Extremes of $L^p$-norm of Vector-valued Gaussian processes with Trend

TL;DR

This paper derives the exact tail asymptotics of the $L^p$ norm of Gaussian vector processes with trend, considering both stationary and non-stationary cases, with applications in engineering, insurance, and statistics.

Contribution

It provides the first precise tail asymptotics for the $L^p$ norm of Gaussian vector processes with trend, including sum and chi-square processes, in both stationary and non-stationary settings.

Findings

Derived exact tail asymptotics for Gaussian vector processes with trend.

Analyzed both locally stationary and non-stationary processes.

Applicable to sum and chi-square Gaussian processes with trend.

Abstract

Let $\boldsymbol{X}(t)=(X_1(t),\ldots,X_d(t))$ be a Gaussian vector process and $g(t)$ be a continuous function. The asymptotics of distribution of $\left\|\boldsymbol{X}(t)\right\|_p$, the $L^p$ norm for Gaussian finite-dimensional vector, have been investigated in numerous literatures. In this contribution we are concerned with the exact tail asymptotics of $\left\|\boldsymbol{X}(t)\right\|^c_p,\ c>0, $ with trend $g(t)$ over $[0,T]$. Both scenarios that $\boldsymbol{X}(t)$ is locally stationary and non-stationary are considered. Important examples include $\sum_{i=1}^d \left|X_i(t)\right|+g(t)$ and chi-square processes with trend, i.e., $\sum_{i=1}^d X_i^2(t)+g(t)$. These results are of interest in applications in engineering, insurance and statistics, etc.

Equations

$$\mathbb{P}\left\{\mathbf{Y}(t)\notin {\mathbf{S}}_u(t),\text{for some}t\in [0,T]\right\},$$

$${\mathbf{S}}_u(t)=\left\{(x_1,\cdots ,x_n)\in {\mathbb{R}}^n:\Vert \mathbf{x}{\Vert }_p\le h(t,u)\right\}$$

$$||\boldsymbol{x}||_{p}=\left\{\begin{array}[]{ll}\left(\sum_{i=1}^{n}|x_{i}|^{p}\right)^{1/p},&\ p\in[1,\infty),\\ \max(|x_{1}|,\ldots,|x_{n}|),&\ p=\infty,\end{array}\right.$$

$$\mathbf{d}=(d_1,\dots ,d_n),1=d_1=\cdots =d_m>d_{m+1}\ge d_{m+2}\ge \cdots \ge d_n>0,1\le m\le n.$$

$$\mathbb{P}\left\{{\exists }_{t\in [0,T]}Z(t)>h(t,u)\right\}$$

$$Z(t):=Z_p(t):={\Vert \mathbf{X}(t)\ast \mathbf{d}\Vert }_p,$$

$$\mathbb{P}\left\{{\exists }_{t\in [0,T]}{Z}_2(t)>u\right\}=\mathbb{P}\left\{\underset{t\in [0,T]}{\sup }{Z}_2(t)>u\right\},\text{as}u\to \infty$$

$$r(s,t)=1-a|t-s{|}^{\alpha }+o(|t-s{|}^{\alpha }),\left|t-s\right|\to 0,\alpha \in (0,2],$$

$$\sigma (t)=1-b(t_0){\left|t-t_0\right|}^2+o({\left|t-t_0\right|}^2),r(s,t)=1-a(t_0){\left|t-s\right|}^2+o({\left|t-s\right|}^2),s,t\to t_0,$$

$$\mathbb{P}\left\{{\exists }_{t\in [0,T]}{Z}_2^c(t)>h(t,u)\right\}=\mathbb{P}\left\{\underset{t\in [0,T]}{\sup }\left(Z_2^c(t)+g(t)\right)>u\right\},c=1,2,$$

$${\mathcal{H}}_{\alpha }=\underset{S_1\to \infty }{\lim }\frac{1}{S_1}{\mathcal{H}}_{\alpha }[0,S_1],\text{with }{\mathcal{H}}_{\alpha }[-S_1,S_2]=\mathbb{E}\left\{\underset{t\in [-S_1,S_2]}{\sup }{e}^{\sqrt{2}{B}_{\alpha }(t)-|t{|}^{\alpha }}\right\}\in (0,\infty ),$$

$${\mathcal{P}}_{\alpha ,a}^f[-S_1,S_2]=\mathbb{E}\left\{\underset{t\in [-S_1,S_2]}{\sup }{e}^{\sqrt{2a}{B}_{\alpha }(t)-a|t{|}^{\alpha }-f(t)}\right\},$$

$${\mathcal{P}}_{\alpha ,a}^f[0,\infty )=\underset{S_2\to \infty }{\lim }{\mathcal{P}}_{\alpha ,a}^f[0,S_2],{\mathcal{P}}_{\alpha ,a}^f(-\infty ,\infty )=\underset{S_1,S_2\to \infty }{\lim }{\mathcal{P}}_{\alpha ,a}^f[-S_1,S_2].$$

$${\mathcal{P}}_{1,a}^{bt}[0,\infty )=1+\frac{a}{b}\text{and}{\mathcal{P}}_{2,a}^{b{t}^2}[0,\infty )=\frac{1}{2}\left(1+\sqrt{1+\frac{a}{b}}\right).$$

$$\mathbb{P}\left\{\underset{t\in [0,T]}{\sup }\left(Z^c(t)+g(t)\right)>u\right\},u\to \infty ,$$

$$\mathbb{P}\left\{\underset{t\in [0,T]}{\sup }{Z}^c(t)>u\right\}=\mathbb{P}\left\{\underset{t\in [0,T]}{\sup }Z(t)>u^{1/c}\right\}=\mathbb{P}\left\{\underset{(t,\mathbf{v})\in [0,T]\times {\mathcal{S}}_q}{\sup }Y(t,\mathbf{v})>u^{1/c}\right\},$$

$${\mathcal{S}}_q=\{\mathbf{v}\in {\mathbb{R}}^n:||\mathbf{v}|{|}_q=1\},$$

$$\mathbf{z}=(z_1,\dots ,z_n),z_i=\pm (d_i/d{)}^{2/(q-2)},d={\left[\sum _{i=1}^n{d}_i^{2p/(2-p)}\right]}^{(2-p)/2p},$$

$$\displaystyle\mathbb{P}\left\{Z^{c}(t_{0})>u\right\}\sim\Psi\left(\frac{u^{1/c}}{d}\right)\left\{\begin{array}[]{ll}2^{n}(2-p)^{(1-n)/2},&\ \text{if}\ \ p\in[1,2),\\ \frac{\sqrt{2\pi}2^{\frac{(2-m)}{2}}u^{\frac{m-1}{c}}}{\Gamma(m/2)}\prod_{i=m+1}^{n}(1-d_{i}^{2})^{-\frac{1}{2}},&\ \text{if}\ \ p=2,\\ 2m,&\ \text{if}\ \ p\in(2,\infty],\end{array}\right.$$

$$\sigma (t)=1-b|t-t_0{|}^{\beta }+o(|t-t_0{|}^{\beta }),t\to t_0$$

$$\displaystyle\mathbb{P}\left\{\sup_{t\in[0,T]}\left(Z^{c}(t)+g(t)\right)>u\right\}\sim\mathbb{P}\left\{Z^{c}(t_{0})>u\right\}\left\{\begin{array}[]{ll}u^{\frac{2}{\alpha^{*}}-\frac{2}{\beta^{*}}}a^{1/\alpha}d^{-2/\alpha}\mathcal{H}_{\alpha}\int_{Q}^{\infty}e^{-f(t)}dt,&\ \text{if}\ \ \alpha^{*}<\beta^{*},\\ \mathcal{P}_{\alpha,ad^{-2}}^{f(t)}[Q,\infty),&\ \text{if}\ \ \alpha^{*}=\beta^{*},\\ 1,&\ \text{if}\ \ \alpha^{*}>\beta^{*},\\ \end{array}\right.$$

$$\mathbb{P}\left\{\underset{t\in [0,T]}{\sup }{Z}^c(t)>u\right\},u\to \infty .$$

$$\mathbb{P}\left\{\underset{t\in [0,T]}{\sup }{Z}^c(t)>u\right\}\sim {\int }_0^T(a(t){)}^{\frac{1}{\alpha }}dt{d}^{-\frac{2}{\alpha }}{\mathcal{H}}_{\alpha }{u}^{\frac{2}{\alpha c}}\mathbb{P}\left\{Z^c(0)>u\right\},u\to \infty ,$$

$$\displaystyle\mathbb{P}\left\{\sup_{t\in[0,T]}\left(Z^{c}(t)+g(t)\right)>u\right\}\sim u^{(\frac{2}{\alpha^{*}}-\frac{2}{\beta^{*}})_{+}}\mathbb{P}\left\{Z^{c}(0)>u\right\}\left\{\begin{array}[]{ll}a^{\frac{1}{\alpha}}d^{-\frac{2}{\alpha}}\mathcal{H}_{\alpha}\int_{Q}^{\infty}e^{-f(t)}dt,&\text{if}\ \alpha^{*}<\beta^{*},\\ \mathcal{P}_{\alpha,ad^{-2}}^{f(t)}[Q,\infty),&\text{if}\ \alpha^{*}=\beta^{*},\\ 1,&\text{if}\ \alpha^{*}>\beta^{*},\end{array}\right.$$

$$\mathbb{P}\left\{\underset{t\in [0,T]}{\sup }\left(Z^c(t)+g(t)\right)>u\right\}\sim {\int }_0^T(a(t){)}^{\frac{1}{\alpha }}{e}^{\frac{g(t)}{2d^2}}dt{d}^{-\frac{2}{\alpha }}{\mathcal{H}}_{\alpha }{u}^{\frac{2}{{\alpha }^{\ast }}}\mathbb{P}\left\{Z^c(0)>u\right\},u\to \infty .$$

$$\mathbb{P}\left\{\underset{t\in [0,T]}{\sup }\left(Z^c(t)+g(t)\right)>u\right\}\sim {\int }_0^T(a(t){)}^{\frac{1}{\alpha }}dt{d}^{-\frac{2}{\alpha }}{\mathcal{H}}_{\alpha }{u}^{\frac{2}{{\alpha }^{\ast }}}\mathbb{P}\left\{Z^c(0)>u\right\},u\to \infty .$$

$$\displaystyle\mathbb{P}\left\{\sup_{t\in[0,1]}Z(t)-\sqrt{1-t}>u\right\}=\mathbb{P}\left\{Z(1)>u\right\}\left\{\begin{array}[]{ll}u^{\frac{2}{\alpha}-2}\left(\frac{1}{2d^{2}}\right)^{1/\alpha}\mathcal{H}_{\alpha}\int_{0}^{\infty}e^{-f(t)}dt,&\ \text{if}\ \ \alpha<1,\\ \mathcal{P}_{\alpha,d^{-2}/2}^{f(t)}[0,\infty),&\ \text{if}\ \ \alpha=1,\\ 1,&\ \text{if}\ \ \alpha>1,\\ \end{array}\right.$$

$$\mathbb{P}\left\{\underset{t\in [0,T]}{\sup }\left(Z_2^2(t)+g(t)\right)>u\right\}\sim {\int }_0^T(a(t){)}^{\frac{1}{\alpha }}{e}^{\frac{g(t)}{2}}dt{\mathcal{H}}_{\alpha }\frac{2^{1-m/2}{\prod }_{i=m+1}^n(1-d_i^2{)}^{-\frac{1}{2}}}{\Gamma (m/2)}{u}^{\frac{m-1}{2}+\frac{1}{\alpha }}{e}^{-\frac{u}{2}},u\to \infty .$$

$$U(t)=u+wt-X(t),t\ge 0,$$

$$X(t)=\sum _{i=1}^n{\left|d_i{B}_{\alpha }^i(t)\right|}^2,t\ge 0,$$

Full text

Extremes of $L^{p}$-norm of Vector-valued Gaussian processes with Trend

Long Bai

Long Bai, Department of Actuarial Science, University of Lausanne

UNIL-Dorigny, 1015 Lausanne, Switzerland

Long.Bai@unil.ch

Abstract: Let $\boldsymbol{X}(t)=(X_{1}(t),\ldots,X_{n}(t))$ be a Gaussian vector process and $g(t)$ be a continuous function. The asymptotics of distribution of $\left\|\boldsymbol{X}(t)\right\|_{p}$, the $L^{p}$ norm for Gaussian finite-dimensional vector, have been investigated in numerous literatures. In this contribution we are concerned with the exact tail asymptotics of $\left\|\boldsymbol{X}(t)\right\|^{c}_{p},\ c>0,$ with trend $g(t)$ over $[0,T]$. Both scenarios that $\boldsymbol{X}(t)$ is locally stationary and non-stationary are considered. Important examples include $\sum_{i=1}^{n}\left\lvert X_{i}(t)\right\rvert+g(t)$ and chi-square processes with trend, i.e., $\sum_{i=1}^{n}X_{i}^{2}(t)+g(t)$. These results are of interest in applications in engineering, insurance and statistics, etc.

Keywords: Tail asymptotics; $L^{p}$-norm; vector-valued Gaussian process; fractional Brownian motion; Pickands constant; Piterbarg constant.

AMS Classification: Primary 60G15; secondary 60G70

1. Introduction

In engineering sciences, extreme values of non-linear functions of multivariate Gaussian processes are of interest in dealing with the safety of structures, see [34] and the references therein. Probabilistic structural analysis to answer the question is: what is the probability that a certain mechanical (or other) structure will survive when it is subject to a random load. The load is then usually defined by some $n$-dimensional vector process $\boldsymbol{Y}(t)=(Y_{1}(t),\ldots,Y_{n}(t)),\ n\geq 1,\ t\in[0,T]$, and one seeks the probability that $\boldsymbol{Y}$ exceeds some more or less well-defined safe region, which is specific for the structure as

[TABLE]

where the time-dependent safety region $\boldsymbol{S}_{u}(t)$ is defined by

[TABLE]

with $h(t,u),\ t,u\geq 0$ some continuous function and $||\cdot||_{p}$, $p\in[1,\infty]$ the $L^{p}$ norm, i.e.,

[TABLE]

in the space $L^{p}_{n}=\{\boldsymbol{x}=(x_{1},\ldots,x_{n}):||\boldsymbol{x}||_{p}<\infty\}$.

Assume that $\boldsymbol{X}(t)=(X_{1}(t),\ldots,X_{n}(t))$ where $X_{i}^{\prime}s$ are independent copies of $X(t)$ a centered Gaussian process which has continuous trajectories, variance function $\sigma^{2}(\cdot)$ and correlation function $r(\cdot,\cdot)$ and

[TABLE]

In the framework of (1), set $\boldsymbol{Y}(t)=\boldsymbol{d}*\boldsymbol{X}(t):=(d_{1}X_{1}(t),\cdots,d_{n}X_{n}(t))$, then we can rewrite (1) as

[TABLE]

where

[TABLE]

and hereafter, we call $Z_{p}(t)$ the $L^{p}$ norm process.

When $p=2$, for a positive constant $c$, as in the convention $Z^{c}_{2}(t)=\left(Z_{2}(t)\right)^{c}$ is called the chi process when $c=1$ and the chi-square process when $c=2$.

Further, as the Gaussian processes, we can introduce the stationary, locally-stationary, and non-stationary $L^{p}$ norm processes according to the stationary, locally-stationary, and non-stationary properties of $X(t)$, respectively.

The investigate of

[TABLE]

is initiated by the studies of high excursions of envelope of a Gaussian process, see e.g., [9] and generalized in [33, 35, 36]. When $X(t)$ is stationary with $\sigma(t)\equiv 1$ and

[TABLE]

[2, 3] develop the Berman’s approach in [10] to obtain an asymptotic behavior of large deviation probabilities of the stationary chi-square processes.

Further, if there exists unique $t_{0}\in[0,T]$ satisfies $\sigma(t_{0})=\sup_{t\in[0,T]}\sigma(t)$ and

[TABLE]

where $b(t_{0})$ and $a(t_{0})$ are positive constants related to $t_{0}$, the tail asymptotic behavior of the non-stationary $Z^{2}_{2}(t)$ and $Z_{p}(t),p\in(1,2)\cup(2,\infty)$ are investigated in [41] and [25], respectively, under the application of the so-called ”double-sum method” in [42].

Some recent contributions are focused on more general scenarios of chi process and chi-square process with $h(t,u)=u-g(t)$, i.e.,

[TABLE]

where the continuous function $g(t)$ is generally considered as a trend or a drift.

When $X_{i},i=1,\ldots,n$ are non-stationary Gaussian processes, $Z_{2}(t)+g(t)$, the non-stationary chi processes with trend, and $Z^{2}_{2}(t)-wt^{\beta},w,\beta>0$, the non-stationary chi-square processes with trend, are studied in [26] and [37], respectively.

When $X_{i},i=1,\ldots,n$ are locally-stationary Gaussian processes, [38] obtains the extreme of the supremum of $Z_{2}^{2}(t)$ with trend, see, e.g., [11, 28] for more details about locally stationary Gaussian processes.

Considering both the locally stationary and non-stationary $L^{p}$ norm processes, the contribution of this paper concerns an exact asymptotic behavior of large deviation probabilities for $Z_{p}^{c}(t)+g(t)$ with $p\in[1,\infty]$, constant $c\in(0,\infty)$ and $g(t),\ t\in[0,T]$ a continuous function, which contains the aforementioned results.

Organisation of the rest of the paper: In Section 2, the notation and some preliminaries are given. Our main results are displayed in Section 3. Following in Section 4 are two applications related to insurance and statistics. Finally, we present the proofs in Section 5 and several lemmas in Section 6.

2. Notation and preliminaries

First we introduce some notation, starting with the well-known Pickands constant $\mathcal{H}_{\alpha}$ defined by

[TABLE]

where $S_{1},S_{2}\in[0,\infty)$ are constants and $B_{\alpha}(t),t\in\mathbb{R}$ is a standard fractional Brownian motion (fBm) with Hurst index $\alpha/2\in(0,1].$ Further, define for $f(\cdot)$ non-negative continuous function.

[TABLE]

and

[TABLE]

The exact values of $\mathcal{P}_{\alpha,a}^{bt^{\alpha}}[0,\infty)$ are known for $\alpha=1$ and $\alpha=2$, namely,

[TABLE]

See [39, 40, 15, 21, 14, 22, 16, 43, 18, 20, 13, 7] for various properties of $\mathcal{H}_{\alpha}$ and $\mathcal{P}_{\alpha,a}^{f}$.

Through this paper $\sim$ means asymptotic equivalence when the argument tends to [math] or $\infty$. We notice that $\Psi(\cdot)$ denotes the tail distribution function of an $\mathcal{N}(0,1)$ random variable and $\Psi(u)\sim\frac{1}{\sqrt{2\pi}u}e^{-\frac{u^{2}}{2}},u\rightarrow\infty$.

For the $L^{p}$ norm process $Z(t)$ in (3) and a continuous function $g(t),t\in\mathbb{R}$, we shall investigate the asymptotics of

[TABLE]

with $c>0$ a constant. As in [25, 41], for $p\in[1,\infty]$, using the duality property of $L^{p}$ norm we find

[TABLE]

where $Y(t,\boldsymbol{v})=\sum_{i=1}^{n}d_{i}v_{i}X_{i}(t)$ is a centered Gaussian field defined on cylinder $[0,T]\times\mathcal{S}_{q}$ with

[TABLE]

where $\frac{1}{p}+\frac{1}{q}=1$ if $q\in(1,\infty)$, $q=\infty$ if $p=1$ and $q=1$ if $p=\infty$.

Lemma 2.1**.**

*On $\mathcal{S}_{q}$ , $\sum_{i=1}^{n}d_{i}^{2}v_{i}^{2}$ attains its maximum $d^{2}$ at:

(i) for $p\in(2,\infty]$ at $2m$ points $\boldsymbol{v}_{+}^{i},\boldsymbol{v}_{-}^{i},i=1,\ldots,m,$ where $\boldsymbol{v}_{+}^{i}=(0,\ldots,0,1,0,\ldots,0)$ ($1$ stands at the i-th position), $\boldsymbol{v}_{-}^{i}=(0,\ldots,0,-1,0,\ldots,0)$ ($-1$ stands at the i-th position), $d=1$;

(ii) for $p=2$ at points on $\{\boldsymbol{v},\boldsymbol{v}\in\mathcal{S}_{q},v_{i}=0,m+1\leq i\leq n\}$, $d=1$;

(iii) for $p\in[1,2)$ at $2^{n}$ points $\boldsymbol{z}$, where*

[TABLE]

( we take all possible $2^{n}$ combinations of signs ”+” and ”-” ), where $z_{i}=\pm(d_{i}/d)^{0}=\pm 1$.

The proof can be easily carried out by method of Lagrangian multipliers or referring to [25] [Lemma 3.1].

Next by [31], we have the following lemma.

Lemma 2.2**.**

For the $L^{p}$ norm process $Z(t)$ in (3), if $\sigma^{2}(t_{0})=\text{Var}(X_{i}(t_{0}))=1,\ i=1,\ldots,n$ for some $t_{0}\in[0,\infty)$, then we have that as $u\rightarrow\infty$

[TABLE]

with the convention $\prod_{i=n+1}^{n}(1-d_{i}^{2})^{-\frac{1}{2}}=1$ and $d$ the same as in Lemma 2.1.

3. Extremes of $L^{p}$ norm processes with trend

In this section, recall that $Z(t)$ in (3) is the $L^{p}$ norm process and $X_{i}(t)$’s are independent copies of $X(t)$ with continuous trajectories, variance functions $\sigma^{2}(\cdot)$ and correlation functions $r(\cdot,\cdot)$.

3.1. Extremes of non-stationary $L^{p}$ norm processes with trend

As in [6], if $X(t)$ is non-stationary, we introduce the following assumptions:

• (i)

$\sigma(\cdot)$ attains its maximum on $[0,T]$ at the unique point $t_{0}\in[0,T]$ and

[TABLE]

for some positive constants $b,\beta$.

• (ii)

$r(s,t)=1-a|t-s|^{\alpha}+o(|t-s|^{\alpha}),\ s,t\rightarrow t_{0}$ for some constants $a>0$ and $\alpha\in(0,2].$

Further, we introduce a bounded measurable trend function $g(t)$ which satisfies

• (iii)

$g(t)\sim-w|t-t_{0}|^{\gamma},\ \ \ \ t\rightarrow t_{0}$ for some constants $\gamma>0,w\geq 0$.

Theorem 3.1**.**

If assumptions (i)-(iii) are satisfied, then for $\boldsymbol{d}$ in (2) and $d$ in Lemma 2.1, we have as $u\rightarrow\infty$

[TABLE]

where $\alpha^{*}=\alpha c$, $\beta^{*}=\min(\beta c,\frac{2\gamma c}{2-c})\mathbb{I}_{\{c<2\}}+\beta c\mathbb{I}_{\{c\geq 2\}}$, $f(t)=\frac{b|t|^{\beta}}{d^{2}}\mathbb{I}_{\{\beta^{*}=\beta c\}}+\frac{w|t|^{\gamma}}{cd^{2}}\mathbb{I}_{\{\beta^{*}=\frac{2\gamma c}{2-c}\}}$, and $Q=-\infty$ if $t_{0}\in(0,T)$, $Q=0$ if $t_{0}\in\{0,T\}$.

Remarks 3.2**.**

i) In Theorem 3.1, if we assume that $w=0$, we get the extremes of centered non-stationary $L^{p}$ norm processes i.e.,

[TABLE]

ii) Following the similar arguments as in the proof of Theorem 3.1, the result in Theorem 3.1 still holds for $w<0,\gamma\geq\frac{(2-c)\beta}{2}$ if $c<2$ and $w<0,\gamma>0$ if $c\geq 2$.

3.2. Extremes of locally stationary $L^{p}$ norm processes with trend

If $X(t)$ is locally stationary, as in [6], we shall suppose that:

• (iv)

$r(s,t)=1-a(t)|t-s|^{\alpha}+o(|t-s|^{\alpha}),\ \ \left\lvert t-s\right\rvert\rightarrow 0,\ \alpha\in(0,2],$ where $a(t)$ are positive continuous function on $[0,T]$.

• (v)

$r(s,t)<1,\ \ \forall\ s,t\in[0,T]\quad\hbox{and}\quad s\neq t.$

Before giving the scenarios with trend, we consider the extremes of the centered locally stationary $L^{p}$ norm processes.

Theorem 3.3**.**

Assume that $\sigma(t)\equiv 1$, i.e., unit variance and covariance function $r(\cdot,\cdot)$ satisfies assumptions (iv) and (v). Then we have for $c>0$

[TABLE]

where $d$ is the same as in Lemma 2.1.

Theorem 3.4**.**

*Assume that $\sigma(t)\equiv 1$, i.e., unit variance and correlation function $r(\cdot,\cdot)$ satisfies assumptions (iv) and (v). Assume that $g(t)\ t\in[0,T]$ is a continuous function which attains its maximum at a unique point $t_{0}\in[0,S]$ satisfying assumption (iii) for some constants $w,\gamma>0$. Further, set $\alpha^{*}=\alpha c,\ \beta^{*}=\frac{2\gamma c}{2-c}\mathbb{I}_{\{c<2\}}$ and $f(t)=\frac{w\left\lvert t\right\rvert^{\gamma}}{cd^{2}}$ and $d$ is the same as in Lemma 2.1.

If $c\in(0,2)$, then we have as $u\rightarrow\infty$*

[TABLE]

*where $a=a(t_{0})$ and $Q=-\infty$ if $t_{0}\in(0,T)$, $Q=0$ if $t_{0}\in\{0,T\}$.

If $c=2$, then we have*

[TABLE]

If $c>2$, then we have

[TABLE]

Remark 3.5**.**

By the proof, we notice that for the case $c=2$ in Theorem 3.4, the result always holds for any continuous function $g(t),t\in[0,1]$. When $c>0$, the result holds for any bounded function $g(t),t\in[0,1]$.

Example 3.6**.**

For $Z(t)$ in (3) with $X_{i}(t)=B^{i}_{\alpha}(t),i=1,\ldots,n$ the independent fractional Brownian motions, we have as $u\rightarrow\infty$

[TABLE]

where $f(t)=\frac{\alpha}{2d^{2}}t+\frac{1}{d^{2}}t^{\frac{1}{2}}$ and $d$ is the same as in Lemma 2.1

Following example is a special case of Theorem 3.4, which is corresponded with [38] [Theorem 2.1].

Example 3.7**.**

In Theorem 3.4, assume that $p=2$, $c=2$ and $g(t),t\in[0,T]$ is a continuous function, then we have

[TABLE]

4. Applications

4.1. Ruin probability of a risk model

In theoretical insurance modelling a surplus process $U(t)$ can be defined by

[TABLE]

see [23], where $u\geq 0$ is the initial reserve, $w>0$ is the rate of premium and the stochastic process $X(t),t\geq 0$ denotes the aggregate claims process. See [45, 17, 27, 8, 6, 5] for more studies on related risk models. Here we investigate

[TABLE]

where $\boldsymbol{d}=(d_{1},\ldots,d_{n})$ is the same as in (2) and $B_{\alpha}^{i}(t)$ are independent fractional Brownian motions. $X(t)$ can be considered as the sum of $n$ independent claims or payments until time $t$. The corresponding ruin probability over a finite-time horizon $[0,1]$ is defined as

[TABLE]

We present next approximation of this ruin probability.

Proposition 4.1**.**

We have as $u\rightarrow\infty$

[TABLE]

Besides in risk modelling, the $L^{p}$ norm processes, especially the chi-square processes, are also widely utilized in hypothesis testing, see [12, 44] and the reference there. Next we give an example.

4.2. The Ornstein-Uhlenbeck chi-square process in Quantitative Trait Locus detection

A Quantitative Trait Locus (QTL) denotes a gene with quantitative effect on a trait. The method used by most of geneticists in order to detect a QTL on a chromosome, is the Interval Mapping proposed by [32]. Using the Haldane distance and modelling in [29], each chromosome is represented by a segment $[0,T]$. The distance on $[0,T]$ is called the genetic distance. At each location $t\in[0,T]$, using the ”genome information” brought by genetic markers, a likelihood ratio test (LRT) is performed, testing the presence of a QTL at this position. [4] prove that when the number of genetic markers and the number of progenies tends to infinity, the limiting process of the LRT process is an Ornstein-Uhlenbeck chi-square process under the null hypothesis of the absence of QTL on the interval $[0,T]$. In order to take decision about the presence of a QTL on $[0,T]$, we need to calculate the critical value for the supremum of an Ornstein-Uhlenbeck chi-square process, i.e.,

[TABLE]

where the Ornstein-Uhlenbeck chi-square process $S(t)$ is

[TABLE]

and $V_{i}(t),\ 1\leq i\leq n$ are independent identically stationary Gaussian processes with covariance function given by

[TABLE]

Proposition 4.2**.**

We have as $u\rightarrow\infty$

[TABLE]

5. Proofs

During the following proofs, $\mathbb{Q}_{i},i\in\mathbb{N}$ are some positive constants which can be different from line by line and for interval $\Delta_{1},\Delta_{2}\subseteq[0,\infty)$ we denote

[TABLE]

and

[TABLE]

Proof of Theorem 3.1: We first present the proof for the case $t_{0}=0$.

Set $\beta^{*}=\min(\beta c,\frac{2\gamma c}{2-c})\mathbb{I}_{\{c<2\}}+\beta c\mathbb{I}_{\{c\geq 2\}},\ \alpha^{*}=\alpha c$, $\delta(u)=\frac{(\ln u)^{\rho}}{u^{2/\beta^{*}}}$ with $\rho>\max\left(\frac{1}{\beta},\frac{1}{\gamma}\right)$ and for $u$ large enough

[TABLE]

with $\mathcal{S}_{q}$ the same as in (5) which is a centered Gaussian field.

We have for some small $\theta>0$ and $u$ large enough

[TABLE]

We first give the upper bounds of $\mathcal{L}_{u}([\delta(u),\theta])$ and $\mathcal{L}_{u}([\theta,T])$.

Set $\sigma_{\theta}:=\sup_{t\in[\theta,T]}\sigma(t)<1$ and $g_{m}=\sup_{t\in[0,T]}g(t)<\infty$. Then by Borell inequality as in [1] and Lemma 2.2 for large $u$

[TABLE]

where $\mathbb{Q}_{1}:=\mathbb{E}\left\{\sup_{(t,\boldsymbol{v})\in[\theta,T]\times\mathcal{S}_{q}}Y(t,\boldsymbol{v})\right\}<\infty$ and

[TABLE]

By assumptions (i) and (iii), we know that for some $\varepsilon_{1}\in(0,1)$

[TABLE]

hold for $t\in[0,\theta]$ when $\theta$ small enough, then

[TABLE]

Denote $\overline{\boldsymbol{X}}(t)=(\overline{X}_{1}(t),\ldots,\overline{X}_{n}(t))$ with $\overline{X}_{i}(t)=\frac{X_{i}(t)}{\sigma(t)},\ t\in[0,\theta]$. By assumption (ii), we have that

[TABLE]

holds for $s,t\in[0,\theta]$ and $\boldsymbol{v},\boldsymbol{v}^{\prime}\in\mathcal{S}_{q}$. Thus it follows from [42] [Theorem 8.1], (5) and Lemma 2.2 that

[TABLE]

Thus by (12), (5) and the fact that $\mathcal{L}_{u}([0,\delta(u)])\geq\mathbb{P}\left\{Z^{c}(0)>u\right\}$ for $u$ positive, we have

[TABLE]

which combined with (11) imply

[TABLE]

Now we focus on the asymptotic of $\mathcal{L}_{u}([0,\delta(u)])$, as $u\rightarrow\infty$.

Denote for any $\lambda>0$ and some $\varepsilon\in(0,1)$

[TABLE]

Case 1: $\beta^{*}>\alpha^{*}$. For $u$ large enough, we have

[TABLE]

where

[TABLE]

In the view of Lemma 6.2 and (13), we have that for some $\epsilon\in[0,1)$,

[TABLE]

as $u\rightarrow\infty,\ \lambda\rightarrow\infty,\ \varepsilon\rightarrow 0,\ \epsilon\rightarrow 0$ where $f(t)=\frac{b|t|^{\beta}}{d^{2}}\mathbb{I}_{\{\beta^{*}=\beta c\}}+\frac{w}{cd^{2}}|t|^{\gamma}\mathbb{I}_{\{\beta^{*}=\frac{2\gamma c}{2-c}\}}$. Similarly, we derive that

[TABLE]

Moreover,

[TABLE]

where $\widehat{\mathcal{G}}_{u,-\varepsilon}(k)=\min(\mathcal{G}_{u,-\varepsilon}(k),\mathcal{G}_{u,-\varepsilon}(k+1))$. By Lemma 6.3, we have

[TABLE]

Combing (20)-(23) with (19), we obtain

[TABLE]

Case 2: $\beta^{*}=\alpha^{*}$. We consider that for $u$ large enough,

[TABLE]

Using (33) of Lemma 6.2 with $u$ replaced by $u^{1/c}$ and (14), we have that

[TABLE]

Similarly,

[TABLE]

Moreover, by Lemma 6.2,

[TABLE]

Inserting (26), (27), and (28) into (25), we have

[TABLE]

Case 3: $\beta^{*}<\alpha^{*}$. Obviously,

[TABLE]

For any $\varepsilon_{2}\in(0,1)$, $[0,\delta(u)]\subseteq[0,u^{-2/\alpha^{*}}\varepsilon_{2}]$ when $u$ large enough. By Lemma 6.2 and the fact that $\sup_{t\in[0,\delta(u)]}g(t)\leq 0$, we obtain

[TABLE]

Together with (30), we get

[TABLE]

Consequently, we have the results according to (18), (24), (29) and (31).

For $t_{0}\in(0,T)$ and $t_{0}=T$, we just need to replace $[0,\delta(u)]$ as $[t_{0}-\delta(u),t_{0}+\delta(u)]$ and $[T-\delta(u),T]$. Thus we complete the proof. $\Box$

Proof of Theorem 3.3: For any $\theta>0$ and $\lambda>0$, set $\alpha^{*}=\alpha c$

[TABLE]

We have

[TABLE]

where

[TABLE]

with

[TABLE]

By Lemma 6.2

[TABLE]

Similarly,

[TABLE]

Further, by Lemma 6.2

[TABLE]

Similarly, by Lemma 6.2

[TABLE]

For any $\theta>0$

[TABLE]

for $(s,t)\in J^{k_{1}}_{l_{1}}(u)\times J^{k_{2}}_{l_{2}}(u),(j_{1},k_{1},j_{2},k_{2})\in\mathcal{L}_{3}$ where $\delta(\theta)>0$ is related to $\theta$. Then by Lemma 6.1

[TABLE]

where $\mathbb{Q}_{3}$ is a large constant. Finally by Lemma 6.3 for $u$ large enough and $\theta$ small enough

[TABLE]

Thus the claim follows. $\Box$

Proof of Theorem 3.4: Through this proof, denote $\mathcal{L}_{u}(\Delta_{1})$ and $\mathcal{L}_{u}(\Delta_{1},\Delta_{2})$ the same as in the proof of Theorem 3.1.

When $c\in(0,2)$, in the proof of Theorem 3.1, if we take $\beta^{*}=\frac{2\gamma c}{2-c}$ and $f(t)=\frac{w\left\lvert t\right\rvert^{\gamma}}{cd^{2}}$, then all argumentations still hold and the results follow.

When $c=2$, for any constant $\theta>0$, we define

[TABLE]

and

[TABLE]

Then

[TABLE]

where

[TABLE]

and by Theorem 3.3

[TABLE]

Similarly,

[TABLE]

Further, we have

[TABLE]

where $\widetilde{M}_{1}(k)=\max(M_{1}(k),M_{1}(k+1))$.

Then for $g_{m}=\sup_{t\in[0,T]}g(t)$ by Lemma 6.1

[TABLE]

Thus, we have

[TABLE]

When $c\in(2,\infty)$, set $M_{1}=\inf_{t\in[0,T]}g(t)$ and $M_{2}=\sup_{t\in[0,T]}g(t)$. Since $g(t)$ is a continuous function, we have $-\infty<M_{1}\leq M_{2}<\infty$. Further, since when $c\in(2,\infty)$,

[TABLE]

holds for any $\mathbb{Q}_{2}>0$. Hence, by Theorem 3.3

[TABLE]

and

[TABLE]

The result follows. $\Box$

6. Appendix

In this section, we give several lemmas which are used in the proofs of the theorems.

Lemma 6.1**.**

let $\boldsymbol{X}(t)=(X_{1}(t)\ldots,X_{n}(t)),\ t\in[0,T],n\geq 1$ be an centered ${\mathbb{R}}^{n}$-valued vector process with independent marginals, which have continuous samples, unit variances and correlation functions satisfying assumption (v). Then for $0<t_{1}<t_{2}<t_{3}<\infty$ and $u$ large enough

[TABLE]

where $D,\delta$ are some constant.

Proof of Lemma 6.1: By assumption (v) and the continuity of $r(t)$, for some $\delta>0$ we have

[TABLE]

holds for any $(s,t)\in[0,t_{1}]\times[t_{2},t_{3}]$. Set $\widetilde{Y}(t,\boldsymbol{v},s,\boldsymbol{w})=\sum_{i=1}^{n}X_{i}(t)d_{i}v_{i}+\sum_{i=1}^{n}X_{i}(s)d_{i}w_{i}$ where $\boldsymbol{v},\boldsymbol{w}\in\mathcal{S}_{q}$ with $\mathcal{S}_{q}=\{\boldsymbol{v}\in\mathbb{R}^{n}:||\boldsymbol{v}||_{q}=1\}$. Since $\widetilde{Y}(t,\boldsymbol{v},s,\boldsymbol{w})$ is a center Gaussian fields, we have further

[TABLE]

for any $(t,\boldsymbol{v},s,\boldsymbol{w})\in[0,t_{1}]\times\mathcal{S}_{q}\times[t_{2},t_{3}]\times\mathcal{S}_{q}$. By Borell inequality,

[TABLE]

where $D$ is some constant such that

[TABLE]

hence the claim follows. $\Box$

Lemma 6.2**.**

let $\boldsymbol{X}(t)=(X_{1}(t)\ldots,X_{n}(t)),\ t\in\mathbb{R},n\geq 1$ be an centered ${\mathbb{R}}^{n}$-valued vector process with independent marginals, which have continuous samples, unit variances and correlation functions satisfying assumption (iv). Set $a:=a(t_{0}),\ t_{0}\in\mathbb{R},$ and $K_{u}$ a family of index sets and $u_{k}$ satisfying that

[TABLE]

If $f(t)$ is a nonnegative continuous function with $f(0)=0,f(t)>0,t\neq 0$ and $\boldsymbol{d}$ is the same as in (2), then we have that for some constants $S_{1},S_{2}\geq 0$ and $\max(S_{1},S_{2})>0$

[TABLE]

and

[TABLE]

If $\lim_{u\rightarrow\infty}\sup_{k\in K_{u}}\left\lvert ku^{-2/\alpha}\right\rvert\leq\theta$ for some small enough $\theta\geq 0$, we have for some constant $S>0$

[TABLE]

*where $\varepsilon_{\theta}\rightarrow 0$, as $\theta\rightarrow 0$.

Specially, if $\theta=0$, we have*

[TABLE]

and

[TABLE]

Proof of Lemma 6.2: Step 1: First we give the proof of (33). When $p=1$, set $\mathcal{W}=\{\boldsymbol{w}=(w_{1},\cdots,w_{n}):w_{i}=\pm 1,\ i=1,\cdots,n\}$. Then we have

[TABLE]

By [6] [Lemma 4.1], we have

[TABLE]

Since for any $\boldsymbol{w}\neq\boldsymbol{w}^{\prime}$

[TABLE]

then by Borell inequality, we have

[TABLE]

Then (33) with $p=1$ is follow.

When $p\in(1,\infty]$, set $Y(t,\boldsymbol{v})=\sum_{i=1}^{n}d_{i}v_{i}X_{i}(t),(t,\boldsymbol{v})\in\mathbb{R}\times\mathcal{S}_{q}$ which is a centered Gaussian field.

Then we have

[TABLE]

Set $\mathcal{S}_{q}^{\delta}=\left\{\boldsymbol{v}\in\mathcal{S}_{q}:d^{2}-\sum_{i=1}^{n}d_{i}^{2}v_{i}^{2}\leq\delta\right\},\delta>0$. Next we prove that as $u\rightarrow\infty$

[TABLE]

Since

[TABLE]

we just need to show as $u\rightarrow\infty$

[TABLE]

In fact, since

[TABLE]

by Borell inequality, we have

[TABLE]

where $\mathbb{Q}_{1}:=\mathbb{E}\left\{\sup_{(t,\boldsymbol{v})\in[-S_{1},S_{2}]\times(\mathcal{S}_{q}\setminus\mathcal{S}^{\delta}_{q})}Y(u^{-2/\alpha}t+t_{0},\boldsymbol{v})\right\}<\infty$.

When $p\in(1,2)\cup(2,\infty]$, by Lemma 2.1, we know $\sigma_{1}^{2}(t,\boldsymbol{v}):=\text{Var}\left(\frac{Y(u^{-2/\alpha}t+t_{0},\boldsymbol{v})}{1+u^{-2}f(t)}\right)$ attains the maximum over $[-S_{1},S_{2}]\times\mathcal{S}_{q}$ at several discrete points, so we can choose $\delta$ small enough such that $\mathcal{D}^{i}_{\delta}=[-S_{1},S_{2}]\times\mathcal{S}^{\delta}_{q}(i)$ with $\mathcal{S}^{\delta}_{q}(i)$ the union of non-overlapping compact neighborhoods of $\boldsymbol{v}_{+}^{i},\boldsymbol{v}_{-}^{i}$ or $\boldsymbol{z}$ in Lemma 2.1. Then as mentioned in [42] or [24][Lemma 2.1]

[TABLE]

where $M$ is the number of the maximum point of $\sigma_{1}^{2}(t,\boldsymbol{v})$.

Case 1) $p\in(1,2)$ and $M=2^{n}$. It is enough to find the asymptotics of single term in (36), for instance, for a point $(0,\boldsymbol{z}),\ z_{i}=(d_{i}/d)^{2/q-2}$. In a neighborhood $\mathcal{S}^{\delta}_{q}(1)$ of $\boldsymbol{z}$, we have

[TABLE]

hence the fields $\frac{Y(u^{-2/\alpha}t+t_{0},\boldsymbol{v})}{1+u^{-2}f(t)}$ can be represented as

[TABLE]

which is defined in $[-S_{1},S_{2}]\times\widetilde{\mathcal{S}}^{\delta}_{q}(1)$ where

[TABLE]

is a small neighborhood of $\widetilde{\boldsymbol{z}}=\left(z_{1},\cdots,z_{n-1}\right)$. On $[-S_{1},S_{2}]\times\widetilde{\mathcal{S}}^{\delta}_{q}(1)$, the variance

[TABLE]

of $Y_{1}(u^{-2/\alpha}t+t_{0},\widetilde{\boldsymbol{v}})$ attains its maximum $d^{2}$ at $(0,\widetilde{\boldsymbol{z}})$ where $\widetilde{\boldsymbol{z}}$ is a interior point of a set $\widetilde{\mathcal{S}}^{\delta}_{q}(1)$. We can write the following Taylor expansion for $\sigma_{1}(t,\widetilde{\boldsymbol{v}})$

[TABLE]

where $\Lambda=(\lambda_{i,j})_{i,j=1,\cdots,n-1}$ is a non-negative define matrix with elements

[TABLE]

We have the following expansion for the correlation function $r_{1}(t,\widetilde{\boldsymbol{v}},s,\widetilde{\boldsymbol{v}}^{\prime})$ of $Y_{1}(u^{-2/\alpha}t+t_{0},\widetilde{\boldsymbol{v}})$

[TABLE]

There exists a non-singular matrix $Q$ such that $Q\Lambda Q^{T}$ is diagonal, and set the diagonal is $(c_{1},\cdots,c_{n-1})$. Then

[TABLE]

and

[TABLE]

Then set $Y_{2}(u^{-2/\alpha}t+t_{0},\widetilde{\boldsymbol{v}})=Y_{1}(u^{-2/\alpha}t+t_{0},Q\widetilde{\boldsymbol{v}})$, defined on a set $[-S_{1},S_{2}]\times(Q^{-1}\widetilde{\mathcal{S}}^{\delta}_{q}(1))$. We know that the point $Q\widetilde{\boldsymbol{z}}$ is a interior point of $Q^{-1}\widetilde{\mathcal{S}}^{\delta}_{q}(1)$. Then the proof follows by similar arguments as in the proof of [42] [Theorem 8.2]. Consequently, we get

[TABLE]

where we use the fact in [30] that

[TABLE]

and

[TABLE]

Case 2) $p\in(2,\infty]$ and $M=2m$. Again we need to find the asymptotics of single term in (36), to wish namely for a maximum point $(0,\boldsymbol{v}^{1}_{+})$, $\boldsymbol{v}^{1}_{+}=(1,0,\cdots,0)$ of variance $\sigma_{1}^{2}(t,\boldsymbol{v})$. hence the fields $\frac{Y(u^{-2/\alpha}t+t_{0},\boldsymbol{v})}{1+u^{-2}f(t)}$ can be represented as

[TABLE]

which is defined in $[-S_{1},S_{2}]\times\widetilde{\mathcal{S}}^{\delta}_{q}(1)$ where

[TABLE]

is a small neighborhood of $\widetilde{\boldsymbol{0}}:=(0,\cdots,0)\in\mathbb{R}^{n-1}$. On $[-S_{1},S_{2}]\times\widetilde{\mathcal{S}}^{\delta}_{q}(1)$, the variance

[TABLE]

of $Y_{1}(u^{-2/\alpha}t+t_{0},\widetilde{\boldsymbol{v}})$ attains its maximum $1$ at $(0,\widetilde{\boldsymbol{0}})$ where $\widetilde{\boldsymbol{0}}$ is a interior point of a set $\widetilde{\mathcal{S}}^{\delta}_{q}(1)$. We can write the following Taylor expansion for $\sigma_{1}(t,\widetilde{\boldsymbol{v}})$

[TABLE]

and the following expansion for the correlation function $r_{1}(t,\widetilde{\boldsymbol{v}},s,\widetilde{\boldsymbol{v}}^{\prime})$ of $Y_{1}(u^{-2/\alpha}t+t_{0},\widetilde{\boldsymbol{v}})$

[TABLE]

Then the proof again follows by similar arguments as in the proof of [42] [Theorem 8.2]. Consequently, we get

[TABLE]

and

[TABLE]

Case 3) $p=2$. By Lemma 2.1, we know that $\sigma_{1}^{2}(t,\boldsymbol{v})$ attains its maximum (equal to 1) over $[-S_{1},S_{2}]\times\mathcal{S}_{q}$ only at points on $\{(0,\boldsymbol{v}),\boldsymbol{v}\in\mathcal{S}_{q},v_{i}=0,m+1\leq i\leq n\}$. The fields $\frac{Y(u^{-2/\alpha}t+t_{0},\boldsymbol{v})}{1+u^{-2}f(t)}$ again can be represented as

[TABLE]

which is defined in $[-S_{1},S_{2}]\times\widetilde{\mathcal{S}}_{q}$ where

[TABLE]

On $[-S_{1},S_{2}]\times\widetilde{\mathcal{S}}_{q}$, the variance

[TABLE]

of $Y_{1}(u^{-2/\alpha}t+t_{0},\widetilde{\boldsymbol{v}})$ attains its maximum $1$ at $\left\{(0,\widetilde{\boldsymbol{v}}),\widetilde{\boldsymbol{v}}\in\widetilde{\mathcal{S}}_{q},v_{i}=0,m+1\leq i\leq n\right\}$. Furthermore, following the arguments as in [41] we conclude that $\sigma_{1}(t,\widetilde{\boldsymbol{v}})$ and the correlation function $r_{1}(t,\boldsymbol{v},s,\widetilde{\boldsymbol{v}})$ of $Y_{1}(u^{-2/\alpha}t+t_{0},\widetilde{\boldsymbol{v}})$ have the following asymptotic expansions:

[TABLE]

and the following expansion for the correlation function $r_{1}(t,\widetilde{\boldsymbol{v}},s,\widetilde{\boldsymbol{v}}^{\prime})$ of $Y_{1}(u^{-2/\alpha}t+t_{0},\widetilde{\boldsymbol{v}})$

[TABLE]

Then the proof follows by similar arguments as in the proof of [37] [Theorem 6.1] with the case $\mu=\nu$. Consequently, we get

[TABLE]

Step 2: Next we proceed to the proof of (34). Setting $a_{u,k}=(a(ku^{-2/\alpha}S+t_{0}))^{1/\alpha}$, then for any $k\in K_{u}$ with $\lim_{u\rightarrow\infty}\sup_{k\in K_{u}}\left\lvert ku^{-2/\alpha}\right\rvert\leq\theta$ and $t\in[-S_{1},S_{2}]$ when $u$ large enough

[TABLE]

holds for some $\varepsilon_{\theta}\in(0,a)$.

Then we have

[TABLE]

and

[TABLE]

We notice that by assumption (iv)

[TABLE]

For $\Pi^{+}(u)$ and $\Pi^{-}(u)$, when $p=1$, (34) follows with the same arguments as in Step 1.

When $p\in(1,\infty]$, for $\Pi^{+}(u)$ and $\Pi^{-}(u)$ we use the similar arguments as in in Step 1 with $Y_{1}(u^{-2/\alpha}(a^{-1}_{u,k}t+kS)+t_{0},\widetilde{\boldsymbol{v}})=Y(u^{-2/\alpha}(a^{-1}_{u,k}t+kS)+t_{0},\widetilde{\boldsymbol{v}})$.

When $p\in(1,2)$,

[TABLE]

and

[TABLE]

When $p\in(2,\infty]$,

[TABLE]

and

[TABLE]

When $p=2$,

[TABLE]

and

[TABLE]

We get that as $u\rightarrow\infty$

[TABLE]

Thus (34) follows.

Further, if letting $\theta\rightarrow 0$ in (34), we get (35).

$\Box$

Lemma 6.3**.**

Assume that Gaussian vector process $\boldsymbol{X}(t)$ with independent marginals which have unit variances, correlation functions $r(t)$ is the same as in Lemma 6.2. Further, set $K_{u}$ a family of index sets and $u_{k}$ satisfying that

[TABLE]

Let $\varepsilon_{0}$ be such that for all $s,t\in[t_{0}-\varepsilon_{0},t_{0}+\varepsilon_{0}]$,

[TABLE]

Then we can find a constant $\mathbb{C}$ such that for all $S>0$ and $T_{2}-T_{1}>S$,

[TABLE]

where $\mathcal{A}_{i}(u_{k})=\{\sup_{t\in[T_{i},T_{i}+S]}Z(u^{-2/\alpha}(t+kS)+t_{0})>u_{k}\},\ i=1,2$, and

[TABLE]

Proof of Lemma 6.3: Through this proof, $\mathbb{C}_{i},i\in\mathbb{N}$ are some positive constant.

When $p=1$, set $\mathcal{W}=\{\boldsymbol{w}=(w_{1},\cdots,w_{n}):w_{i}=\pm 1,\ i=1,\cdots,n\}$. We have by [19][Theorem 3.1] for $u$ large enough

[TABLE]

When $p\in(1,\infty]$, set $Y_{u}(t,\boldsymbol{v})=\sum_{i=1}^{n}d_{i}v_{i}X_{i}(u^{-2/\alpha^{*}}(t+kS)+t_{0}),(t,\boldsymbol{v})\in\mathbb{R}\times\mathcal{S}_{q}$ which is a centered Gaussian field and $\mathcal{S}_{q}^{\delta}=\{\boldsymbol{v}\in\mathcal{S}_{q}:d^{2}-\sum_{i=1}^{n}d_{i}^{2}v_{i}^{2}\leq\delta\},\delta>0$.

Below for $\Delta_{1},\Delta_{2}\subseteq\mathbb{R}^{n+1}$, denote

[TABLE]

We have

[TABLE]

and

[TABLE]

as $u\rightarrow\infty$ where the last second inequality follows from Borell inequality and the fact that

[TABLE]

Similarly, we have

[TABLE]

Then we just need to focus on

[TABLE]

We split $\mathcal{S}^{\delta}_{q}$ into sets of small diameters $\{\partial\mathcal{S}_{i},0\leq i\leq\mathcal{N}^{*}\}$, where

[TABLE]

Further, we see that $\Pi(u)\leq\Pi_{1}(u)+\Pi_{2}(u)$ with

[TABLE]

where ${\partial\mathcal{S}_{i}\cap\partial\mathcal{S}_{l}\neq\emptyset}$ means $\partial\mathcal{S}_{i},\partial\mathcal{S}_{l}$ are identical or adjacent, and ${\partial\mathcal{S}_{i}\cap\partial\mathcal{S}_{l}=\emptyset}$ means $\partial\mathcal{S}_{i},\partial\mathcal{S}_{l}$ are neither identical nor adjacent. Denote the distance of two set $\boldsymbol{A},\boldsymbol{B}\in\mathbb{R}^{n}$ as

[TABLE]

if $\partial\mathcal{S}_{i}\cap\partial\mathcal{S}_{l}=\emptyset$, then there exists some small positive constant $\rho_{0}$ (independent of $i,l$) such that $\rho(\partial\mathcal{S}_{i},\partial\mathcal{S}_{l})>\rho_{0}$. Next we estimate $\Pi_{1}(u)$. For any $u\geq 0$

[TABLE]

where $Z_{u}(t,\boldsymbol{v},s,\boldsymbol{w})=Y_{u}(t,\boldsymbol{v})+Y_{u}(s,\boldsymbol{w}),\ t,s\geq 0,\boldsymbol{v},\boldsymbol{w}\in\mathbb{R}^{n}$.

When $u$ is sufficiently large for $(t,s)\in[T_{1},T_{1}+S]\times[T_{2},T_{2}+S],\boldsymbol{v}\in\partial\mathcal{S}_{i}\subset[-2,2]^{n},\boldsymbol{w}\in\partial\mathcal{S}_{i}\subset[-2,2]^{n}$, with $\rho(\partial\mathcal{S}_{i},\partial\mathcal{S}_{l})>\rho_{0}$ we have

[TABLE]

for some $\delta_{0}>0$. Therefore, it follows from the Borell inequality that

[TABLE]

with

[TABLE]

Now we consider $\Pi_{2}(u)$. Similar to the argumentation as in Step1 of the proof of Lemma 6.2. we set $\widetilde{Y}_{u}(t,\widetilde{\boldsymbol{v}})=Y_{u}(t,Q\widetilde{\boldsymbol{v}})$ and $\widetilde{Z}_{u}(t,\widetilde{\boldsymbol{v}},s,\widetilde{\boldsymbol{w}})=\widetilde{Y}_{u}(t,\widetilde{\boldsymbol{v}})+\widetilde{Y}_{u}(s,\widetilde{\boldsymbol{w}})$ with $\widetilde{\boldsymbol{v}},\widetilde{\boldsymbol{w}}\in\mathbb{R}^{n-1}$. Since for $(t,s)\in[T_{1},T_{1}+S]\times[T_{2},T_{2}+S],\widetilde{\boldsymbol{v}}\in[-2,2]^{n-1},\widetilde{\boldsymbol{w}}\in[-2,2]^{n-1}$, we have

[TABLE]

Set

[TABLE]

Borrowing the arguments of the proof in [42] [Lemma 6.3] we show that

[TABLE]

Moreover, since when $p\in(1,2)$,

[TABLE]

When $p\in(2,\infty)$,

[TABLE]

When $p=2$,

[TABLE]

Then we have

[TABLE]

Therefore

[TABLE]

Set $\zeta(t,s,\widetilde{\boldsymbol{v}},\widetilde{\boldsymbol{w}}),t,s\geq 0,\boldsymbol{v},\boldsymbol{w}\in\mathbb{R}^{n-1}$ is a stationary Gaussian field with unit variance and correlation function

[TABLE]

Then

[TABLE]

Then following the similar argumentation as in [26], we have

[TABLE]

where $M=0$ when $p\in(1,2)\cup(2,\infty]$ and $M=m$ when $p=2$. Thus we have

[TABLE]

Thus we complete the proof.

$\Box$

Proof of Eaxmple 3.6: Note that the variance function $\sigma^{2}(t)$ of $B_{\alpha}(t)$ attain its maximum over $[0,1]$ at $t=1$ and

[TABLE]

For $g(t)=-(1-t)^{1/2},\ t\in[0,1]$, by Theorem 3.1 with $c=1$ we get the result.

Proof of Proposition 4.1: Note that the variance function $\sigma^{2}(t)$ of $B_{\alpha}(t)$ attains its maximum over $[0,1]$ at $t=1$ and (37) is satisfied. Since

[TABLE]

for $g(t)=w(1-t),\ t\in[0,1]$, by Theorem 3.1 and Remarks 3.2 ii) with $c=2$ and $p=2$ we get the result.

$\Box$

Proof of Proposition 4.2: Note that $V_{i}(t),\ 1\leq i\leq n$ are stationary with unit variance and correlation function $r(s,t)$ satisfies

[TABLE]

By Theorem 3.3 with $c=2$ and $p=2$ we get the result.

$\Box$

Acknowledgement:Thanks to the referees for their comments and suggestions which significantly improved the manuscript. Thanks to Prof. Enkelejd Hashorva for his suggestions. Thanks to Swiss National Science Foundation Grant no. 200021-166274.