\mathbf{O}}X(\mathbf{t})-b_{\mathbf{T}}^{{}^{\prime}}\big{)}\leq
y\right\\}\bigg{|}$ $\displaystyle\ \ \ \ \ \leq
P\left\\{\max_{\mathbf{t}\in\mathbf{I_{T}}\backslash\mathbf{O}}X(\mathbf{t})>b_{\mathbf{T}}+x/a_{\mathbf{T}}\right\\}+P\left\\{\max_{\mathbf{t}\in\mathfrak{R}(\delta_{1})\times\mathfrak{R}(\delta_{2})\cap\mathbf{I_{T}}\backslash\mathbf{O}}X(\mathbf{t})>b_{\mathbf{T}}^{{}^{\prime}}+y/a_{\mathbf{T}}\right\\}$
(9)

By Theorem 7.2 of Piterbarg (1996) (denote by $mes(\cdot)$ the Lebesgue
measure)

$\displaystyle
P\left\\{\max_{\mathbf{t}\in\mathbf{I_{T}}\backslash\mathbf{O}}X(\mathbf{t})>b_{\mathbf{T}}+x/a_{\mathbf{T}}\right\\}$
$\displaystyle=$ $\displaystyle
O(1)mes(\mathbf{I_{T}}\backslash\mathbf{O})(b_{\mathbf{T}}+x/a_{\mathbf{T}})^{2/\alpha_{1}+2/\alpha_{2}}\Psi(b_{\mathbf{T}}+x/a_{\mathbf{T}})$
$\displaystyle=$ $\displaystyle
O(1)\frac{mes(\mathbf{I_{T}}\backslash\mathbf{O})}{T_{1}T_{2}}$
$\displaystyle\leq$ $\displaystyle
O(1)\frac{n_{1}T_{1}^{b}(T_{2}^{a}+T_{2}^{b})+n_{2}T_{2}^{b}(T_{1}^{a}+T_{1}^{b})}{T_{1}T_{2}}\rightarrow
0$

as $\mathbf{T}\rightarrow\infty$, by the choice of $a_{\mathbf{T}}$ and
$b_{\mathbf{T}}$. In light of (16) and (4.2) in the Appendix for a sparse grid
and Pickands grid, respectively, we can get the same estimation for the second
probability in the right-hand side of (3), hence the proof is complete. $\Box$

For the proofs we need also the following auxiliary grids
$\mathfrak{R}(q_{i})$ with $q_{i}=\gamma_{i}(2\log
T_{1}T_{2})^{-1/\alpha_{i}}$ and $\gamma_{i}>0$, $i=1,2$.

Lemma 3.2. Suppose that the grids $\mathfrak{R}(p_{i})$, $i=1,2$ are sparse
grids or Pickands grids. For any $B>0$, we have for all $x,y\in[-B,B]$

$\displaystyle\bigg{|}P\left\\{a_{\mathbf{T}}\big{(}\max_{\mathbf{t}\in\mathbf{O}}X(\mathbf{t})-b_{\mathbf{T}}\big{)}\leq
x,a_{\mathbf{T}}\big{(}\max_{\mathbf{t}\in\mathfrak{R}(p_{1})\times\mathfrak{R}(p_{2})\cap\mathbf{O}}X(\mathbf{t})-b_{\mathbf{T}}^{{}^{\prime}}\big{)}\leq
y\right\\}$ $\displaystyle\ \ \ \ \
-P\left\\{a_{\mathbf{T}}\big{(}\max_{\mathbf{t}\in\mathfrak{R}(q_{1})\times\mathfrak{R}(q_{2})\cap\mathbf{O}}X(\mathbf{t})-b_{T}\big{)}\leq
x,a_{\mathbf{T}}\big{(}\max_{\mathbf{t}\in\mathfrak{R}(p_{1})\times\mathfrak{R}(p_{2})\cap\mathbf{O}}X(\mathbf{t})-b_{\mathbf{T}}^{{}^{\prime}}\big{)}\leq
y\right\\}\bigg{|}\rightarrow 0$

as $\mathbf{T}\rightarrow\infty$ and $\gamma_{i}\downarrow 0$, where
$b_{\mathbf{T}}^{{}^{\prime}}=b_{\mathbf{T}}^{\mathbf{p}}$ for sparse grids
and $b_{\mathbf{T}}^{{}^{\prime}}=b_{\mathbf{a},\mathbf{T}}$ for Pickands
grids.

Proof: It follows from Lemma A4. $\Box$

The following lemma plays a crucial role in the proofs of Theorems 2.1 and
2.2.

Lemma 3.3. Suppose that the grids $\mathfrak{R}(p_{i})$, $i=1,2$ are sparse
grids or Pickands grids. For any $B>0$ we have for all $x,y\in[-B,B]$,

$\displaystyle\bigg{|}P\left\\{a_{\mathbf{T}}\big{(}\max_{\mathbf{t}\in\mathfrak{R}(q_{1})\times\mathfrak{R}(q_{2})\cap\mathbf{O}}X(\mathbf{t})-b_{T}\big{)}\leq
x,a_{\mathbf{T}}\big{(}\max_{\mathbf{t}\in\mathfrak{R}(p_{1})\times\mathfrak{R}(p_{2})\cap\mathbf{O}}X(\mathbf{t})-b_{\mathbf{T}}^{{}^{\prime}}\big{)}\leq
y\right\\}$ $\displaystyle\ \ \ \ \
-P\left\\{a_{\mathbf{T}}\big{(}\max_{\mathbf{t}\in\mathfrak{R}(q_{1})\times\mathfrak{R}(q_{2})\cap\mathbf{O}}\xi_{\mathbf{T}}(\mathbf{t})-b_{\mathbf{T}}\big{)}\leq
x,a_{\mathbf{T}}\big{(}\max_{\mathbf{t}\in\mathfrak{R}(p_{1})\times\mathfrak{R}(p_{2})\cap\mathbf{O}}\xi_{\mathbf{T}}(\mathbf{t})-b_{\mathbf{T}}^{{}^{\prime}}\big{)}\leq
y\right\\}\bigg{|}\rightarrow 0$

as $\mathbf{T}\rightarrow\infty$, where
$b_{\mathbf{T}}^{{}^{\prime}}=b_{\mathbf{T}}^{\mathbf{p}}$ for sparse grids
and $b_{\mathbf{T}}^{{}^{\prime}}=b_{\mathbf{a},\mathbf{T}}$ for Pickands
grids.

Proof: For the sake of simplicity, let
$u_{\mathbf{T}}=b_{\mathbf{T}}+x/a_{\mathbf{T}}$,
$u_{\mathbf{T}}^{\prime}=b_{\mathbf{T}}^{\prime}+y/a_{\mathbf{T}}$. Using the
Normal Comparison Lemma (see eg. Leadbetter et al. (1983) and Piterbarg
(1996)), we have

$\displaystyle\bigg{|}P\left\\{a_{\mathbf{T}}\big{(}\max_{\mathbf{t}\in\mathfrak{R}(q_{1})\times\mathfrak{R}(q_{2})\cap\mathbf{O}}X(\mathbf{t})-b_{\mathbf{T}}\big{)}\leq
x,a_{\mathbf{T}}\big{(}\max_{\mathbf{t}\in\mathfrak{R}(p_{1})\times\mathfrak{R}(p_{2})\cap\mathbf{O}}X(\mathbf{t})-b_{\mathbf{T}}^{{}^{\prime}}\big{)}\leq
y\right\\}$ $\displaystyle\ \ \ \ \
-P\left\\{a_{\mathbf{T}}\big{(}\max_{\mathbf{t}\in\mathfrak{R}(q_{1})\times\mathfrak{R}(q_{2})\cap\mathbf{O}}\xi_{\mathbf{T}}(\mathbf{t})-b_{\mathbf{T}}\big{)}\leq
x,a_{\mathbf{T}}\big{(}\max_{\mathbf{t}\in\mathfrak{R}(p_{1})\times\mathfrak{R}(p_{2})\cap\mathbf{O}}\xi_{\mathbf{T}}(\mathbf{t})-b_{\mathbf{T}}^{{}^{\prime}}\big{)}\leq
y\right\\}\bigg{|}$
$\displaystyle\leq\sum_{\mathbf{kq}\in\mathbf{O_{i}},\mathbf{lq}\in\mathbf{O_{j}}\atop\mathbf{kq}\neq\mathbf{lq},\mathbf{1}\leq\mathbf{i,j}\leq\mathbf{n}}|r(\mathbf{kq},\mathbf{lq})-\varrho(\mathbf{kq},\mathbf{lq})|\int_{0}^{1}\frac{1}{\sqrt{1-r^{(h)}(\mathbf{kq},\mathbf{lq})}}\exp\left(-\frac{u_{\mathbf{T}}^{2}}{1+r^{(h)}(\mathbf{kq},\mathbf{lq})}\right)dh$
$\displaystyle+\sum_{\mathbf{\mathbf{kp}}\in\mathbf{O_{i}},\mathbf{\mathbf{lp}}\in\mathbf{O_{j}}\atop\mathbf{kp}\neq\mathbf{lp},\mathbf{1}\leq\mathbf{i,j}\leq\mathbf{n}}|r(\mathbf{kp},\mathbf{lp})-\varrho(\mathbf{kp},\mathbf{lp})|\int_{0}^{1}\frac{1}{\sqrt{1-r^{(h)}(\mathbf{kp},\mathbf{lp})}}\exp\left(-\frac{u_{\mathbf{T}}^{\prime
2}}{1+r^{(h)}(\mathbf{kp},\mathbf{lp})}\right)dh$
$\displaystyle+\sum_{\mathbf{kq}\in\mathbf{O_{i}},\mathbf{lp}\in\mathbf{O_{j}}\atop\mathbf{kq}\neq\mathbf{lp},\mathbf{1}\leq\mathbf{i,j}\leq\mathbf{n}}|r(\mathbf{kq},\mathbf{lp})-\varrho(\mathbf{kq},\mathbf{lp})|\int_{0}^{1}\frac{1}{\sqrt{1-r^{(h)}(\mathbf{kq},\mathbf{lp})}}\exp\left(-\frac{u_{\mathbf{T}}^{\prime
2}+u_{\mathbf{T}}^{2}}{2(1+r^{(h)}(\mathbf{kq},\mathbf{lp}))}\right)dh,$

where
$r^{(h)}(\mathbf{kq},\mathbf{lq})=hr(\mathbf{kq},\mathbf{lq})+(1-h)\varrho(\mathbf{kq},\mathbf{lq})$.
Now, the lemma follows from Lemmas B1-B3 in the Appendix B. $\Box$

Lemma 3.4. Suppose that the grids $\mathfrak{R}(p_{i})$, $i=1,2$ are sparse
grids or Pickands grids. For any $B>0$ we have for all $x,y\in[-B,B]$ and the
grids $\mathfrak{R}(q_{i})$ with $q_{i}=\gamma_{i}(2\log
T_{1}T_{2})^{-1/\alpha_{i}}$ and $\gamma_{i}>0$, $i=1,2$

$\displaystyle\bigg{|}P\left\\{a_{\mathbf{T}}\big{(}\max_{\mathbf{t}\in\mathfrak{R}(q_{1})\times\mathfrak{R}(q_{2})\cap\mathbf{O}}\xi_{\mathbf{T}}(\mathbf{t})-b_{\mathbf{T}}\big{)}\leq
x,a_{\mathbf{T}}\big{(}\max_{\mathbf{t}\in\mathfrak{R}(p_{1})\times\mathfrak{R}(p_{2})\cap\mathbf{O}}\xi_{\mathbf{T}}(\mathbf{t})-b_{\mathbf{T}}^{{}^{\prime}}\big{)}\leq
y\right\\}$ $\displaystyle\ \ \ \ \ \
-\int_{-\infty}^{+\infty}\prod_{i_{1}=1}^{n_{1}}\prod_{i_{2}=1}^{n_{2}}P\left\\{\max_{\mathbf{t}\in\mathbf{O_{i}}}\eta(\mathbf{t})\leq
u_{\mathbf{T}}^{*},\max_{\mathbf{t}\in\mathbf{O_{i}}}\eta(\mathbf{t})\leq
u_{\mathbf{T}}^{*^{\prime}}\right\\}\phi(z)dz\bigg{|}\rightarrow 0,$

as $\gamma_{i}\downarrow 0$, where

$\displaystyle
u_{\mathbf{T}}^{*}:=\frac{b_{\mathbf{T}}+x/a_{\mathbf{T}}-\rho^{1/2}(\mathbf{T})z}{(1-\rho(\mathbf{T}))^{1/2}}=\frac{x+r-\sqrt{2r}z}{a_{\mathbf{T}}}+b_{\mathbf{T}}+o(a_{\mathbf{T}}^{-1}),$
(10)

and

$\displaystyle
u_{\mathbf{T}}^{*^{\prime}}:=\frac{b_{\mathbf{T}}^{{}^{\prime}}+y/a_{\mathbf{T}}-\rho^{1/2}(\mathbf{T})z}{(1-\rho(\mathbf{T}))^{1/2}}=\frac{y+r-\sqrt{2r}z}{a_{\mathbf{T}}}+b_{\mathbf{T}}^{{}^{\prime}}+o(a_{\mathbf{T}}^{-1})$
(11)

with $b_{\mathbf{T}}^{{}^{\prime}}=b_{\mathbf{T}}^{\mathbf{p}}$ for sparse
grids and $b_{\mathbf{T}}^{{}^{\prime}}=b_{\mathbf{a},\mathbf{T}}$ for
Pickands grids.

Proof: First, by the definition of
$\\{\xi_{\mathbf{T}}(\mathbf{t}),\mathbf{0}\leq\mathbf{t}\leq\mathbf{T}\\}$,
we have

$\displaystyle
P\left\\{a_{\mathbf{T}}\big{(}\max_{\mathbf{t}\in\mathfrak{R}(q_{1})\times\mathfrak{R}(q_{2})\cap\mathbf{O}}\xi_{\mathbf{T}}(\mathbf{t})-b_{\mathbf{T}}\big{)}\leq
x,a_{\mathbf{T}}\big{(}\max_{\mathbf{t}\in\mathfrak{R}(p_{1})\times\mathfrak{R}(p_{2})\cap\mathbf{O}}\xi_{\mathbf{T}}(\mathbf{t})-b_{\mathbf{T}}^{{}^{\prime}}\big{)}\leq
y\right\\}$ $\displaystyle\ \
=\int_{-\infty}^{+\infty}P\left\\{\max_{\mathbf{t}\in\mathfrak{R}(q_{1})\times\mathfrak{R}(q_{2})\cap\mathbf{O}}\eta(\mathbf{t})\leq
u_{\mathbf{T}}^{*},\max_{\mathbf{t}\in\mathfrak{R}(p_{1})\times\mathfrak{R}(p_{2})\cap\mathbf{O}}\eta(\mathbf{t})\leq
u_{\mathbf{T}}^{*^{\prime}}\right\\}\phi(z)dz$ $\displaystyle\ \
=\int_{-\infty}^{+\infty}\prod_{i_{1}=1}^{n_{1}}\prod_{i_{2}=1}^{n_{2}}P\left\\{\max_{\mathbf{t}\in\mathfrak{R}(q_{1})\times\mathfrak{R}(q_{2})\cap\mathbf{O_{i}}}\eta(\mathbf{t})\leq
u_{\mathbf{T}}^{*},\max_{t\in\mathfrak{R}(p_{1})\times\mathfrak{R}(p_{2})\cap\mathbf{O_{i}}}\eta(\mathbf{t})\leq
u_{\mathbf{T}}^{*^{\prime}}\right\\}\phi(z)dz.$ (12)

As for the discrete case, see page 137 on Leadbetter et al. (1983), a direct
calculation leads to

$\displaystyle u_{\mathbf{T}}^{*}$ $\displaystyle=$
$\displaystyle\frac{x+r-\sqrt{2r}z}{a_{\mathbf{T}}}+b_{\mathbf{T}}+o(a_{\mathbf{T}}^{-1})$

and

$\displaystyle u_{\mathbf{T}}^{*^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{y+r-\sqrt{2r}z}{a_{\mathbf{T}}}+b_{\mathbf{T}}^{{}^{\prime}}+o(a_{\mathbf{T}}^{-1}).$

Next, by Lemma A4 and the dominated convergence theorem, we have

$\displaystyle\bigg{|}\int_{-\infty}^{+\infty}\prod_{i_{1}=1}^{n_{1}}\prod_{i_{2}=1}^{n_{2}}P\left\\{\max_{\mathbf{t}\in\mathfrak{R}(q_{1})\times\mathfrak{R}(q_{2})\cap\mathbf{O_{i}}}\eta(\mathbf{t})\leq
u_{\mathbf{T}}^{*},\max_{\mathbf{t}\in\mathfrak{R}(p_{1})\times\mathfrak{R}(p_{2})\cap\mathbf{O_{i}}}\eta(\mathbf{t})\leq
u_{\mathbf{T}}^{*^{\prime}}\right\\}\phi(z)dz$ $\displaystyle\ \ \ \ \ \
-\int_{-\infty}^{+\infty}\prod_{i_{1}=1}^{n_{1}}\prod_{i_{2}=1}^{n_{2}}P\left\\{\max_{\mathbf{t}\in\mathbf{O_{i}}}\eta(\mathbf{t})\leq
u_{\mathbf{T}}^{*},\max_{\mathbf{t}\in\mathfrak{R}(p_{1})\times\mathfrak{R}(p_{2})\cap\mathbf{O_{i}}}\eta(\mathbf{t})\leq
u_{\mathbf{T}}^{*^{\prime}}\right\\}\phi(z)dz\bigg{|}\rightarrow 0$ (13)

as $\gamma_{i}\downarrow 0$. Lemma 3.4 now follows by combining (3) with (3).
$\Box$

Proof of Theorem 2.1. From Lemmas 3.1-3.4, we known that in order to prove
Theorem 2.1, it suffice to show that

$\displaystyle\bigg{|}\int_{-\infty}^{+\infty}\prod_{i_{1}=1}^{n_{1}}\prod_{i_{2}=1}^{n_{2}}P\left\\{\max_{\mathbf{t}\in\mathbf{O_{i}}}\eta(\mathbf{t})\leq
u_{\mathbf{T}}^{*},\max_{\mathbf{t}\in\mathfrak{R}(p_{1})\times\mathfrak{R}(p_{2})\cap\mathbf{O_{i}}}\eta(\mathbf{t})\leq
u_{\mathbf{T}}^{*^{\prime}}\right\\}\phi(z)dz$ $\displaystyle\ \ \ \ \ \ \
-\int_{-\infty}^{+\infty}\exp\left(-\big{(}e^{-x-r+\sqrt{2r}z}+e^{-y-r+\sqrt{2r}z}\big{)}\right)\phi(z)dz\bigg{|}\rightarrow
0$

as $\mathbf{T}\rightarrow\infty$, where $u_{\mathbf{T}}^{*}$ and
$u_{\mathbf{T}}^{*^{\prime}}$ are defined in Lemma 3.4. Using the homogeneity
of $\\{\eta(\mathbf{t}),\mathbf{t}\geq\mathbf{0}\\}$,

$\displaystyle\prod_{i_{1}=1}^{n_{1}}\prod_{i_{2}=1}^{n_{2}}P\left\\{\max_{\mathbf{t}\in\mathbf{O_{i}}}\eta(\mathbf{t})\leq
u_{\mathbf{T}}^{*},\max_{\mathbf{t}\in\mathfrak{R}(p_{1})\times\mathfrak{R}(p_{2})\cap\mathbf{O_{i}}}\eta(\mathbf{t})\leq
u_{\mathbf{T}}^{*^{\prime}}\right\\}$
$\displaystyle=\left(P\left\\{\max_{\mathbf{t}\in[0,T_{1}^{a}]\times[0,T_{2}^{a}]}\eta(\mathbf{t})\leq
u_{\mathbf{T}}^{*},\max_{\mathbf{t}\in\mathfrak{R}(p_{1})\times\mathfrak{R}(p_{2})\cap[0,T_{1}^{a}]\times[0,T_{2}^{a}]}\eta(\mathbf{t})\leq
u_{\mathbf{T}}^{*^{\prime}}\right\\}\right)^{n_{1}n_{2}}$
$\displaystyle=\exp\left(n_{1}n_{2}\log\left(P\left\\{\max_{\mathbf{t}\in[0,T_{1}^{a}]\times[0,T_{2}^{a}]}\eta(\mathbf{t})\leq
u_{\mathbf{T}}^{*},\max_{\mathbf{t}\in\mathfrak{R}(p_{1})\times\mathfrak{R}(p_{2})\cap[0,T_{1}^{a}]\times[0,T_{2}^{a}]}\eta(\mathbf{t})\leq
u_{\mathbf{T}}^{*^{\prime}}\right\\}\right)\right)$
$\displaystyle=\exp\left(-n_{1}n_{2}\left(1-P\left\\{\max_{\mathbf{t}\in[0,T_{1}^{a}]\times[0,T_{2}^{a}]}\eta(\mathbf{t})\leq
u_{\mathbf{T}}^{*},\max_{\mathbf{t}\in\mathfrak{R}(p_{1})\times\mathfrak{R}(p_{2})\cap[0,T_{1}^{a}]\times[0,T_{2}^{a}]}\eta(\mathbf{t})\leq
u_{\mathbf{T}}^{*^{\prime}}\right\\}\right)+R_{\mathbf{n}}\right).$

Since

$P_{\mathbf{n}}=:P\left\\{\max_{\mathbf{t}\in[0,T_{1}^{a}]\times[0,T_{2}^{a}]}\eta(\mathbf{t})\leq
u_{\mathbf{T}}^{*},\max_{\mathbf{t}\in\mathfrak{R}(p_{1})\times\mathfrak{R}(p_{2})\cap[0,T_{1}^{a}]\times[0,T_{2}^{a}]}\eta(\mathbf{t})\leq
u_{\mathbf{T}}^{*^{\prime}}\right\\}\rightarrow 1,$

as $\mathbf{T}\rightarrow\infty$, we get that the remainder $R_{\mathbf{n}}$
can be estimated as $R_{\mathbf{n}}=o(n_{1}n_{2}(1-P_{\mathbf{n}}))$. Using
Lemma A2, (10) and (11), we get that

$\displaystyle
n_{1}n_{2}\left(1-P\left\\{\max_{\mathbf{t}\in[0,T_{1}^{a}]\times[0,T_{2}^{a}]}\eta(\mathbf{t})\leq
u_{\mathbf{T}}^{*},\max_{\mathbf{t}\in\mathfrak{R}(p_{1})\times\mathfrak{R}(p_{2})\cap[0,T_{1}^{a}]\times[0,T_{2}^{a}]}\eta(\mathbf{t})\leq
u_{\mathbf{T}}^{*^{\prime}}\right\\}\right)$ $\displaystyle\thicksim
n_{1}n_{2}T_{1}^{a}T_{2}^{a}T_{1}^{-1}T_{2}^{-1}\left(e^{-x-r+\sqrt{2r}z}+e^{-y-r+\sqrt{2r}z}\right)$
$\displaystyle\thicksim e^{-x-r+\sqrt{2r}z}+e^{-y-r+\sqrt{2r}z},$

which combined with the dominated convergence theorem completes the proof of
Theorem 2.1. $\Box$

Proof of Theorem 2.2. The proof of the first assertion of Theorem 2.2 can be
found in Subsection 4.2. Next, we give the proof of the second assertion. In
view of Lemmas 3.1-3.4 in order to establish the proof we need to show

$\displaystyle\bigg{|}\int_{-\infty}^{+\infty}\prod_{i_{1}=1}^{n_{1}}\prod_{i_{2}=1}^{n_{2}}P\left\\{\max_{\mathbf{t}\in\mathbf{O_{i}}}\eta(\mathbf{t})\leq
u_{\mathbf{T}}^{*},\max_{t\in\mathfrak{R}(p_{1})\times\mathfrak{R}(p_{2})\cap\mathbf{O_{i}}}\eta(\mathbf{t})\leq
u_{\mathbf{T}}^{*^{\prime}}\right\\}\phi(z)dz$ $\displaystyle\ \ \ \ \ \ \
-\int_{-\infty}^{+\infty}\exp\left(-\big{(}e^{-x-r+\sqrt{2r}z}+e^{-y-r+\sqrt{2r}z}-H_{a,\alpha}^{\log
H_{a,\alpha}+x,\log
H_{\alpha}+y}e^{-r+\sqrt{2r}z}\big{)}\right)\phi(z)dz\bigg{|}\rightarrow 0$

as $\mathbf{T}\rightarrow\infty$, where $u_{\mathbf{T}}^{*}$ and
$u_{\mathbf{T}}^{*^{\prime}}$ are defined in Lemma 3.4. Similar to the proof
of Theorem 2.1, using Lemma A3, we get

$\displaystyle
n_{1}n_{2}\left(1-P\left\\{\max_{\mathbf{t}\in[0,T_{1}^{a}]\times[0,T_{2}^{a}]}\eta(\mathbf{t})\leq
u_{\mathbf{T}}^{*},\max_{\mathbf{t}\in\mathfrak{R}(p_{1})\times\mathfrak{R}(p_{2})\cap[0,T_{1}^{a}]\times[0,T_{2}^{a}]}\eta(\mathbf{t})\leq
u_{\mathbf{T}}^{*^{\prime}}\right\\}\right)$
$\displaystyle=n_{1}n_{2}P\left\\{\max_{\mathbf{t}\in[0,T_{1}^{a}]\times[0,T_{2}^{a}]}\eta(\mathbf{t})>u_{\mathbf{T}}^{*}\right\\}+n_{1}n_{2}P\left\\{\max_{\mathbf{t}\in\mathfrak{R}(p_{1})\times\mathfrak{R}(p_{2})\cap[0,T_{1}^{a}]\times[0,T_{2}^{a}]}\eta(\mathbf{t})>u_{\mathbf{T}}^{*^{\prime}}\right\\}$
$\displaystyle\ \
-n_{1}n_{2}P\left\\{\max_{\mathbf{t}\in[0,T_{1}^{a}]\times[0,T_{2}^{a}]}\eta(\mathbf{t})>u_{\mathbf{T}}^{*},\max_{\mathbf{t}\in\mathfrak{R}(p_{1})\times\mathfrak{R}(p_{2})\cap[0,T_{1}^{a}]\times[0,T_{2}^{a}]}\eta(\mathbf{t})>u_{\mathbf{T}}^{*^{\prime}}\right\\}$
$\displaystyle\thicksim
n_{1}n_{2}T_{1}^{a}T_{2}^{a}T_{1}^{-1}T_{2}^{-1}\left(e^{-x-r+\sqrt{2r}z}+e^{-y-r+\sqrt{2r}z}\right)$
$\displaystyle\ \
-n_{1}n_{2}P\left\\{\max_{\mathbf{t}\in[0,T_{1}^{a}]\times[0,T_{2}^{a}]}\eta(\mathbf{t})>u_{\mathbf{T}}^{*},\max_{\mathbf{t}\in\mathfrak{R}(p_{1})\times\mathfrak{R}(p_{2})\cap[0,T_{1}^{a}]\times[0,T_{2}^{a}]}\eta(\mathbf{t})>u_{\mathbf{T}}^{*^{\prime}}\right\\},$

as $\mathbf{T}\rightarrow\infty$. To transform the last term, using (10) and
(11), we get

$\displaystyle u_{\mathbf{T}}^{*}$ $\displaystyle=$
$\displaystyle\frac{x+r-\sqrt{2r}z}{a_{\mathbf{T}}}+b_{\mathbf{T}}+o(a_{\mathbf{T}}^{-1})$
$\displaystyle=$ $\displaystyle
u_{\mathbf{T}}^{*^{\prime}}+b_{\mathbf{T}}-b_{a,\mathbf{T}}+(x-y)/a_{\mathbf{T}}$
$\displaystyle=$ $\displaystyle
u_{\mathbf{T}}^{*^{\prime}}+\frac{\log(\mathcal{H}_{\alpha_{1}}\mathcal{H}_{\alpha_{2}})-\log(\mathcal{H}_{a_{1},\alpha_{1}}\mathcal{H}_{a_{2},\alpha_{2}})+x-y}{u_{\mathbf{T}}^{*^{\prime}}}+O\left((\log\log(T_{1}T_{2}))^{2}(\log
T_{1}T_{2})^{-3/2}\right).$

Observing that $u_{\mathbf{T}}^{*^{\prime}}\thicksim(2\log T_{1}T_{2})^{1/2}$,
we see that the reminder $O(\cdot)$ plays a negligible role. Therefore, by
(21) in Appendix A

$\displaystyle
n_{1}n_{2}P\left\\{\max_{\mathbf{t}\in[0,T_{1}^{a}]\times[0,T_{2}^{a}]}\eta(\mathbf{t})>u_{\mathbf{T}}^{*},\max_{\mathbf{t}\in\mathfrak{R}(p_{1})\times\mathfrak{R}(p_{2})\cap[0,T_{1}^{a}]\times[0,T_{2}^{a}]}\eta(\mathbf{t})>u_{\mathbf{T}}^{*^{\prime}}\right\\}$
$\displaystyle=n_{1}n_{2}T_{1}^{a}T_{2}^{a}\mathcal{H}_{\mathbf{a},\alpha_{1},\alpha_{2}}^{0,Z_{x,y}}(u_{T}^{*^{\prime}})^{2/\alpha_{1}+2/\alpha_{2}}\Psi(u_{T}^{*^{\prime}})(1+o(1))$
$\displaystyle=n_{1}n_{2}T_{1}^{a}T_{2}^{a}(T_{1}T_{2})^{-1}\mathcal{H}_{\mathbf{a},\alpha_{1},\alpha_{2}}^{0,Z_{x,y}}(\mathcal{H}_{a_{1},\alpha_{1}}\mathcal{H}_{a_{2},\alpha_{2}})^{-1}e^{-y-r+\sqrt{2r}z}(1+o(1)),$

where
$Z_{x,y}=\log(\mathcal{H}_{\alpha_{1}}\mathcal{H}_{\alpha_{2}})-\log(\mathcal{H}_{a_{1},\alpha_{1}}\mathcal{H}_{a_{2},\alpha_{2}})+x-y$.
Next, changing the variables in the definition of
$\mathcal{H}_{\mathbf{a},\alpha_{1},\alpha_{2}}^{x,y}$ we get that
$\mathcal{H}_{\mathbf{a},\alpha_{1},\alpha_{2}}^{0,Z_{x,y}}(\mathcal{H}_{a_{1},\alpha_{1}}\mathcal{H}_{a_{2},\alpha_{2}})^{-1}e^{-y}=\mathcal{H}_{\mathbf{a},\alpha_{1},\alpha_{2}}^{\log(\mathcal{H}_{\alpha_{1}}\mathcal{H}_{\alpha_{2}})+x,\log(\mathcal{H}_{a_{1},\alpha_{1}}\mathcal{H}_{a_{2},\alpha_{2}})+y}$.
This and the dominated convergence theorem conclude the proof of Theorem 2.2.
$\Box$

Proof of Theorem 2.3. In view of Lemma A4 we have

$\displaystyle\bigg{|}P\left\\{a_{\mathbf{T}}\big{(}M_{\mathbf{T}}-b_{\mathbf{T}}\big{)}\leq
x,a_{\mathbf{T}}\big{(}M_{\mathbf{T}}^{\mathbf{p}}-b_{\mathbf{T}}\big{)}\leq
y\right\\}-P\left\\{a_{\mathbf{T}}\big{(}M_{\mathbf{T}}-b_{\mathbf{T}}\big{)}\leq
x,a_{\mathbf{T}}\big{(}M_{\mathbf{T}}-b_{\mathbf{T}}\big{)}\leq
y\right\\}\bigg{|}$
$\displaystyle\leq\bigg{|}P\left\\{a_{\mathbf{T}}\big{(}M_{\mathbf{T}}^{\mathbf{p}}-b_{\mathbf{T}}\big{)}\leq
y\right\\}-P\left\\{a_{\mathbf{T}}\big{(}M_{\mathbf{T}}-b_{\mathbf{T}}\big{)}\leq
y\right\\}\bigg{|}\rightarrow 0,\quad\mathbf{T}\rightarrow\infty.$

Next, applying Corollary 2.1, we get

$\displaystyle
P\left\\{a_{\mathbf{T}}\big{(}M_{\mathbf{T}}-b_{\mathbf{T}}\big{)}\leq
x,a_{\mathbf{T}}\big{(}M_{\mathbf{T}}-b_{\mathbf{T}}\big{)}\leq y\right\\}$
$\displaystyle=$ $\displaystyle
P\\{a_{\mathbf{T}}\big{(}M_{\mathbf{T}}-b_{\mathbf{T}}\big{)}\leq\min(x,y)\\}$
$\displaystyle\rightarrow$
$\displaystyle\int_{-\infty}^{+\infty}\exp\left(-e^{-\min(x,y)-r+\sqrt{2r}z}\right)\phi(z)dz,\quad\mathbf{T}\rightarrow\infty,$

hence the proof is complete. $\Box$

##  4 Appendix A

In this section, we give some auxiliary results, which extend Lemmas 1-4 of
Piterbarg (2004) from stationary Gaussian processes to Gaussian random fields.
The ideas of the proofs are very close to that of the above mentioned lemmas.
In the following subsections, we suppose that Assumptions A1 and A2 hold.

###  4.1 Sparse grid

In this subsection, suppose $\mathfrak{R}(p_{i})$, $i=1,2$ are sparse grids.
We will use the notations $u=\sqrt{2\log T_{1}T_{2}}$, so that
$p_{i}=p_{i}(u)=l_{i}(u)u^{-2/\alpha_{i}}$, $i=1,2$, where
$l_{i}(u)\rightarrow\infty$ as $u\rightarrow\infty$, with $p_{i}(u)\leq p_{0}$
for some positive $p_{0}$, in particular, $p_{i}(u)=p_{0}$. Let
$\mathbf{L_{p}}=[-p_{1},p_{1}]\times[-p_{2},p_{2}]$. First we consider the
following probability

$P(u,x)=P\left(X(\mathbf{0})>u,\max_{\mathbf{t}\in\mathbf{L_{p}}}X(\mathbf{t})>u+\frac{(\frac{2}{\alpha_{1}}+\frac{2}{\alpha_{2}})\log
u+\log(p_{1}p_{2})+x}{u}\right),$

where $x$ is varies in a closed interval, say, $x\in[-A,A]$ with $A<\infty$.
For simplicity, we denote

$v:=\sqrt{(\frac{2}{\alpha_{1}}+\frac{2}{\alpha_{2}})\log
u+\log(p_{1}p_{2})}.$

By (4) (see also Theorem 7.1 of Piterbarg (1996)), we have

$\displaystyle
P\left(\max_{\mathbf{t}\in\mathbf{L_{p}}}X(\mathbf{t})>u+\frac{v^{2}+x}{u}\right)$
$\displaystyle=$ $\displaystyle
4p_{1}p_{2}\mathcal{H}_{\alpha_{1}}\mathcal{H}_{\alpha_{2}}\left(u+\frac{v^{2}+x}{u}\right)^{2/\alpha_{1}+2/\alpha_{2}}\Psi\left(u+\frac{v^{2}+x}{u}\right)(1+o(1))$
(14) $\displaystyle=$ $\displaystyle
4\mathcal{H}_{\alpha_{1}}\mathcal{H}_{\alpha_{2}}e^{-x}\Psi(u)(1+o(1))$

as $u\rightarrow\infty$.

Lemma A1. We have $P(u,x)=o(\Psi(u))$ as $u\rightarrow\infty$.

Proof: Write $w=\left((\frac{2}{\alpha_{1}}+\frac{2}{\alpha_{2}})\log
u+\log(p_{1}p_{2})+x\right)/u$. We have,

$P(u,x)\leq
P\left(\max_{\mathbf{t}\in\mathbf{L_{p}}}(X(\mathbf{0})+X(\mathbf{t}))>2u+w\right).$

Let $p_{1}^{\prime},p_{2}^{\prime}$ be so small that $1-r(\mathbf{t})\leq
2(|t_{1}|^{\alpha_{1}}+|t_{2}|^{\alpha_{2}})$ for all
$\mathbf{t}\in\mathbf{I_{p^{\prime}}}=[-p_{1}^{\prime},p_{1}^{\prime}]\times[-p_{2}^{\prime},p_{2}^{\prime}]$.
If $\mathbf{L_{p}}\cap\mathbf{L_{p^{\prime}}}\neq\emptyset$, write

$\displaystyle
P\left(\max_{\mathbf{t}\in\mathbf{L_{p}}}(X(\mathbf{0})+X(\mathbf{t}))>2u+w\right)$
$\displaystyle\leq$ $\displaystyle
P\left(\max_{\mathbf{t}\in\mathbf{L_{p^{\prime}}}}(X(\mathbf{0})+X(\mathbf{t}))>2u+w\right)$
(15) $\displaystyle+$ $\displaystyle
P\left(\max_{\mathbf{t}\in\mathbf{L_{p}}\diagdown\mathbf{L_{p^{\prime}}}}(X(\mathbf{0})+X(\mathbf{t}))>2u+w\right).$

The variance of the field $X(\mathbf{0})+X(\mathbf{t})$,
$\mathbf{t}\in\mathbf{L_{p}}\diagdown\mathbf{L_{p^{\prime}}}$, is less than
$4-\varepsilon$ for sufficiently small $\varepsilon>0$ and that

$E[(X(\mathbf{0})+X(\mathbf{t}))-(X(\mathbf{0})+X(\mathbf{s}))]^{2}=2(|t_{1}-s_{1}|^{\alpha_{1}}+|t_{2}-s_{2}|^{\alpha_{1}})(1+o(1))$

as $\mathbf{t}-\mathbf{s}\rightarrow\mathbf{0}$, so by Theorem 8.1 of
Piterbarg (1996), for all sufficiently large $u$ and some positive
$\varepsilon^{\prime}<\varepsilon$,

$\displaystyle
P\left(\max_{\mathbf{t}\in\mathbf{L_{p}}\diagdown\mathbf{L_{p^{\prime}}}}(X(\mathbf{0})+X(\mathbf{t}))>2u+w\right)$
$\displaystyle\leq$ $\displaystyle
Cp_{1}p_{2}(2u+w)^{2/\alpha_{1}+2/\alpha_{2}}\Psi\left(\frac{2u+w}{\sqrt{4-\varepsilon}}\right)$
$\displaystyle\leq$ $\displaystyle
C(u)^{2/\alpha_{1}+2/\alpha_{2}-1}\exp\left(-\frac{u^{2}}{2-\varepsilon^{\prime}/2}\right)$
$\displaystyle=$ $\displaystyle o(\Psi(u)),$

as $u\rightarrow\infty$.
We will apply Theorem 8.2 of Piterbarg (1996), for the first probability in
the right-hand part of (15). To this end, by some simple calculations, we get
for the correlation function of the field $X(\mathbf{0})+X(\mathbf{t})$,
$\mathbf{t}\in\mathbf{L_{p^{\prime}}}$

$\displaystyle
1-\frac{E(X(\mathbf{0})+X(\mathbf{t}))(X(\mathbf{0})+X(\mathbf{s}))}{\sqrt{E(X(\mathbf{0})+X(\mathbf{t}))^{2}E(X(\mathbf{0})+X(\mathbf{s}))^{2}}}$
$\displaystyle\leq$
$\displaystyle\frac{1-r(\mathbf{t}-\mathbf{s})}{2\sqrt{1+r(\mathbf{t})}\sqrt{1+r(\mathbf{s})}}$
$\displaystyle\leq$
$\displaystyle\frac{|t_{1}-s_{1}|^{\alpha_{1}}+|t_{2}-s_{2}|^{\alpha_{2}}}{2(2-\delta_{1}^{\prime\alpha_{1}}-\delta_{2}^{\prime\alpha_{2}})}$
$\displaystyle\leq$ $\displaystyle
1-\exp(-|t_{1}-s_{1}|^{\alpha_{1}}-|t_{2}-s_{2}|^{\alpha_{2}}),$

where we assume an additionally that
$p_{1}^{\prime\alpha_{1}}+p_{2}^{\prime\alpha_{2}}\leq 3/2$. For the variance
of the field $X(\mathbf{0})+X(\mathbf{t})$,
$\mathbf{t}\in\mathbf{L_{p^{\prime}}}$ we have

$Var(X(\mathbf{0})+X(\mathbf{t}))=2+2r(\mathbf{t})=4-2(|t_{1}|^{\alpha_{1}}+|t_{2}|^{\alpha_{2}})(1+o(1))$

as $\mathbf{t}\rightarrow 0$, and the point $\mathbf{t}=\mathbf{0}$ is the
unique point of maximum of variance of the field
$X(\mathbf{0})+X(\mathbf{t})$. By Slepian’s inequality

$\displaystyle
P\left(\max_{\mathbf{t}\in\mathbf{L_{p^{\prime}}}}(X(\mathbf{0})+X(\mathbf{t}))>2u+w\right)$
$\displaystyle=P\left(\max_{\mathbf{t}\in\mathbf{L_{p^{\prime}}}}\frac{X(\mathbf{0})+X(\mathbf{t})}{\sqrt{E(X(\mathbf{0})+X(\mathbf{t}))^{2}}}\sqrt{E(X(\mathbf{0})+X(\mathbf{t}))^{2}}>2u+w\right)$
$\displaystyle\leq
P\left(\max_{\mathbf{t}\in\mathbf{L_{p^{\prime}}}}Y(\mathbf{t})\sqrt{E(X(\mathbf{0})+X(\mathbf{t}))^{2}}>2u+w\right),$

where $Y(\mathbf{t})$ is a Gaussian zero mean homogeneous field with
covariance function $\exp(-[|t_{1}|^{\alpha_{1}}+|t_{2}|^{\alpha_{2}}])$, and
thus the conditions of Theorem 8.2 of Piterbarg (1996), for the case $(ii)$
holds. By this theorem, for some constants $C,C^{\prime},$

$\displaystyle
P\left(\max_{\mathbf{t}\in\mathbf{L_{p^{\prime}}}}Y(\mathbf{t})\sqrt{E(X(\mathbf{0})+X(\mathbf{t}))^{2}}>2u+w\right)$
$\displaystyle=C\Psi(u+w/2)(1+o(1))$
$\displaystyle=C^{\prime}u^{-1}\exp(-u^{2}/2-uw)(1+o(1))$
$\displaystyle=C^{\prime}\Psi(u)\exp\left(-\frac{1}{2}\big{(}(\frac{2}{\alpha_{1}}+\frac{2}{\alpha_{2}})\log
u+\log(p_{1}p_{2})+x\big{)}\right)(1+o(1))$
$\displaystyle=C^{\prime}\Psi(u)e^{-1/2x}\left(u^{\frac{2}{\alpha_{1}}+\frac{2}{\alpha_{2}}}p_{1}p_{2}\right)^{-1/2}(1+o(1))$
$\displaystyle=C^{\prime}\Psi(u)e^{-1/2x}\left((2\log
T_{1}T_{2})^{\frac{1}{\alpha_{1}}+\frac{1}{\alpha_{2}}}p_{1}p_{2}\right)^{-1/2}(1+o(1)).$

Since $(2\log T_{1}T_{2})^{1/\alpha_{i}}p_{i}\rightarrow\infty$ for sparse
grids $\mathfrak{R}(p_{i})$, we get the assertion of the lemma. $\Box$

Now we consider the probability

$P_{\mathbf{S}}(u,x)=P\left(\max_{\mathbf{t}\in\mathbf{I_{S}}\cap\mathfrak{R}(p_{1})\times\mathfrak{R}(p_{2})}X(\mathbf{t})>u,\max_{\mathbf{t}\in\mathbf{I_{S}}}X(\mathbf{t})>u+\frac{v^{2}+x}{u}\right)$

when we will allow $S_{1}S_{2}$ tends to infinity with $u$ but not too fast.
Define

$\delta(\varepsilon)=\inf_{\max\\{|t_{1}|,|t_{2}|\\}\geq\varepsilon}(1-r(\mathbf{t})).$

Note that $\delta(\varepsilon)$ is positive for all positive $\varepsilon$.

Lemma A2. Let $S_{i}=S_{i}(u)\geq 2p_{i}$ for all $u$, $i=1,2$ and
$S_{1}S_{2}u^{2/\alpha_{1}+2/\alpha_{2}}=o(\exp(u^{2}\delta(\varepsilon)/8))$
as $u\rightarrow\infty$. Then there exists an $\varepsilon>0$ such that

$\displaystyle
P\left(\max_{\mathbf{t}\in\mathbf{I_{S}}\cap\mathfrak{R}(p_{1})\times\mathfrak{R}(p_{2})}X(\mathbf{t})>u\right)\thicksim
S_{1}S_{2}p_{1}^{-1}p_{2}^{-1}\Psi(u),$ (16) $\displaystyle
P\left(\max_{\mathbf{t}\in\mathbf{I_{S}}}X(\mathbf{t})>u+\frac{v^{2}+x}{u}\right)\thicksim
S_{1}S_{2}p_{1}^{-1}p_{2}^{-1}e^{-x}\mathcal{H}_{\alpha_{1}}\mathcal{H}_{\alpha_{2}}\Psi(u),$
(17)

as $u\rightarrow\infty$ and

$\displaystyle
P_{\mathbf{S}}(u,x)=o\left(P\left(\max_{\mathbf{t}\in\mathbf{I_{S}}\cap\mathfrak{R}(p_{1})\times\mathfrak{R}(p_{2})}X(\mathbf{t})>u\right)+P\left(\max_{\mathbf{t}\in\mathbf{I_{S}}}X(\mathbf{t})>u+\frac{v^{2}+x}{u}\right)\right)$

as $u\rightarrow\infty$ so that

$\displaystyle
1-P\left(\max_{\mathbf{t}\in\mathbf{I_{S}}\cap\mathfrak{R}(p_{1})\times\mathfrak{R}(p_{2})}X(\mathbf{t})\leq
u,\max_{\mathbf{t}\in\mathbf{I_{S}}}X(\mathbf{t})\leq
u+\frac{v^{2}+x}{u}\right)$ $\displaystyle\thicksim
P\left(\max_{\mathbf{t}\in\mathbf{I_{S}}\cap\mathfrak{R}(p_{1})\times\mathfrak{R}(p_{2})}X(\mathbf{t})>u\right)+P\left(\max_{\mathbf{t}\in\mathbf{I_{S}}}X(\mathbf{t})>u+\frac{v^{2}+x}{u}\right)$
$\displaystyle\thicksim
S_{1}S_{2}p_{1}^{-1}p_{2}^{-1}\Psi(u)(1+e^{-x}\mathcal{H}_{\alpha_{1}}\mathcal{H}_{\alpha_{2}})$
(18)

as $u\rightarrow\infty$.

Proof: The relation (17) is in fact a special case of Theorem 7.2 of Piterbarg
(1996) and relation (16) can be proved by the same way. Now we prove that for
a sparse grid the double probability $P_{\mathbf{S}}(u,x)$ tends to zero
faster than right-hand part of (17). Let
$\mathbf{J_{l}}=[(l_{1}-1)p_{1},(l_{1}+1)p_{1}]\times[(l_{2}-1)p_{2},(l_{2}+1)p_{2}]$,
where $l_{i}=0,1,2,\cdots,[S_{i}/p_{i}]$, $i=1,2$. We have

$\displaystyle P_{\mathbf{S}}(u,x)$ $\displaystyle\leq$
$\displaystyle\sum_{k_{1},l_{1}=0}^{[S_{1}/p_{1}]}\sum_{k_{2},l_{2}=0}^{[S_{2}/p_{2}]}P\left(X(k_{1}p_{1},k_{2}p_{2})>u,\max_{\mathbf{t}\in\mathbf{J_{l}}}X(\mathbf{t})>u+\frac{v^{2}+x}{u}\right)=:\sum_{k_{1},l_{1}=0}^{[S_{1}/p_{1}]}\sum_{k_{2},l_{2}=0}^{[S_{2}/p_{2}]}P_{\mathbf{k},\mathbf{l}}$
(19) $\displaystyle=$ $\displaystyle\sum_{k_{1},l_{1}=0,|k_{1}-l_{1}|\leq
1}^{[S_{1}/p_{1}]}\sum_{k_{2},l_{2}=0,|k_{2}-l_{2}|\leq
1}^{[S_{2}/p_{2}]}P_{\mathbf{k},\mathbf{l}}+\sum_{k_{1},l_{1}=0,|k_{1}-l_{1}|\leq
1}^{[S_{1}/p_{1}]}\sum_{k_{2},l_{2}=0,|k_{2}-l_{2}|>1}^{[S_{2}/p_{2}]}P_{\mathbf{k},\mathbf{l}}$
$\displaystyle+\sum_{k_{1},l_{1}=0,|k_{1}-l_{1}|>1}^{[S_{1}/p_{1}]}\sum_{k_{2},l_{2}=0,|k_{2}-l_{2}|\leq
1}^{[S_{2}/p_{2}]}P_{\mathbf{k},\mathbf{l}}+\sum_{k_{1},l_{1}=0,|k_{1}-l_{1}|>1}^{[S_{1}/p_{1}]}\sum_{k_{2},l_{2}=0,|k_{2}-l_{2}|>1}^{[S_{2}/p_{2}]}P_{\mathbf{k},\mathbf{l}}.$

The members of the first term on the right-hand side of (19) can be estimated
by Lemma A1, so that

$\displaystyle\sum_{k_{1},l_{1}=0,|k_{1}-l_{1}|\leq
1}^{[S_{1}/p_{1}]}\sum_{k_{2},l_{2}=0,|k_{2}-l_{2}|\leq
1}^{[S_{2}/p_{2}]}P_{\mathbf{k},\mathbf{l}}=S_{1}S_{2}p_{1}^{-1}p_{2}^{-1}o(\Psi(u))$
(20)

as $u\rightarrow\infty$. Let $\mathbf{m}=(m_{1},m_{2})$ with
$m_{i}=0,1,2,\cdots,[S_{i}/p_{i}]$, $i=1,2$. Now consider the probability
$P_{\mathbf{k},\mathbf{k+m}}=P_{\mathbf{0},\mathbf{m}}$ for
$\max\\{m_{1},m_{2}\\}>1$. We have, using Theorem 8.1 of Piterbarg (1996),

$\displaystyle P_{\mathbf{0},\mathbf{m}}$ $\displaystyle\leq$ $\displaystyle
P\left(\max_{\mathbf{t}\in\mathbf{J_{m}}}(X(\mathbf{0})+X(\mathbf{t}))>2u+\frac{v^{2}+x}{u}\right)$
$\displaystyle\leq$ $\displaystyle
Cp_{1}p_{2}u^{\frac{2}{\alpha_{1}}+\frac{2}{\alpha_{2}}-1}\exp\left(-\frac{(2u+(v^{2}+x)/u)^{2}}{2\max_{\mathbf{t}\in\mathbf{J_{m}}}(2+2r(\mathbf{t}))}\right)$
$\displaystyle\leq$ $\displaystyle
Cp_{1}p_{2}u^{\frac{2}{\alpha_{1}}+\frac{2}{\alpha_{2}}-1}\exp\left(-\frac{u^{2}+v^{2}}{2(1-\frac{1}{2}\min_{\mathbf{t}\in\mathbf{J_{m}}}(1-r(\mathbf{t}))}\right)$
$\displaystyle\leq$ $\displaystyle
Cp_{1}p_{2}u^{\frac{2}{\alpha_{1}}+\frac{2}{\alpha_{2}}-1}\exp\left(-\frac{1}{2}(u^{2}+v^{2})(1+\frac{1}{2}\min_{\mathbf{t}\in\mathbf{J_{m}}}(1-r(\mathbf{t}))\right)$
$\displaystyle\leq$ $\displaystyle
Cp_{1}^{1/2}p_{2}^{1/2}u^{\frac{1}{\alpha_{1}}+\frac{1}{\alpha_{2}}-1}\exp\left(-\frac{1}{2}u^{2}\right)\exp\left(-\frac{1}{4}u^{2}\min_{\mathbf{t}\in\mathbf{J_{m}}}(1-r(\mathbf{t}))\right)$
$\displaystyle\leq$ $\displaystyle
Cp_{1}^{1/2}p_{2}^{1/2}u^{\frac{1}{\alpha_{1}}+\frac{1}{\alpha_{2}}}\Psi(u)\exp\left(-\frac{1}{4}u^{2}\min_{\mathbf{t}\in\mathbf{J_{m}}}(1-r(\mathbf{t}))\right).$

Let $\varepsilon$ be such that
$1-r(\mathbf{t})\geq\frac{1}{2}(|t_{1}|^{\alpha_{1}}+|t_{2}|^{\alpha_{2}})$
for all
$\mathbf{t}\in(-\varepsilon,\varepsilon)\times(-\varepsilon,\varepsilon)$.
Then

$P_{\mathbf{0},\mathbf{m}}\leq
Cp_{1}^{1/2}p_{2}^{1/2}u^{\frac{1}{\alpha_{1}}+\frac{1}{\alpha_{2}}}\Psi(u)\exp\left(-\frac{1}{8}u^{2}\delta(\varepsilon)\right)$

for $\max\\{|(m_{1}-1)p_{1}|,|(m_{2}-1)p_{2}|\\}>\varepsilon$ and

$P_{\mathbf{0},\mathbf{m}}\leq
Cp_{1}^{1/2}p_{2}^{1/2}u^{\frac{1}{\alpha_{1}}+\frac{1}{\alpha_{2}}}\Psi(u)\exp\left(-\frac{1}{8}u^{2}[|(m_{1}-1)p_{1}|^{\alpha_{1}}+|(m_{2}-1)p_{2}|^{\alpha_{2}}]\right)$

for $\max\\{|(m_{1}-1)p_{1}|,|(m_{2}-1)p_{2}|\\}\leq\varepsilon$. Thus,
letting $\mathbf{i}=\mathbf{l}-\mathbf{k}$, for the second sum we have

$\displaystyle\sum_{k_{1},l_{1}=0,|k_{1}-l_{1}|\leq
1}^{[S_{1}/p_{1}]}\sum_{k_{2},l_{2}=0,|k_{2}-l_{2}|>1}^{[S_{2}/p_{2}]}P_{\mathbf{k},\mathbf{l}}$
$\displaystyle\leq$ $\displaystyle
4S_{1}S_{2}p_{1}^{-1}p_{2}^{-1}\sum_{i_{1}=0}^{1}\sum_{i_{2}=2}^{[S_{2}/p_{2}]}p_{\mathbf{0},\mathbf{i}}$
$\displaystyle\leq$ $\displaystyle
CS_{1}S_{2}p_{1}^{-1/2}p_{2}^{-1/2}u^{\frac{1}{\alpha_{1}}+\frac{1}{\alpha_{2}}}\Psi(u)\bigg{\\{}S_{2}p_{2}^{-1}\exp\left(-\frac{1}{8}u^{2}\delta(\varepsilon)\right)$
$\displaystyle+\sum_{i_{1}-1=-1}^{0}\sum_{i_{2}-1=1}^{\varepsilon/p_{2}}\exp\left(-\frac{1}{8}u^{2}[|(i_{1}-1)p_{1}|^{\alpha_{1}}+|(i_{2}-1)p_{2}|^{\alpha_{2}}]\right)\bigg{\\}}$
$\displaystyle\leq$ $\displaystyle CS_{1}S_{2}p_{1}^{-1}p_{2}^{-1}\Psi(u)o(1)$

as $u\rightarrow\infty$. Similarly,

$\sum_{k_{1},l_{1}=0,|k_{1}-l_{1}|>1}^{[S_{1}/p_{1}]}\sum_{k_{2},l_{2}=0,|k_{2}-l_{2}|\leq
1}^{[S_{2}/p_{2}]}P_{\mathbf{k},\mathbf{l}}\leq
CS_{1}S_{2}p_{1}^{-1}p_{2}^{-1}\Psi(u)o(1)$

as $u\rightarrow\infty$. For the fourth sum, we have

$\displaystyle\sum_{k_{1},l_{1}=0,|k_{1}-l_{1}|>1}^{[S_{1}/p_{1}]}\sum_{k_{2},l_{2}=0,|k_{2}-l_{2}|>1}^{[S_{2}/p_{2}]}P_{\mathbf{k},\mathbf{l}}$
$\displaystyle\leq$ $\displaystyle
4S_{1}S_{2}p_{1}^{-1}p_{2}^{-1}\sum_{i_{1}=2}^{[S_{1}/\delta_{1}]}\sum_{i_{2}=2}^{[S_{2}/p_{2}]}P_{\mathbf{0},\mathbf{i}}$
$\displaystyle\leq$ $\displaystyle
CS_{1}S_{2}p_{1}^{-1/2}p_{2}^{-1/2}u^{\frac{1}{\alpha_{1}}+\frac{1}{\alpha_{2}}}\Psi(u)\bigg{\\{}S_{1}p_{1}^{-1}S_{2}p_{2}^{-1}\exp\left(-\frac{1}{8}u^{2}\delta(\varepsilon)\right)$
$\displaystyle+\sum_{i_{1}-1=1}^{\varepsilon/p_{1}}\sum_{i_{2}-1=1}^{\varepsilon/p_{2}}\exp\left(-\frac{1}{8}u^{2}[|(i_{1}-1)p_{1}|^{\alpha_{1}}+|(i_{2}-1)p_{2}|^{\alpha_{2}}]\right)\bigg{\\}}$
$\displaystyle\leq$ $\displaystyle
CS_{1}S_{2}p_{1}^{-1}p_{2}^{-1}\Psi(u)o(1),$

as $u\rightarrow\infty$. Now we can easily prove the relation (17). We have
for all $\mathbf{k}$ and $\mathbf{l}$

$P\left(X(k_{1}p_{1},k_{2}p_{2})>u,X(l_{1}p_{1},l_{2}p_{2})>u\right)\leq
P_{\mathbf{k},\mathbf{l}},$

hence

$\sum_{k_{1},l_{1}=0,k_{1}\neq
l_{1}}^{[S_{1}/p_{1}]}\sum_{k_{2},l_{2}=0,k_{2}\neq
l_{2}}^{[S_{2}/p_{2}]}P\left(X(k_{1}p_{1},k_{2}p_{2})>u,X(l_{1}p_{1},l_{2}p_{2})>u\right)\leq\sum_{k_{1},l_{1}=0,k_{1}\neq
l_{1}}^{[S_{1}/p_{1}]}\sum_{k_{2},l_{2}=0,k_{2}\neq
l_{2}}^{[S_{2}/p_{2}]}P_{\mathbf{k},\mathbf{l}},$

from which it follows that double sum in the above left-hand side tends to
zero faster than $S_{1}S_{2}p_{1}^{-1}p_{2}^{-1}\Psi(u)$ as
$u\rightarrow\infty$. Thus, both the assertions of Lemma A2 are proved. $\Box$

###  4.2 Pickands grid

Lat $\mathbf{a}=(a_{1},a_{2})>(0,0)$. In this subsection suppose that
$\mathfrak{R}(p_{i})$, $i=1,2$ are Pickands grids, ie.,
$\mathfrak{R}(p_{i})=\\{a_{i}ku^{-2/\alpha_{i}},k\in\mathbb{N}\\}$. We will
evaluate the asymptotic behavior of the probability

$P_{\mathbf{S}}^{\prime}(u,x)=P\left(\max_{\mathbf{t}\in\mathbf{I_{S}}\cap\mathfrak{R}(p_{1})\times\mathfrak{R}(p_{2})}X(\mathbf{t})>u,\max_{\mathbf{t}\in\mathbf{I_{S}}}X(\mathbf{t})>u+\frac{x}{u}\right).$

As in the previous subsection, we begin with a short interval. Let
$\lambda_{i}>a_{i}$. Then it can be proved quite similar to the proof of Lemma
6.1 of Piterbarg (1996), that

$P_{(\lambda_{1}u^{-2/\alpha_{1}},\lambda_{2}u^{-2/\alpha_{2}})}^{\prime}(u,x)\thicksim\mathcal{H}_{\mathbf{a},\alpha_{1},\alpha_{2}}^{0,x}\Psi(u)$

as $u\rightarrow\infty$, where

$H_{\mathbf{d},\alpha_{1},\alpha_{2}}^{0,x}(\lambda_{1},\lambda_{2})=\int_{-\infty}^{+\infty}e^{s}P\left(\max_{(k_{1}d_{1},k_{2}d_{2})\in[0,\lambda_{1}]\times[0,\lambda_{2}]}\sqrt{2}\chi(k_{1}d_{1},k_{2}d_{2})>s,\max_{(t_{1},t_{2})\in[0,\lambda_{1}]\times[0,\lambda_{2}]}\sqrt{2}\chi(t_{1},t_{2})>s+x\right)ds.$

It also can be proved in a similar way as for Lemma 6.1 and Theorem 7.2 of
Piterbarg (1996) that

$\mathcal{H}_{\mathbf{a},\alpha_{1},\alpha_{2}}^{0,x}:=\lim_{\lambda_{1}\rightarrow\infty\atop\lambda_{2}\rightarrow\infty}\mathcal{H}_{\mathbf{a},\alpha_{1},\alpha_{2}}^{0,x}(\lambda_{1},\lambda_{2})/(\lambda_{1}\lambda_{2})\in(0,\infty)$

and that there exsits $\kappa\in(0,1/2)$ such that for any $S_{i}=S_{i}(u)$
with $S_{1}S_{2}u^{-(2/\alpha_{1}+2/\alpha_{2})}\rightarrow\infty$ and
$S_{1}S_{2}=O(\exp(\kappa u^{2}))$ as $u\rightarrow\infty$ with

$\displaystyle P_{\mathbf{S}}^{\prime}(u,x)\thicksim
S_{1}S_{2}\mathcal{H}_{\mathbf{a},\alpha_{1},\alpha_{2}}^{0,x}u^{2/\alpha_{1}+2/\alpha_{2}}\Psi(u)$
(21)

as $u\rightarrow\infty$, respectively. From here we have for Pickands grids,

Lemma A3. For any $\mathbf{a}=(a_{1},a_{2})$ and
$\mathfrak{R}(p_{i})=\\{a_{i}ku^{-2/\alpha_{i}},k\in\mathbb{N}\\}$,

$\displaystyle
1-P\left(\max_{\mathbf{t}\in\mathbf{I_{S}}\cap\mathfrak{R}(p_{1})\times\mathfrak{R}(p_{2})}X(\mathbf{t})\leq
u,\max_{\mathbf{t}\in\mathbf{I_{S}}}X(\mathbf{t})\leq
u+\frac{v^{2}+x}{u}\right)$
$\displaystyle=P\left(\max_{\mathbf{t}\in\mathbf{I_{S}}\cap\mathfrak{R}(p_{1})\times\mathfrak{R}(p_{2})}X(\mathbf{t})>u\right)+P\left(\max_{\mathbf{t}\in\mathbf{I_{S}}}X(\mathbf{t})>u+\frac{v^{2}+x}{u}\right)$
$\displaystyle\ \ \ \
-P\left(\max_{\mathbf{t}\in\mathbf{I_{S}}\cap\mathfrak{R}(p_{1})\times\mathfrak{R}(p_{2})}X(\mathbf{t})>u,\max_{\mathbf{t}\in\mathbf{I_{S}}}X(\mathbf{t})>u+\frac{x}{u}\right)$
$\displaystyle\thicksim
S_{1}S_{2}\mathcal{H}_{\alpha_{1}}\mathcal{H}_{\alpha_{2}}\left(u+\frac{v^{2}+x}{u}\right)^{2/\alpha_{1}+2/\alpha_{2}}\Psi\left(u+\frac{v^{2}+x}{u}\right)+S_{1}S_{2}\mathcal{H}_{a_{1},\alpha_{1}}\mathcal{H}_{a_{2},\alpha_{2}}u^{2/\alpha_{1}+2/\alpha_{2}}\Psi(u)$
$\displaystyle\ \ \ \
-S_{1}S_{2}\mathcal{H}_{\mathbf{a},\alpha_{1},\alpha_{2}}^{0,x}u^{2/\alpha_{1}+2/\alpha_{2}}\Psi(u)$
(22)

as $u\rightarrow\infty$.

###  4.3 Dense grid

In this subsection, we state a lemma for the dense grid case which is
important for our proofs.

Lemma A4. Let $S_{i}=S_{i}(u)$ with
$S_{1}S_{2}u^{-(2/\alpha_{1}+2/\alpha_{2})}\rightarrow\infty$ and
$S_{1}S_{2}=O(\exp(\kappa u^{2}))$ with $\kappa\in(0,1/2]$ as
$u\rightarrow\infty$. For any $\mathbf{a}=(a_{1},a_{2})$ and
$\mathfrak{R}(p_{i})=\\{a_{i}ku^{-2/\alpha_{i}},k\in\mathbb{N}\\}$, we have

$\displaystyle
P\left(\max_{\mathbf{t}\in\mathbf{I_{S}}\cap\mathfrak{R}(p_{1})\times\mathfrak{R}(p_{2})}X(\mathbf{t})\leq
u\right)-P\left(\max_{\mathbf{t}\in\mathbf{I_{S}}}X(\mathbf{t})\leq
u\right)=g(a_{1},a_{2})\mathcal{H}_{\alpha_{1}}\mathcal{H}_{\alpha_{2}}S_{1}S_{2}u^{2/\alpha_{1}+2/\alpha_{2}}\Psi(u),$
(23)

where $g(a_{1},a_{2})\rightarrow 0$ as $\mathbf{a}\rightarrow\mathbf{0}$.

Proof: Lemma 1 of Dȩbicki et al. (2014) shows that (23) holds for some fixed
$S_{i}>0$. By the homogeneity of $X(\mathbf{t})$, it is easy to extend (23) to
the case $S_{1}S_{2}=O(\exp(\kappa u^{2}))$ with $\kappa\in(0,1/2]$, see eg.
the proof of Lemma 12.3.2 of Leadbetter (1983) for more details. $\Box$

##  5 Appendix B

In this section, we give three technical lemmas which are used for the proof
of Lemma 3.1. Recall that $u_{\mathbf{T}}=b_{\mathbf{T}}+x/a_{\mathbf{T}}$,
$u_{\mathbf{T}}^{\prime}=b_{\mathbf{T}}^{\prime}+y/a_{\mathbf{T}}$, where
$b_{\mathbf{T}}^{{}^{\prime}}=b_{\mathbf{T}}^{\mathbf{p}}$ for sparse grids
and $b_{\mathbf{T}}^{{}^{\prime}}=b_{\mathbf{a},\mathbf{T}}$ for Pickands
grids, and
$r^{(h)}(\mathbf{kq},\mathbf{lq})=hr(\mathbf{kq},\mathbf{lq})+(1-h)\varrho(\mathbf{kq},\mathbf{lq})$
with $h\in[0,1]$. Let

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

and

$\vartheta(\mathbf{\mathbf{z}})=\sup_{\mathbf{0}\leq\mathbf{s,t}\leq\mathbf{T},\atop\\{|s_{1}-t_{1}|>z_{1}\\}\cup\\{|s_{2}-t_{2}|>z_{2}\\}}\\{\varpi(\mathbf{t},\mathbf{s})\\}.$

It is easy to see from Assumptions A1 and A2 that for any $\varepsilon_{1}>0$
and $\varepsilon_{2}>0$

$\vartheta(\varepsilon_{1},\varepsilon_{2})<1$

for all sufficiently large $\mathbf{T}$. Further, let $a,b$ be such that

$0<b<a<\big{(}1-\vartheta(\varepsilon,\varepsilon)\big{)}/\big{(}1+\vartheta(\varepsilon,\varepsilon)\big{)}<1$

for all sufficiently large $\mathbf{T}$ and for some $\varepsilon>0$ which
will be chosen in the blow.

Lemma B1. Under the conditions of Lemma 3.3, we have

$\displaystyle\sum_{\mathbf{kq}\in\mathbf{O_{i}},\mathbf{lq}\in\mathbf{O_{j}}\atop\mathbf{kq}\neq\mathbf{lq},\mathbf{1}\leq\mathbf{i,j}\leq\mathbf{n}}|r(\mathbf{kq},\mathbf{lq})-\varrho(\mathbf{kq},\mathbf{lq})|\int_{0}^{1}\frac{1}{\sqrt{1-r^{(h)}(\mathbf{kq},\mathbf{lq})}}\exp\left(-\frac{u_{\mathbf{T}}^{2}}{1+r^{(h)}(\mathbf{kq},\mathbf{lq})}\right)dh\rightarrow
0$ (24)

as $\mathbf{T}\rightarrow\infty$.

Proof: Recall that $\mathfrak{R}(q_{i})$, $i=1,2$ are Pickands grids. First,
we consider the case that $\mathbf{kq},\mathbf{lq}$ in the same interval
$\mathbf{O_{i}}$. Split the sum (24) into two parts as

$\displaystyle\sum_{\mathbf{kq},\mathbf{lq}\in\mathbf{O_{i}},\mathbf{kq}\neq\mathbf{lq},\mathbf{i}=\mathbf{1},\cdots,\mathbf{n},\atop\max\\{|l_{1}q_{1}-k_{1}q_{1}|,|l_{2}q_{2}-k_{2}q_{2}|\\}\leq\varepsilon}+\sum_{\mathbf{kq},\mathbf{lq}\in\mathbf{O_{i}},\mathbf{kq}\neq\mathbf{lq},\mathbf{i}=\mathbf{1},\cdots,\mathbf{n},\atop\max\\{|l_{1}q_{1}-k_{1}q_{1}|,|l_{2}q_{2}-k_{2}q_{2}|\\}>\varepsilon}=:J_{\mathbf{T},1}+J_{\mathbf{T},2}.$
(25)

We deal with $J_{\mathbf{T},1}$ and note that in this case, by the definition
of the field $\xi_{\mathbf{T}}(\mathbf{t})$, we have
$\varrho(\mathbf{kq},\mathbf{lq})-r(\mathbf{kq},\mathbf{lq})=\rho(\mathbf{T})(1-r(\mathbf{kq},\mathbf{lq}))$.
By Assumption A1 we can choose small enough $\varepsilon>0$ such that
$\varrho(\mathbf{kq},\mathbf{lq})=r(\mathbf{kq},\mathbf{lq})+(1-r(\mathbf{kq},\mathbf{lq}))\rho(\mathbf{T})\sim
r(\mathbf{kq},\mathbf{lq})$ for sufficiently large $\mathbf{T}$ and
$\max\\{|l_{1}q_{1}-k_{1}q_{1}|,|l_{2}q_{2}-k_{2}q_{2}|\\}\leq\varepsilon$. It
follows from Assumption A1 again that for all
$|t_{i}|\leq\varepsilon<2^{-1/\alpha_{i}}$,

$\displaystyle\frac{1}{2}(|t_{1}|^{\alpha_{1}}+|t_{2}|^{\alpha_{2}})\leq
1-r(\mathbf{t})\leq 2(|t_{1}|^{\alpha_{1}}+|t_{2}|^{\alpha_{2}})$ (26)

and the definition of $u_{\mathbf{T}}$ implies

$\displaystyle u^{2}_{\mathbf{T}}=2\log T_{1}T_{2}-\log\log
T_{1}T_{2}+(\frac{2}{\alpha_{1}}+\frac{2}{\alpha_{2}})\log\log
T_{1}T_{2}+O(1).$ (27)

Consequently, since further $q_{i}=\gamma_{i}(\log
T_{1}T_{2})^{-1/\alpha_{i}}$ we obtain

$\displaystyle J_{\mathbf{T},1}$ $\displaystyle\leq$ $\displaystyle
C\sum_{\mathbf{kq},\mathbf{lq}\in\mathbf{O_{i}},\mathbf{kq}\neq\mathbf{lq},\mathbf{i}=\mathbf{1},\cdots,\mathbf{n},\atop\max\\{|l_{1}q_{1}-k_{1}q_{1}|,|l_{2}q_{2}-k_{2}q_{2}|\\}\leq\varepsilon}|r(\mathbf{kq},\mathbf{lq})-\varrho(\mathbf{kq},\mathbf{lq})|\frac{1}{\sqrt{1-r(\mathbf{kq},\mathbf{lq})}}\exp\left(-\frac{u^{2}_{\mathbf{T}}}{1+r(\mathbf{kq},\mathbf{lq})}\right)$
(28) $\displaystyle\leq$ $\displaystyle
C\frac{T_{1}}{q_{1}}\frac{T_{2}}{q_{2}}\rho(\mathbf{T})\sum_{0<k_{1}q_{1}\leq\varepsilon,0<k_{2}q_{2}\leq\varepsilon}|1-r(\mathbf{kq})|\frac{1}{\sqrt{1-r(\mathbf{kq})}}\exp\left(-\frac{u^{2}_{\mathbf{T}}}{2}\right)\exp\left(-\frac{(1-r(\mathbf{kq}))u^{2}_{\mathbf{T}}}{2(1+r(\mathbf{kq}))}\right)$
$\displaystyle\leq$ $\displaystyle
C\frac{T_{1}}{q_{1}}\frac{T_{2}}{q_{2}}\rho(\mathbf{T})T_{1}^{-1}T_{2}^{-1}(\log
T_{1}T_{2})^{1/2-1/\alpha_{1}-1/\alpha_{2}}\sum_{0<k_{1}q_{1}\leq\varepsilon,0<k_{2}q_{2}\leq\varepsilon}\sqrt{1-r(\mathbf{kq})}\exp\left(-\frac{(1-r(\mathbf{kq}))u^{2}_{\mathbf{T}}}{2(1+r(\mathbf{kq}))}\right)$
$\displaystyle\leq$ $\displaystyle C(\log
T_{1}T_{2})^{-1/2}\sum_{0<k_{1}q_{1}\leq\varepsilon,0<k_{2}q_{2}\leq\varepsilon}[(kq_{1})^{\alpha_{1}}+(kq_{2})^{\alpha_{2}}]^{1/2}\exp\left(-\frac{1}{4}[(kq_{1})^{\alpha_{1}}+(kq_{2})^{\alpha_{2}}]\log(T_{1}T_{2})\right)$
$\displaystyle\leq$ $\displaystyle C(\log
T_{1}T_{2})^{-1/2}\sum_{0<k_{1}q_{1}\leq\varepsilon,0<k_{2}q_{2}\leq\varepsilon}\exp\left(-\frac{1}{4}[(kq_{1})^{\alpha_{1}}+(kq_{2})^{\alpha_{2}}]\log(T_{1}T_{2})\right)$
$\displaystyle\leq$ $\displaystyle C(\log
T_{1}T_{2})^{-1/2}\sum_{k_{1}=1}^{\infty}e^{-\frac{1}{4}(k_{1}\gamma_{1})^{\alpha_{1}}}\sum_{k_{2}=1}^{\infty}e^{-\frac{1}{4}(k_{2}\gamma_{2})^{\alpha_{2}}}$
$\displaystyle\leq$ $\displaystyle C(\log T_{1}T_{2})^{-1/2},$

which shows $J_{\mathbf{T},1}\rightarrow 0$ as $\mathbf{T}\rightarrow\infty$.
Using the fact that $u_{\mathbf{T}}\thicksim(2\log T_{1}T_{2})^{1/2}$, we
obtain

$\displaystyle J_{\mathbf{T},2}$ $\displaystyle\leq$ $\displaystyle
C\sum_{\mathbf{kq},\mathbf{lq}\in\mathbf{O_{i}},\mathbf{kq}\neq\mathbf{lq},\mathbf{i}=\mathbf{1},\cdots,\mathbf{n},\atop\max\\{|l_{1}q_{1}-k_{1}q_{1}|,|l_{2}q_{2}-k_{2}q_{2}|\\}>\varepsilon}|r(\mathbf{kq},\mathbf{lq})-\varrho(\mathbf{kq},\mathbf{lq})|\exp\left(-\frac{u^{2}_{\mathbf{T}}}{1+\varpi(\mathbf{kq},\mathbf{lq})}\right)$
(29) $\displaystyle\leq$ $\displaystyle
C\frac{T_{1}}{q_{1}}\frac{T_{2}}{q_{2}}\sum_{0\leq k_{1}q_{1}\leq
T_{1}^{a},0\leq k_{2}q_{2}\leq
T_{2}^{a},\mathbf{i}=\mathbf{1},\cdots,\mathbf{n},\atop\max\\{k_{1}q_{1},k_{2}q_{2}\\}>\varepsilon}\exp\left(-\frac{u^{2}_{\mathbf{T}}}{1+\vartheta(\varepsilon,\varepsilon)}\right)$
$\displaystyle\leq$ $\displaystyle
C\frac{T_{1}}{q_{1}}\frac{T_{2}}{q_{2}}\exp\left(-\frac{u^{2}_{\mathbf{T}}}{1+\vartheta(\varepsilon,\varepsilon)}\right)\sum_{0\leq
k_{1}q_{1}\leq T_{1}^{a},0\leq k_{2}q_{2}\leq T_{2}^{a}}1$ $\displaystyle\leq$
$\displaystyle
C\frac{T_{1}}{q_{1}}\frac{T_{2}}{q_{2}}(T_{1}T_{2})^{-\frac{2}{1+\vartheta(\varepsilon,\varepsilon)}}\sum_{0\leq
k_{1}q_{1}\leq T_{1}^{a},0\leq k_{2}q_{2}\leq T_{2}^{a}}1$ $\displaystyle\leq$
$\displaystyle
C(T_{1}T_{2})^{a-\frac{1-\vartheta(\varepsilon,\varepsilon)}{1+\vartheta(\varepsilon,\varepsilon)}}(\log
T_{1}T_{2})^{2/\alpha_{1}+2/\alpha_{2}}.$

Thus, $J_{\mathbf{T},2}\rightarrow 0$ as $\mathbf{T}\rightarrow\infty$ since
$a<\frac{1-\vartheta(\varepsilon,\varepsilon)}{1+\vartheta(\varepsilon,\varepsilon)}$.

Second, we deal with the case that $\mathbf{kq}\in\mathbf{O_{i}}$ and
$\mathbf{lq}\in\mathbf{O_{j}}$, $\mathbf{i}\neq\mathbf{j}$. Note that in this
case, the distance between the points in any two rectangles $\mathbf{O_{i}}$
and $\mathbf{O_{j}}$ is large than $T_{1}^{b}$ or $T_{2}^{b}$ and
$\varrho(\mathbf{kq},\mathbf{lq})=\rho(\mathbf{T})$ for
$\mathbf{kq}\in\mathbf{O_{i}}$ and $\mathbf{lq}\in\mathbf{O_{j}}$,
$\mathbf{i}\neq\mathbf{j}$.

Obviously, the sum in (24) is smaller than

$\displaystyle
C\sum_{\mathbf{kq}\in\mathbf{O_{i}},\mathbf{lq}\in\mathbf{O_{j}}\atop\mathbf{kq}\neq\mathbf{lq},\mathbf{1}\leq\mathbf{i}\neq\mathbf{j}\leq\mathbf{n}}|r(\mathbf{kq},\mathbf{lq})-\rho(\mathbf{T})|\exp\left(-\frac{u^{2}_{\mathbf{T}}}{1+\varpi(\mathbf{kq},\mathbf{lq})}\right).$
(30)

Split the sum of (30) into three parts, the first for
$|k_{1}q_{1}-l_{1}q_{1}|>0$ and $|k_{2}q_{2}-l_{2}q_{2}|>0$, the second for
$k_{1}q_{1}-l_{1}q_{1}=0$ and $|k_{2}q_{2}-l_{2}q_{2}|>0$, the third for
$k_{2}q_{2}-l_{2}q_{2}=0$ and $|k_{1}q_{1}-l_{1}q_{1}|>0$ and denote them by
$S_{\mathbf{T},i}$, $i=1,2,3,$ respectively. Let $\beta$ be such that
$0<b<a<\beta<\frac{1-\vartheta(\varepsilon,\varepsilon)}{1+\vartheta(\varepsilon,\varepsilon)}$
for all sufficiently large $\mathbf{T}$.

We consider the term $S_{\mathbf{T},1}$ and split it into two parts as

$\displaystyle
S_{\mathbf{T},1}=C\sum_{\mathbf{kq}\in\mathbf{O_{i}},\mathbf{lq}\in\mathbf{O_{j}},\mathbf{kq}\neq\mathbf{lq},\mathbf{1}\leq\mathbf{i}\neq\mathbf{j}\leq\mathbf{n}\atop|k_{1}q_{1}-l_{1}q_{1}||k_{2}q_{2}-l_{2}q_{2}|\leq(T_{1}T_{2})^{\beta}}+C\sum_{\mathbf{kq}\in\mathbf{O_{i}},\mathbf{lq}\in\mathbf{O_{j}},\mathbf{kq}\neq\mathbf{lq},\mathbf{1}\leq\mathbf{i}\neq\mathbf{j}\leq\mathbf{n}\atop|k_{1}q_{1}-l_{1}q_{1}||k_{2}q_{2}-l_{2}q_{2}|>(T_{1}T_{2})^{\beta}}=:S_{\mathbf{T},11}+S_{\mathbf{T},12}.$

For $S_{\mathbf{T},11}$, with the similar derivation as for (29), we have

$\displaystyle S_{\mathbf{T},11}$ $\displaystyle\leq$ $\displaystyle
C\frac{T_{1}}{q_{1}}\frac{T_{2}}{q_{2}}\sum_{0\leq k_{1}q_{1}\leq T_{1},0\leq
k_{2}q_{2}\leq T_{2},\atop
k_{1}q_{1}k_{2}q_{2}\leq(T_{1}T_{2})^{\beta}}\exp\left(-\frac{u^{2}_{\mathbf{T}}}{1+\vartheta(\varepsilon,\varepsilon)}\right)$
(31) $\displaystyle\leq$ $\displaystyle
C(T_{1}T_{2})^{\beta-\frac{1-\vartheta(\varepsilon,\varepsilon)}{1+\vartheta(\varepsilon,\varepsilon)}}(\log
T_{1}T_{2})^{2/\alpha_{1}+2/\alpha_{2}}.$

Consequently, since
$\beta<\frac{1-\vartheta(\varepsilon,\varepsilon)}{1+\vartheta(\varepsilon,\varepsilon)}$,
we have $S_{\mathbf{T},11}\rightarrow 0$ as $\mathbf{T}\rightarrow\infty$.
For $S_{\mathbf{T},12}$, we need more precise estimation. Let’s define

$\omega_{1}(\mathbf{t})=\max\\{|r(\mathbf{t})|,|\rho(\mathbf{T})|\\}$

and

$\theta_{1}(\mathbf{\mathbf{z}})=\sup_{\mathbf{0}\leq\mathbf{t}\leq\mathbf{T},\atop|t_{1}t_{2}|>z_{1}z_{2}}\\{\omega_{1}(\mathbf{t})\\}.$

By the Assumption A3, there exist constants $C>0$ and $K>0$ such that

$\theta_{1}(\mathbf{t})\log\left(t_{1}t_{2}\right)\leqslant K$

for all $\mathbf{T}$ sufficiently large and $\mathbf{t}$ satisfying
$t_{1}t_{2}\geq C$. Thus for all $\mathbf{T}$ large enough and for
$(k_{1}q_{1},k_{2}q_{2})$ such that
$k_{1}q_{1}k_{2}q_{2}\geq(T_{1}T_{2})^{\beta}$, $\theta_{1}(\mathbf{kq})\leq
K/\log(T_{1}T_{2})^{\beta}$. Now making use of (27), we obtain

$\displaystyle\frac{(T_{1}T_{2})^{2}}{q_{1}^{2}q_{2}^{2}\log(T_{1}T_{2})}\exp\left(-\frac{u_{\mathbf{T}}^{2}}{1+\theta_{1}(T_{1}^{\beta},T_{2}^{\beta})}\right)$
$\displaystyle\leq\frac{(T_{1}T_{2})^{2}}{q_{1}^{2}q_{2}^{2}\log(T_{1}T_{2})}\exp\left(-\frac{u_{\mathbf{T}}^{2}}{1+K/\log(T_{1}T_{2})^{\beta}}\right)$
$\displaystyle\thicksim\frac{(T_{1}T_{2})^{2}}{q_{1}^{2}q_{2}^{2}\log(T_{1}T_{2})}\left((T_{1}T_{2})^{-2}(\log
T_{1}T_{2})(\log
T_{1}T_{2})^{-(2/\alpha_{1}+2/\alpha_{2})}\right)^{\frac{1}{1+K/\log(T_{1}T_{2})^{\beta}}}$
$\displaystyle\leq
O(1)(T_{1}T_{2})^{(2K/\log(T_{1}T_{2})^{\beta})/(1+K/\log(T_{1}T_{2})^{\beta})}(\log
T_{1}T_{2})^{((2/\alpha_{1}+2/\alpha_{2}-1)K/\log(T_{1}T_{2})^{\beta})/(1+K/\log(T_{1}T_{2})^{\beta})}$
$\displaystyle=O(1).$ (32)

Therefore, by a similar argument as for the proof of Lemma 6.4.1 of Leadbetter
et al. (1983) we obtain

$\displaystyle S_{\mathbf{T},12}$ $\displaystyle\leq$ $\displaystyle
C\frac{T_{1}}{q_{1}}\frac{T_{2}}{q_{2}}\sum_{\mathbf{0}\leq\mathbf{kq}\leq\mathbf{T},\mathbf{kq}\neq\mathbf{0}\atop
k_{1}q_{1}k_{2}q_{2}>(T_{1}T_{2})^{\beta}}|r(\mathbf{kq})-\rho(\mathbf{T})|\exp\left(-\frac{u^{2}_{\mathbf{T}}}{1+\theta_{1}(T_{1}^{\beta},T_{2}^{\beta})}\right)$
(33) $\displaystyle\leq$ $\displaystyle
C\frac{T_{1}}{q_{1}}\frac{T_{2}}{q_{2}}\exp\left(-\frac{u^{2}_{\mathbf{T}}}{1+\theta_{1}(T_{1}^{\beta},T_{2}^{\beta})}\right)\sum_{\mathbf{0}\leq\mathbf{kq}\leq\mathbf{T},\mathbf{kq}\neq\mathbf{0}\atop
k_{1}q_{1}k_{2}q_{2}>(T_{1}T_{2})^{\beta}}|r(\mathbf{kq})-\rho(\mathbf{T})|$
$\displaystyle=$ $\displaystyle
C\frac{(T_{1}T_{2})^{2}}{q_{1}^{2}q_{2}^{2}\log(T_{1}T_{2})}\exp\left(-\frac{u_{\mathbf{T}}^{2}}{1+\theta_{1}(T_{1}^{\beta},T_{2}^{\beta})}\right)\cdot\frac{q_{1}q_{2}\log(T_{1}T_{2})}{T_{1}T_{2}}\sum_{\mathbf{0}\leq\mathbf{kq}\leq\mathbf{T},\mathbf{kq}\neq\mathbf{0}\atop
k_{1}q_{1}k_{2}q_{2}>(T_{1}T_{2})^{\beta}}|r(\mathbf{kq})-\rho(\mathbf{T})|$
$\displaystyle\leq$ $\displaystyle
C\frac{q_{1}q_{2}\log(T_{1}T_{2})}{T_{1}T_{2}}\sum_{\mathbf{0}\leq\mathbf{kq}\leq\mathbf{T},\mathbf{kq}\neq\mathbf{0}\atop
k_{1}q_{1}k_{2}q_{2}>(T_{1}T_{2})^{\beta}}|r(\mathbf{kq})-\rho(\mathbf{T})|$
$\displaystyle\leq$ $\displaystyle C\frac{q_{1}q_{2}}{\beta
T_{1}T_{2}}\sum_{\mathbf{0}\leq\mathbf{kq}\leq\mathbf{T},\mathbf{kq}\neq\mathbf{0}\atop
k_{1}q_{1}k_{2}q_{2}>(T_{1}T_{2})^{\beta}}|r(\mathbf{kq})\log(k_{1}q_{1}k_{2}q_{2})-r|$
$\displaystyle+Cr\frac{q_{1}q_{2}}{T_{1}T_{2}}\sum_{\mathbf{0}\leq\mathbf{kq}\leq\mathbf{T},\mathbf{kq}\neq\mathbf{0}\atop
k_{1}q_{1}k_{2}q_{2}>(T_{1}T_{2})^{\beta}}\left|1-\frac{\log(T_{1}T_{2})}{\log(k_{1}q_{1}k_{2}q_{2})}\right|.$

By Assumption A3, the first term on the right-hand-side of (33) tends to 0 as
$\mathbf{T}\to\infty$. Furthermore, the second term of the right-hand-side of
(33) also tends to 0 by an integral estimate as follows ( see also the proof
of Lemma 6.4.1 of Leadbetter et al. (1983))

$\displaystyle\frac{q_{1}q_{2}}{T_{1}T_{2}}\sum_{\mathbf{0}\leq\mathbf{kq}\leq\mathbf{T},\mathbf{kq}\neq\mathbf{0}\atop
k_{1}q_{1}k_{2}q_{2}>(T_{1}T_{2})^{\beta}}\left|1-\frac{\log(T_{1}T_{2})}{\log(k_{1}q_{1}k_{2}q_{2})}\right|$
$\displaystyle\leq\frac{r}{\log(T_{1}T_{2})^{\beta}}\frac{q_{1}q_{2}}{T_{1}T_{2}}\sum\left|\log(k_{1}q_{1}k_{2}q_{2})-\log(T_{1}T_{2})\right|$
$\displaystyle=\frac{r}{\log(T_{1}T_{2})^{\beta}}\frac{q_{1}q_{2}}{T_{1}T_{2}}\sum\left|\log\left(\frac{k_{1}q_{1}k_{2}q_{2}}{T_{1}T_{2}}\right)\right|$
$\displaystyle=O\left(\frac{r}{\log(T_{1}T_{2})^{\beta}}\int_{0}^{1}\int_{0}^{1}\log(xy)dxdy\right),$

which shows that $S_{\mathbf{T},12}\rightarrow 0$ as
$\mathbf{T}\rightarrow\infty$. Thus, $S_{\mathbf{T},1}\rightarrow 0$ as
$\mathbf{T}\rightarrow\infty$.

We consider the term $S_{\mathbf{T},2}$ and we will discuss it for two cases,
the first for $(T_{1}T_{2})^{\beta}>T_{2}$, and the second for
$(T_{1}T_{2})^{\beta}\leq T_{2}$.

For the case $(T_{1}T_{2})^{\beta}>T_{2}$, by the same arguments as for (29),
we have

$\displaystyle S_{\mathbf{T},2}$ $\displaystyle=$ $\displaystyle
C\frac{T_{1}}{q_{1}}\frac{T_{2}}{q_{2}}\sum_{0\leq k_{2}q_{2}\leq
T_{2},k_{1}q_{1}=0}|r(0,k_{2}q_{2})-\rho(\mathbf{T})|\exp\left(-\frac{u^{2}_{\mathbf{T}}}{1+\vartheta(T_{1}^{b},T_{2}^{b})}\right)$
$\displaystyle\leq$ $\displaystyle
C\frac{T_{1}}{q_{1}}\frac{T_{2}}{q_{2}}\sum_{0\leq k_{2}q_{2}\leq
T_{2},k_{1}q_{1}=0}\exp\left(-\frac{u^{2}_{\mathbf{T}}}{1+\vartheta(\varepsilon,\varepsilon)}\right)$
$\displaystyle\leq$ $\displaystyle
C(T_{1}T_{2})^{\beta-\frac{1-\vartheta(\varepsilon,\varepsilon)}{1+\vartheta(\varepsilon,\varepsilon)}}(\log
T_{1}T_{2})^{1/\alpha_{1}+2/\alpha_{2}}.$

Therefore, $S_{\mathbf{T},2}\rightarrow 0$ as $\mathbf{T}\rightarrow\infty$ in
view of
$\beta<\frac{1-\vartheta(\varepsilon,\varepsilon)}{1+\vartheta(\varepsilon,\varepsilon)}$.

For the second case $(T_{1}T_{2})^{\beta}\leq T_{2}$, split $S_{\mathbf{T},2}$
into two parts as

$\displaystyle
S_{\mathbf{T},2}=C\sum_{\mathbf{kq}\in\mathbf{O_{i}},\mathbf{lq}\in\mathbf{O_{j}},\mathbf{kq}\neq\mathbf{lq},\mathbf{1}\leq\mathbf{i}\neq\mathbf{j}\leq\mathbf{n}\atop
0<|k_{2}q_{2}-l_{2}q_{2}|\leq(T_{1}T_{2})^{\beta},k_{1}q_{1}=l_{1}q_{1}}+C\sum_{\mathbf{kq}\in\mathbf{O_{i}},\mathbf{lq}\in\mathbf{O_{j}},\mathbf{kq}\neq\mathbf{lq},\mathbf{1}\leq\mathbf{i}\neq\mathbf{j}\leq\mathbf{n}\atop(T_{1}T_{2})^{\beta}<|k_{2}q_{2}-l_{2}q_{2}|\leq
T_{2},k_{1}q_{1}=l_{1}q_{1}}=:S_{\mathbf{T},21}+S_{\mathbf{T},22}.$

For $S_{\mathbf{T},21}$, similarly to the derivation of (29) again, we have

$\displaystyle S_{\mathbf{T},21}$ $\displaystyle\leq$ $\displaystyle
C\frac{T_{1}}{q_{1}}\frac{T_{2}}{q_{2}}\sum_{0\leq
k_{2}q_{2}\leq(T_{1}T_{2})^{\beta},k_{1}q_{1}=0}|r(0,k_{2}q_{2})-\rho(\mathbf{T})|\exp\left(-\frac{u^{2}_{\mathbf{T}}}{1+\vartheta(T_{1}^{b},T_{2}^{b})}\right)$
$\displaystyle\leq$ $\displaystyle
C\frac{T_{1}}{q_{1}}\frac{T_{2}}{q_{2}}\sum_{0\leq
k_{2}q_{2}\leq(T_{1}T_{2})^{\beta},k_{1}q_{1}=0}\exp\left(-\frac{u^{2}_{\mathbf{T}}}{1+\vartheta(\varepsilon,\varepsilon)}\right)$
$\displaystyle\leq$ $\displaystyle
C(T_{1}T_{2})^{\beta-\frac{1-\vartheta(\varepsilon,\varepsilon)}{1+\vartheta(\varepsilon,\varepsilon)}}(\log
T_{1}T_{2})^{1/\alpha_{1}+2/\alpha_{2}},$

which shows that $S_{\mathbf{T},21}\rightarrow 0$ as
$\mathbf{T}\rightarrow\infty$.

For bound the term $S_{\mathbf{T},22}$, we need to define

$\omega_{2}(\mathbf{t})=\max\\{|r(0,t_{2})|,|\rho(\mathbf{T})|\\}$

and

$\theta_{2}(\mathbf{\mathbf{z}})=\sup_{\mathbf{0}\leq\mathbf{t}\leq\mathbf{T},\atop|t_{2}|>z_{1}z_{2}}\\{\omega(\mathbf{t})\\}.$

By Assumption A3 again, we have also $\theta_{2}(\mathbf{kq})\leq
K/\log(T_{1}T_{2})^{\beta}$ and $r(0,k_{2}q_{2})\log(T_{1}T_{2})\leq C$ for
$k_{1}q_{1}=0$ and $k_{2}q_{2}>(T_{1}T_{2})^{\beta}$. So by the same arguments
as for (5), we have

$\frac{(T_{1}T_{2})^{2}}{q_{1}^{2}q_{2}^{2}\log(T_{1}T_{2})}\exp\left(-\frac{u_{\mathbf{T}}^{2}}{1+\theta_{2}(T_{1}^{\beta},T_{2}^{\beta})}\right)=O(1)$

and we thus have

$\displaystyle S_{\mathbf{T},22}$ $\displaystyle\leq$ $\displaystyle
C\frac{T_{1}}{q_{1}}\frac{T_{2}}{q_{2}}\sum_{(T_{1}T_{2})^{\beta}<k_{2}q_{2}\leq
T_{2},k_{1}q_{1}=0}|r(0,k_{2}q_{2})-\rho(\mathbf{T})|\exp\left(-\frac{u^{2}_{\mathbf{T}}}{1+\theta_{2}(T_{1}^{\beta},T_{2}^{\beta})}\right)$
$\displaystyle=$ $\displaystyle
C\frac{(T_{1}T_{2})^{2}}{q_{1}^{2}q_{2}^{2}\log(T_{1}T_{2})}\exp\left(-\frac{u_{\mathbf{T}}^{2}}{1+\theta_{2}(T_{1}^{\beta},T_{2}^{\beta})}\right)\cdot\frac{q_{1}q_{2}\log(T_{1}T_{2})}{T_{1}T_{2}}\sum_{(T_{1}T_{2})^{\beta}<k_{2}q_{2}\leq
T_{2},k_{1}q_{1}=0}|r(0,k_{2}q_{2})-\rho(\mathbf{T})|$ $\displaystyle\leq$
$\displaystyle
C\frac{q_{1}q_{2}\log(T_{1}T_{2})}{T_{1}T_{2}}\sum_{(T_{1}T_{2})^{\beta}<k_{2}q_{2}\leq
T_{2},k_{1}q_{1}=0}|r(0,k_{2}q_{2})-\rho(\mathbf{T})|$ $\displaystyle\leq$
$\displaystyle
C\frac{q_{1}q_{2}\log(T_{1}T_{2})}{T_{1}T_{2}}\sum_{(T_{1}T_{2})^{\beta}<k_{2}q_{2}\leq
T_{2},k_{1}q_{1}=0}(|r(0,k_{2}q_{2})|+\rho(\mathbf{T}))$ $\displaystyle\leq$
$\displaystyle
C\frac{q_{1}q_{2}\log(T_{1}T_{2})}{T_{1}T_{2}}\frac{T_{2}}{q_{2}}\frac{1}{\log(T_{1}T_{2})}$
$\displaystyle=$ $\displaystyle C\frac{q_{1}}{T_{1}},$

which implies $S_{\mathbf{T},22}\rightarrow 0$ as
$\mathbf{T}\rightarrow\infty$. Thus we have proved that
$S_{\mathbf{T},2}\rightarrow 0$ as $\mathbf{T}\rightarrow\infty$. By the same
arguments, we can show that $S_{\mathbf{T},3}\rightarrow 0$ as
$\mathbf{T}\rightarrow\infty$. The proof of the lemma is complete. $\Box$

Lemma B2. Under the conditions of Lemma 3.3, we have

$\displaystyle\sum_{\mathbf{\mathbf{kp}}\in\mathbf{O_{i}},\mathbf{\mathbf{lp}}\in\mathbf{O_{j}}\atop\mathbf{kp}\neq\mathbf{lp},\mathbf{1}\leq\mathbf{i,j}\leq\mathbf{n}}|r(\mathbf{kp},\mathbf{lp})-\varrho(\mathbf{kp},\mathbf{lp})|\int_{0}^{1}\frac{1}{\sqrt{1-r^{(h)}(\mathbf{kp},\mathbf{lp})}}\exp\left(-\frac{u_{\mathbf{T}}^{\prime
2}}{1+r^{(h)}(\mathbf{kp},\mathbf{lp})}\right)dh\rightarrow 0$ (34)

as $\mathbf{T}\rightarrow\infty$.

Proof: The proof is the same as that of Lemma B1, we omit the details.

Lemma B3. Under the conditions of Lemma 3.3, we have

$\displaystyle\sum_{\mathbf{kq}\in\mathbf{O_{i}},\mathbf{lp}\in\mathbf{O_{j}}\atop\mathbf{kq}\neq\mathbf{lp},\mathbf{1}\leq\mathbf{i,j}\leq\mathbf{n}}|r(\mathbf{kq},\mathbf{lp})-\varrho(\mathbf{kq},\mathbf{lp})|\int_{0}^{1}\frac{1}{\sqrt{1-r^{(h)}(\mathbf{kq},\mathbf{lp})}}\exp\left(-\frac{u_{\mathbf{T}}^{\prime
2}+u_{\mathbf{T}}^{2}}{2(1+r^{(h)}(\mathbf{kq},\mathbf{lp}))}\right)dh\rightarrow
0$ (35)

as $\mathbf{T}\rightarrow\infty$.

Proof: Recall that $\mathfrak{R}(p_{i})$, $i=1,2$ can be sparse grids or
Pickands grids. First, we consider the case that $\mathbf{kq},\mathbf{lp}$ in
the same interval $\mathbf{O_{i}}$. Split the sum in (35) into two parts as

$\displaystyle\sum_{\mathbf{kq},\mathbf{lp}\in\mathbf{O_{i}},\mathbf{kq}\neq\mathbf{lp},\mathbf{i}=\mathbf{1},\cdots,\mathbf{n},\atop\max\\{|l_{1}p_{1}-k_{1}q_{1}|,|l_{2}p_{2}-k_{2}q_{2}|\\}\leq\varepsilon}+\sum_{\mathbf{kq},\mathbf{lp}\in\mathbf{O_{i}},\mathbf{kq}\neq\mathbf{lp},\mathbf{i}=\mathbf{1},\cdots,\mathbf{n},\atop\max\\{|l_{1}p_{1}-k_{1}q_{1}|,|l_{2}p_{2}-k_{2}q_{2}|\\}>\varepsilon}=:W_{\mathbf{T},1}+W_{\mathbf{T},2}.$
(36)

We deal with $W_{\mathbf{T},1}$. For $\mathbf{kq},\mathbf{lp}$ in the same
interval $\mathbf{O_{i}}$, we have
$\varrho(\mathbf{kq},\mathbf{lp})-r(\mathbf{kq},\mathbf{lq})=\rho(\mathbf{T})(1-r(\mathbf{kq},\mathbf{lq}))$.
By Assumption A1 we can also choose small enough $\varepsilon>0$ such that
$\varrho(\mathbf{kq},\mathbf{lp})=r(\mathbf{kq},\mathbf{lp})+(1-r(\mathbf{kq},\mathbf{lp}))\rho(\mathbf{T})\sim
r(\mathbf{kq},\mathbf{lp})$ for sufficiently large $\mathbf{T}$ and
$\max\\{|l_{1}p_{1}-k_{1}q_{1}|,|l_{2}p_{2}-k_{2}q_{2}|\\}\leq\varepsilon$. By
the definitions of $u_{\mathbf{T}}$ and $u_{\mathbf{T}}^{\prime}$, we have

$\displaystyle
v_{\mathbf{T}}^{2}:=\frac{1}{2}(u_{\mathbf{T}}^{2}+(u_{\mathbf{T}}^{\prime})^{2})=2\log(T_{1}T_{2})-\log\log(T_{1}T_{2})+\log(p_{1}^{-1}p_{2}^{-1})+(1/\alpha_{1}+1/\alpha_{2})\log\log(T_{1}T_{2})+O(1).$
(37)

Consequently, in view of (26), we obtain

$\displaystyle W_{\mathbf{T},1}$ $\displaystyle\leq$ $\displaystyle
C\sum_{\mathbf{kq},\mathbf{lp}\in\mathbf{O_{i}},\mathbf{kq}\neq\mathbf{lp},\mathbf{i}=\mathbf{1},\cdots,\mathbf{n},\atop\max\\{|l_{1}p_{1}-k_{1}q_{1}|,|l_{2}p_{2}-k_{2}q_{2}|\\}\leq\varepsilon}|r(\mathbf{kq},\mathbf{lp})-\varrho(\mathbf{kq},\mathbf{lp})|\frac{1}{\sqrt{1-r(\mathbf{kq},\mathbf{lp})}}\exp\left(-\frac{v_{\mathbf{T}}^{2}}{1+r(\mathbf{kq},\mathbf{lq})}\right)$
(38) $\displaystyle\leq$ $\displaystyle
C\rho(\mathbf{T})\sum_{\mathbf{kq},\mathbf{lp}\in\mathbf{O_{i}},\mathbf{kq}\neq\mathbf{lp},\mathbf{i}=\mathbf{1},\cdots,\mathbf{n},\atop\max\\{|l_{1}p_{1}-k_{1}q_{1}|,|l_{2}p_{2}-k_{2}q_{2}|\\}\leq\varepsilon}\sqrt{1-r(\mathbf{kq},\mathbf{lp})}\exp\left(-\frac{v^{2}_{\mathbf{T}}}{2}\right)\exp\left(-\frac{(1-r(\mathbf{kq},\mathbf{lp}))v^{2}_{\mathbf{T}}}{(1+r(\mathbf{kq},\mathbf{lp}))}\right)$
$\displaystyle\leq$ $\displaystyle
C\rho(\mathbf{T})(T_{1}T_{2})^{-1}(p_{1}p_{2})^{1/2}(\log
T_{1}T_{2})^{1/2-1/2\alpha_{1}-1/2\alpha_{2}}\times$
$\displaystyle\sum_{\mathbf{kq},\mathbf{lp}\in\mathbf{O_{i}},\mathbf{kq}\neq\mathbf{lp},\mathbf{i}=\mathbf{1},\cdots,\mathbf{n},\atop\max\\{|l_{1}p_{1}-k_{1}q_{1}|,|l_{2}p_{2}-k_{2}q_{2}|\\}\leq\varepsilon}\sqrt{1-r(\mathbf{kq}-\mathbf{lp})}\exp\left(-\frac{(1-r(\mathbf{kq}-\mathbf{lp}))u^{2}_{\mathbf{T}}}{2(1+r(\mathbf{kq}-\mathbf{lp}))}\right)$
$\displaystyle\leq$ $\displaystyle C(T_{1}T_{2})^{-1}(p_{1}p_{2})^{1/2}(\log
T_{1}T_{2})^{-1/2-1/2\alpha_{1}-1/2\alpha_{2}}\times$
$\displaystyle\sum_{\mathbf{kq},\mathbf{lp}\in\mathbf{O_{i}},\mathbf{kq}\neq\mathbf{lp},\mathbf{i}=\mathbf{1},\cdots,\mathbf{n},\atop\max\\{|l_{1}p_{1}-k_{1}q_{1}|,|l_{2}p_{2}-k_{2}q_{2}|\\}\leq\varepsilon}\sqrt{|l_{1}p_{1}-k_{1}q_{1}|^{\alpha_{1}}+|l_{2}p_{2}-k_{2}q_{2}|^{\alpha_{2}}}\times$
$\displaystyle\exp\left(-\frac{(|l_{1}p_{1}-k_{1}q_{1}|^{\alpha_{1}}+|l_{2}p_{2}-k_{2}q_{2}|^{\alpha_{2}})v^{2}_{\mathbf{T}}}{8}\right).$

Noting that $q_{i}=\gamma_{i}\log(T_{1}T_{2})^{1/\alpha_{i}}$ and
$\mathfrak{R}(p_{i})$, $i=1,2$ are sparse grids or Pickands grids, a direct
calculation shows that

$\displaystyle\sum_{\mathbf{kq},\mathbf{lp}\in\mathbf{O_{i}},\mathbf{kq}\neq\mathbf{lp},\mathbf{i}=\mathbf{1},\cdots,\mathbf{n},\atop\max\\{|l_{1}p_{1}-k_{1}q_{1}|,|l_{2}p_{2}-k_{2}q_{2}|\\}\leq\varepsilon}\sqrt{|l_{1}p_{1}-k_{1}q_{1}|^{\alpha_{1}}+|l_{2}p_{2}-k_{2}q_{2}|^{\alpha_{2}}}\times$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \
\exp\left(-\frac{(|l_{1}p_{1}-k_{1}q_{1}|^{\alpha_{1}}+|l_{2}p_{2}-k_{2}q_{2}|^{\alpha_{2}})v^{2}_{\mathbf{T}}}{8}\right)$
$\displaystyle\ \ \ \ \ \leq
CT_{1}T_{2}p_{1}^{-1}p_{2}^{-1}\sum_{0<k_{1}q_{1}<\varepsilon,0<k_{2}q_{2}<\varepsilon}\exp\left(-\frac{1}{4}[(k_{1}q_{1})^{\alpha_{1}}+(k_{2}q_{2})^{\alpha_{2}}]\log(T_{1}T_{2})\right)$
$\displaystyle\ \ \ \ \ \leq
CT_{1}T_{2}p_{1}^{-1}p_{2}^{-1}\sum_{k_{1}=1}^{\infty}e^{-\frac{1}{4}(k_{1}\gamma_{1})^{\alpha_{1}}}\sum_{k_{2}=1}^{\infty}e^{-\frac{1}{4}(k_{2}\gamma_{2})^{\alpha_{2}}}$
$\displaystyle\ \ \ \ \ \leq CT_{1}T_{2}p_{1}^{-1}p_{2}^{-1},$

which combine with (38) shows that $W_{\mathbf{T},1}\rightarrow 0$ as
$\mathbf{T}\rightarrow\infty$.
Using the fact that $v_{\mathbf{T}}\thicksim(2\log T_{1}T_{2})^{1/2}$, we
obtain

$\displaystyle W_{\mathbf{T},2}$ $\displaystyle\leq$ $\displaystyle
C\sum_{\mathbf{kq},\mathbf{lp}\in\mathbf{O_{i}},\mathbf{kq}\neq\mathbf{lp},\mathbf{i}=\mathbf{1},\cdots,\mathbf{n},\atop\max\\{|l_{1}p_{1}-k_{1}q_{1}|,|l_{2}p_{2}-k_{2}q_{2}|\\}>\varepsilon}|r(\mathbf{kq},\mathbf{lp})-\varrho(\mathbf{kq},\mathbf{lp})|\exp\left(-\frac{v^{2}_{\mathbf{T}}}{1+\varpi(\mathbf{kq},\mathbf{lp})}\right)$
(39) $\displaystyle\leq$ $\displaystyle
C\sum_{\mathbf{kq},\mathbf{lp}\in\mathbf{O_{i}},\mathbf{kq}\neq\mathbf{lp},\mathbf{i}=\mathbf{1},\cdots,\mathbf{n},\atop\max\\{|l_{1}p_{1}-k_{1}q_{1}|,|l_{2}p_{2}-k_{2}q_{2}|\\}>\varepsilon}\exp\left(-\frac{v^{2}_{\mathbf{T}}}{1+\vartheta(\varepsilon,\varepsilon)}\right)$
$\displaystyle\leq$ $\displaystyle
C\frac{T_{1}}{p_{1}}\frac{T_{2}}{p_{2}}\exp\left(-\frac{u^{2}_{\mathbf{T}}}{1+\vartheta(\varepsilon,\varepsilon)}\right)\sum_{0\leq
k_{1}q_{1}\leq T_{1}^{a},0\leq k_{2}q_{2}\leq T_{2}^{a}}1$ $\displaystyle\leq$
$\displaystyle
C\frac{T_{1}}{p_{1}}\frac{T_{2}}{p_{2}}(T_{1}T_{2})^{-\frac{2}{1+\vartheta(\varepsilon,\varepsilon)}}\sum_{0\leq
k_{1}q_{1}\leq T_{1}^{a},0\leq k_{2}q_{2}\leq T_{2}^{a}}1$ $\displaystyle\leq$
$\displaystyle
C(T_{1}T_{2})^{a-\frac{1-\vartheta(\varepsilon,\varepsilon)}{1+\vartheta(\varepsilon,\varepsilon)}}(\log
T_{1}T_{2})^{1/\alpha_{1}+1/\alpha_{2}}(p_{1}p_{2})^{-1}.$

Thus, $W_{\mathbf{T},2}\rightarrow 0$ as $\mathbf{T}\rightarrow\infty$ by
virtue of
$a<\frac{1-\vartheta(\varepsilon,\varepsilon)}{1+\vartheta(\varepsilon,\varepsilon)}$
again.

Second, we deal with the case that $\mathbf{kq}\in\mathbf{O_{i}}$ and
$\mathbf{lp}\in\mathbf{O_{j}}$, $\mathbf{i}\neq\mathbf{j}$. Note that in this
case, the distance between the points in any two rectangles $\mathbf{O_{i}}$
and $\mathbf{O_{j}}$ is large than $T_{1}^{b}$ or $T_{2}^{b}$ and
$\varrho(\mathbf{kq},\mathbf{lp})=\rho(\mathbf{T})$ for
$\mathbf{kq}\in\mathbf{O_{i}}$ and $\mathbf{lp}\in\mathbf{O_{j}}$,
$\mathbf{i}\neq\mathbf{j}$.

Obviously, the sum in (35) is at most

$\displaystyle
C\sum_{\mathbf{kq}\in\mathbf{O_{i}},\mathbf{lp}\in\mathbf{O_{j}}\atop\mathbf{kq}\neq\mathbf{lp},\mathbf{1}\leq\mathbf{i}\neq\mathbf{j}\leq\mathbf{n}}|r(\mathbf{kq},\mathbf{lp})-\rho(\mathbf{T})|\exp\left(-\frac{v^{2}_{\mathbf{T}}}{1+\varpi(\mathbf{kq},\mathbf{lp})}\right).$
(40)

Split the sum of (40) into three parts, the first for
$|k_{1}q_{1}-l_{1}p_{1}|>0$ and $|k_{2}q_{2}-l_{2}p_{2}|>0$, the second for
$k_{1}q_{1}-l_{1}p_{1}=0$ and $|k_{2}q_{2}-l_{2}p_{2}|>0$, the third for
$k_{2}q_{2}-l_{2}p_{2}=0$ and $|k_{1}q_{1}-l_{1}p_{1}|>0$ and denote them by
$M_{\mathbf{T},i}$, $i=1,2,3,$ respectively. Let $\beta$ be chosen as before,
ie,
$0<b<a<\beta<\frac{1-\vartheta(\varepsilon,\varepsilon)}{1+\vartheta(\varepsilon,\varepsilon)}$
for all sufficiently large $\mathbf{T}$.

We consider the term $M_{\mathbf{T},1}$ and split it into two parts as

$\displaystyle
M_{\mathbf{T},1}=C\sum_{\mathbf{kq}\in\mathbf{O_{i}},\mathbf{lp}\in\mathbf{O_{j}},\mathbf{kq}\neq\mathbf{lp},\mathbf{1}\leq\mathbf{i}\neq\mathbf{j}\leq\mathbf{n}\atop|k_{1}q_{1}-l_{1}q_{1}||k_{2}q_{2}-l_{2}q_{2}|\leq(T_{1}T_{2})^{\beta}}+C\sum_{\mathbf{kq}\in\mathbf{O_{i}},\mathbf{lp}\in\mathbf{O_{j}},\mathbf{kq}\neq\mathbf{lp},\mathbf{1}\leq\mathbf{i}\neq\mathbf{j}\leq\mathbf{n}\atop|k_{1}q_{1}-l_{1}q_{1}||k_{2}q_{2}-l_{2}q_{2}|>(T_{1}T_{2})^{\beta}}=:M_{\mathbf{T},11}+M_{\mathbf{T},12}.$

For $M_{\mathbf{T},11}$, with the similar derivation as for (39), we have

$\displaystyle M_{\mathbf{T},11}$ $\displaystyle\leq$ $\displaystyle
C\frac{T_{1}}{p_{1}}\frac{T_{2}}{p_{2}}\sum_{0\leq k_{1}q_{1}\leq T_{1},0\leq
k_{2}q_{2}\leq T_{2},\atop
k_{1}q_{1}k_{2}q_{2}\leq(T_{1}T_{2})^{\beta}}\exp\left(-\frac{v^{2}_{\mathbf{T}}}{1+\vartheta(\varepsilon,\varepsilon)}\right)$
(41) $\displaystyle\leq$ $\displaystyle
C(T_{1}T_{2})^{\beta-\frac{1-\vartheta(\varepsilon,\varepsilon)}{1+\vartheta(\varepsilon,\varepsilon)}}(\log
T_{1}T_{2})^{1/\alpha_{1}+1/\alpha_{2}}(p_{1}p_{2})^{-1}.$

Thus, $M_{\mathbf{T},11}\rightarrow 0$ as $\mathbf{T}\rightarrow\infty$, since
$\beta<\frac{1-\vartheta(\varepsilon,\varepsilon)}{1+\vartheta(\varepsilon,\varepsilon)}$.
For $M_{\mathbf{T},12}$, we need also more precise estimation. Recall that

$\omega_{1}(\mathbf{t})=\max\\{|r(\mathbf{t})|,|\rho(\mathbf{T})|\\}\ \
\mbox{and}\ \
\theta_{1}(\mathbf{\mathbf{z}})=\sup_{\mathbf{0}\leq\mathbf{t}\leq\mathbf{T},\atop|t_{1}t_{2}|>z_{1}z_{2}}\\{\omega_{1}(\mathbf{t})\\}.$

Now using (37) again, by the same arguments as for (5), we obtain

$\displaystyle\frac{(T_{1}T_{2})^{2}}{q_{1}q_{2}p_{1}p_{2}\log(T_{1}T_{2})}\exp\left(-\frac{v_{\mathbf{T}}^{2}}{1+\theta_{1}(T_{1}^{\beta},T_{2}^{\beta})}\right)$
$\displaystyle\leq\frac{(T_{1}T_{2})^{2}}{q_{1}q_{2}p_{1}p_{2}\log(T_{1}T_{2})}\exp\left(-\frac{v_{\mathbf{T}}^{2}}{1+K/\log(T_{1}T_{2})^{\beta}}\right)$
$\displaystyle\thicksim\frac{(T_{1}T_{2})^{2}}{q_{1}q_{2}p_{1}p_{2}\log(T_{1}T_{2})}\left((T_{1}T_{2})^{-2}(\log
T_{1}T_{2})(\log
T_{1}T_{2})^{-(1/\alpha_{1}+1/\alpha_{2})}(p_{1}p_{2})^{-1}\right)^{\frac{1}{1+K/\log(T_{1}T_{2})^{\beta}}}$
$\displaystyle=O(1)$ (42)

and then we thus have

$\displaystyle M_{\mathbf{T},12}$ $\displaystyle\leq$ $\displaystyle
C\sum_{\mathbf{kq}\in\mathbf{O_{i}},\mathbf{lp}\in\mathbf{O_{j}},\mathbf{kq}\neq\mathbf{lp},\mathbf{1}\leq\mathbf{i}\neq\mathbf{j}\leq\mathbf{n}\atop|k_{1}q_{1}-l_{1}q_{1}||k_{2}q_{2}-l_{2}q_{2}|>(T_{1}T_{2})^{\beta}}|r(\mathbf{kq}-\mathbf{lp})-\rho(\mathbf{T})|\exp\left(-\frac{v^{2}_{\mathbf{T}}}{1+\theta_{1}(T_{1}^{\beta},T_{2}^{\beta})}\right)$
(43) $\displaystyle=$ $\displaystyle
C\frac{(T_{1}T_{2})^{2}}{q_{1}q_{2}p_{1}p_{2}\log(T_{1}T_{2})}\exp\left(-\frac{v_{\mathbf{T}}^{2}}{1+\theta_{1}(T_{1}^{\beta},T_{2}^{\beta})}\right)\times$
$\displaystyle\ \ \
\frac{q_{1}q_{2}p_{1}p_{2}\log(T_{1}T_{2})}{(T_{1}T_{2})^{2}}\sum_{\mathbf{kq}\in\mathbf{O_{i}},\mathbf{lp}\in\mathbf{O_{j}},\mathbf{kq}\neq\mathbf{lp},\mathbf{1}\leq\mathbf{i}\neq\mathbf{j}\leq\mathbf{n}\atop|k_{1}q_{1}-l_{1}q_{1}||k_{2}q_{2}-l_{2}q_{2}|>(T_{1}T_{2})^{\beta}}|r(\mathbf{kq}-\mathbf{lp})-\rho(\mathbf{T})|$
$\displaystyle\leq$ $\displaystyle
C\frac{q_{1}q_{2}p_{1}p_{2}\log(T_{1}T_{2})}{(T_{1}T_{2})^{2}}\sum_{\mathbf{kq}\in\mathbf{O_{i}},\mathbf{lp}\in\mathbf{O_{j}},\mathbf{kq}\neq\mathbf{lp},\mathbf{1}\leq\mathbf{i}\neq\mathbf{j}\leq\mathbf{n}\atop|k_{1}q_{1}-l_{1}q_{1}||k_{2}q_{2}-l_{2}q_{2}|>(T_{1}T_{2})^{\beta}}|r(\mathbf{kq}-\mathbf{lp})-\rho(\mathbf{T})|$
$\displaystyle\leq$ $\displaystyle
C\frac{q_{1}q_{2}p_{1}p_{2}}{\beta(T_{1}T_{2})^{2}}\sum_{\mathbf{kq}\in\mathbf{O_{i}},\mathbf{lp}\in\mathbf{O_{j}},\mathbf{kq}\neq\mathbf{lp},\mathbf{1}\leq\mathbf{i}\neq\mathbf{j}\leq\mathbf{n}\atop|k_{1}q_{1}-l_{1}q_{1}||k_{2}q_{2}-l_{2}q_{2}|>(T_{1}T_{2})^{\beta}}|r(\mathbf{kq}-\mathbf{lp})\log((k_{1}q_{1}-l_{1}p_{1})(k_{2}q_{2}-l_{2}p_{2}))-r|$
$\displaystyle+Cr\frac{q_{1}q_{2}p_{1}p_{2}}{(T_{1}T_{2})^{2}}\sum_{\mathbf{kq}\in\mathbf{O_{i}},\mathbf{lp}\in\mathbf{O_{j}},\mathbf{kq}\neq\mathbf{lp},\mathbf{1}\leq\mathbf{i}\neq\mathbf{j}\leq\mathbf{n}\atop|k_{1}q_{1}-l_{1}q_{1}||k_{2}q_{2}-l_{2}q_{2}|>(T_{1}T_{2})^{\beta}}\left|1-\frac{\log(T_{1}T_{2})}{\log((k_{1}q_{1}-l_{1}p_{1})(k_{2}q_{2}-l_{2}p_{2}))}\right|.$

By Assumption A3, the first term on the right-hand-side of (43) tends to 0 as
$\mathbf{T}\to\infty$. Furthermore, the second term of the right-hand-side of
(43) also tends to 0 by an integral estimate as for Lemma B1. Thus
$M_{\mathbf{T},12}\rightarrow 0$ as $\mathbf{T}\rightarrow\infty$ and then
$M_{\mathbf{T},1}\rightarrow 0$ as $\mathbf{T}\rightarrow\infty$.

We consider the term $M_{\mathbf{T},2}$ now. As the the proof of the previous
lemma, we also discuss it for two cases, the first for
$(T_{1}T_{2})^{\beta}>T_{2}$, and the second for $(T_{1}T_{2})^{\beta}\leq
T_{2}$.

For the case $(T_{1}T_{2})^{\beta}>T_{2}$, by the same arguments as for (39),
we have

$\displaystyle M_{\mathbf{T},2}$ $\displaystyle=$ $\displaystyle
C\sum_{\mathbf{kq}\in\mathbf{O_{i}},\mathbf{lp}\in\mathbf{O_{j}},\mathbf{kq}\neq\mathbf{lp},\mathbf{1}\leq\mathbf{i}\neq\mathbf{j}\leq\mathbf{n}\atop
k_{1}q_{1}-l_{1}p_{1}=0}|r(0,k_{2}q_{2}-l_{2}p_{2})-\rho(\mathbf{T})|\exp\left(-\frac{v^{2}_{\mathbf{T}}}{1+\vartheta(T_{1}^{b},T_{2}^{b})}\right)$
$\displaystyle\leq$ $\displaystyle
C\exp\left(-\frac{v^{2}_{\mathbf{T}}}{1+\vartheta(\varepsilon,\varepsilon)}\right)\sum_{\mathbf{kq}\in\mathbf{O_{i}},\mathbf{lp}\in\mathbf{O_{j}},\mathbf{kq}\neq\mathbf{lp},\mathbf{1}\leq\mathbf{i}\neq\mathbf{j}\leq\mathbf{n}\atop
k_{1}q_{1}-l_{1}p_{1}=0}|r(0,k_{2}q_{2}-l_{2}p_{2})-\rho(\mathbf{T})|$
$\displaystyle\leq$ $\displaystyle
C(T_{1}T_{2})^{\beta-\frac{1-\vartheta(\varepsilon,\varepsilon)}{1+\vartheta(\varepsilon,\varepsilon)}}(\log
T_{1}T_{2})^{1/\alpha_{2}}(p_{1}p_{2})^{-1},$

which shows that $M_{\mathbf{T},2}\rightarrow 0$ as
$\mathbf{T}\rightarrow\infty$, where in the last step, we use the fact that
the number of $(k_{1},l_{1})$ such that $k_{1}q_{1}-l_{1}p_{1}=0$ does not
exceed $T_{1}/p_{1}$ and $(T_{1}T_{2})^{\beta}>T_{2}$.

For the second case $(T_{1}T_{2})^{\beta}\leq T_{2}$, split $M_{\mathbf{T},2}$
into two parts as

$\displaystyle
M_{\mathbf{T},2}=C\sum_{\mathbf{kq}\in\mathbf{O_{i}},\mathbf{lp}\in\mathbf{O_{j}},\mathbf{kq}\neq\mathbf{lp},\mathbf{1}\leq\mathbf{i}\neq\mathbf{j}\leq\mathbf{n}\atop
0<|k_{2}q_{2}-l_{2}p_{2}|\leq(T_{1}T_{2})^{\beta},k_{1}q_{1}=l_{1}p_{1}}+C\sum_{\mathbf{kq}\in\mathbf{O_{i}},\mathbf{lp}\in\mathbf{O_{j}},\mathbf{kq}\neq\mathbf{lp},\mathbf{1}\leq\mathbf{i}\neq\mathbf{j}\leq\mathbf{n}\atop(T_{1}T_{2})^{\beta}<|k_{2}q_{2}-l_{2}p_{2}|\leq
T_{2},k_{1}q_{1}=l_{1}p_{1}}=:M_{\mathbf{T},21}+M_{\mathbf{T},22}.$

For $M_{\mathbf{T},21}$, similarly to the derivation of (39) again, we have

$\displaystyle M_{\mathbf{T},21}$ $\displaystyle\leq$ $\displaystyle
C\sum_{\mathbf{kq}\in\mathbf{O_{i}},\mathbf{lp}\in\mathbf{O_{j}},\mathbf{kq}\neq\mathbf{lp},\mathbf{1}\leq\mathbf{i}\neq\mathbf{j}\leq\mathbf{n}\atop
0<|k_{2}q_{2}-l_{2}p_{2}|\leq(T_{1}T_{2})^{\beta},k_{1}q_{1}-l_{1}p_{1}=0}|r(0,k_{2}q_{2}-l_{2}p_{2})-\rho(\mathbf{T})|\exp\left(-\frac{v^{2}_{\mathbf{T}}}{1+\vartheta(T_{1}^{b},T_{2}^{b})}\right)$
$\displaystyle\leq$ $\displaystyle
C\exp\left(-\frac{v^{2}_{\mathbf{T}}}{1+\vartheta(\varepsilon,\varepsilon)}\right)\sum_{\mathbf{kq}\in\mathbf{O_{i}},\mathbf{lp}\in\mathbf{O_{j}},\mathbf{kq}\neq\mathbf{lp},\mathbf{1}\leq\mathbf{i}\neq\mathbf{j}\leq\mathbf{n}\atop
0<|k_{2}q_{2}-l_{2}p_{2}|\leq(T_{1}T_{2})^{\beta},k_{1}q_{1}-l_{1}p_{1}=0}|r(0,k_{2}q_{2}-l_{2}p_{2})-\rho(\mathbf{T})|$
$\displaystyle\leq$ $\displaystyle
C(T_{1}T_{2})^{\beta-\frac{1-\vartheta(\varepsilon,\varepsilon)}{1+\vartheta(\varepsilon,\varepsilon)}}(\log
T_{1}T_{2})^{1/\alpha_{2}}(p_{1}p_{2})^{-1},$

which shows that $M_{\mathbf{T},21}\rightarrow 0$ as
$\mathbf{T}\rightarrow\infty$.

For $M_{\mathbf{T},22}$, we recall that

$\omega_{2}(\mathbf{t})=\max\\{|r(0,t_{2})|,|\rho(\mathbf{T})|\\}\ \
\mbox{and}\ \
\theta_{2}(\mathbf{\mathbf{z}})=\sup_{\mathbf{0}\leq\mathbf{t}\leq\mathbf{T},\atop|t_{2}|>z_{1}z_{2}}\\{\omega(\mathbf{t})\\}.$

So by the same arguments as for (5), we have

$\frac{(T_{1}T_{2})^{2}}{q_{1}q_{2}p_{1}p_{2}\log(T_{1}T_{2})}\exp\left(-\frac{v_{\mathbf{T}}^{2}}{1+\theta_{2}(T_{1}^{\beta},T_{2}^{\beta})}\right)=O(1)$

and we thus have

$\displaystyle M_{\mathbf{T},22}$ $\displaystyle\leq$ $\displaystyle
C\sum_{\mathbf{kq}\in\mathbf{O_{i}},\mathbf{lp}\in\mathbf{O_{j}},\mathbf{kq}\neq\mathbf{lp},\mathbf{1}\leq\mathbf{i}\neq\mathbf{j}\leq\mathbf{n}\atop(T_{1}T_{2})^{\beta}<|k_{2}q_{2}-l_{2}p_{2}|,k_{1}q_{1}-l_{1}p_{1}=0}|r(0,k_{2}q_{2}-l_{2}p_{2})-\rho(\mathbf{T})|\exp\left(-\frac{v^{2}_{\mathbf{T}}}{1+\theta_{2}(T_{1}^{\beta},T_{2}^{\beta})}\right)$
$\displaystyle=$ $\displaystyle
C\frac{(T_{1}T_{2})^{2}}{q_{1}q_{2}p_{1}p_{2}\log(T_{1}T_{2})}\exp\left(-\frac{v_{\mathbf{T}}^{2}}{1+\theta_{2}(T_{1}^{\beta},T_{2}^{\beta})}\right)\times$
$\displaystyle\frac{q_{1}q_{2}p_{1}p_{2}\log(T_{1}T_{2})}{(T_{1}T_{2})^{2}}\sum_{\mathbf{kq}\in\mathbf{O_{i}},\mathbf{lp}\in\mathbf{O_{j}},\mathbf{kq}\neq\mathbf{lp},\mathbf{1}\leq\mathbf{i}\neq\mathbf{j}\leq\mathbf{n}\atop(T_{1}T_{2})^{\beta}<|k_{2}q_{2}-l_{2}p_{2}|,k_{1}q_{1}-l_{1}p_{1}=0}|r(0,k_{2}q_{2}-l_{2}p_{2})-\rho(\mathbf{T})|$
$\displaystyle\leq$ $\displaystyle
C\frac{q_{1}q_{2}p_{1}p_{2}\log(T_{1}T_{2})}{(T_{1}T_{2})^{2}}\sum_{\mathbf{kq}\in\mathbf{O_{i}},\mathbf{lp}\in\mathbf{O_{j}},\mathbf{kq}\neq\mathbf{lp},\mathbf{1}\leq\mathbf{i}\neq\mathbf{j}\leq\mathbf{n}\atop(T_{1}T_{2})^{\beta}<|k_{2}q_{2}-l_{2}p_{2}|,k_{1}q_{1}-l_{1}p_{1}=0}|r(0,k_{2}q_{2}-l_{2}p_{2})-\rho(\mathbf{T})|$
$\displaystyle\leq$ $\displaystyle
C\frac{q_{1}q_{2}p_{1}p_{2}\log(T_{1}T_{2})}{(T_{1}T_{2})^{2}}\sum_{\mathbf{kq}\in\mathbf{O_{i}},\mathbf{lp}\in\mathbf{O_{j}},\mathbf{kq}\neq\mathbf{lp},\mathbf{1}\leq\mathbf{i}\neq\mathbf{j}\leq\mathbf{n}\atop(T_{1}T_{2})^{\beta}<|k_{2}q_{2}-l_{2}p_{2}|,k_{1}q_{1}-l_{1}p_{1}=0}(|r(0,k_{2}q_{2}-l_{2}p_{2})|+\rho(\mathbf{T}))$
$\displaystyle\leq$ $\displaystyle
C\frac{q_{1}q_{2}p_{1}p_{2}\log(T_{1}T_{2})}{(T_{1}T_{2})^{2}}\frac{T_{2}^{2}T_{1}}{p_{1}q_{2}p_{2}}\frac{1}{\log(T_{1}T_{2})}$
$\displaystyle=$ $\displaystyle C\frac{q_{1}}{T_{1}},$

which implies $M_{\mathbf{T},22}\rightarrow 0$ as
$\mathbf{T}\rightarrow\infty$. Now we have showed that
$M_{\mathbf{T},2}\rightarrow 0$ as $\mathbf{T}\rightarrow\infty$. By the same
arguments, we can show that $M_{\mathbf{T},3}\rightarrow 0$ as
$\mathbf{T}\rightarrow\infty$. The proof of the lemma is complete. $\Box$

Acknowledgement: We would like to thank the referee for several important
suggestions and corrections.

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