Convergence rate of extremes of normal triangular arrays

Abstract

In this paper, we mainly discuss the convergence rate of the limit distribution of a class of normal stationary triangular arrays, and point out that the convergence rate of this triangular array is not faster than that of the extremes of random variables in the i.i.d. case.

Graphical abstract

Keywords

Stationary normal sequence / extreme index / convergence rate

1 Introduction

There have been numerous studies on the asymptotic limit distribution of the maximum of a sequence of random variables. Assume that the first sequence of random variables \left\{{\xi }_n,n\ge 0\right\} is a stationary standard normal sequence with correlation coefficient {\rho }_j=Cov\left({\xi }_0,{\xi }_j\right), and the second sequence of random variables is an independent standard normal sequence. [1] proved that when

(1.1){\rho }_j\mathrm{l}\mathrm{n}j\to 0, j\to \mathrm{\infty }

is satisfied, the limit distribution of the maximum of the first sequence {\mathrm{m}\mathrm{a}\mathrm{x}}_{1\le i\le n}{\xi }_i is the same as that of the maximum of the second sequence (Refer to [6]). [8] proved that condition (1.1) is indispensable to obtain the above conclusion. However, the conclusion in [1] does not consider the clustering case. [10] proved that for any sequence v_n satisfying the convergence of n(1-\Phi (v_n)) to a certain constant (\Phi \left(x\right) represents a distribution function with standard normal distribution), the convergence rate of

\underset{u\ge v_n}{\mathrm{s}\mathrm{u}\mathrm{p}}\left|P\left(\underset{1\le i\le n}{\mathrm{m}\mathrm{a}\mathrm{x}}{\xi }_i\le u\right)-{\Phi }^n\left(u\right)\right|

is n^{-\frac{1-\rho }{1+\rho }}, where \rho =\mathrm{m}\mathrm{a}\mathrm{x}\left\{0,{\rho }_1,{\rho }_2,\dots \right\}.

To avoid the clustering situation, we take a standard normal triangular array \{{\xi }_{ni}, i=1,2,\dots ,n;n=1, 2,\dots \}into account, and for any fixed n, \left\{{\xi }_{ni},i\ge 1\right\} is a stationary standard normal sequence. Inspired by [3, 4] on the limit distribution of two-dimensional triangular arrays, a new condition was introduced in [5], that is, when n\to \mathrm{\infty },

(1.2)\left(1-{\rho }_{n,j}\right)\ln n\to {\delta }_j\in \left(0,\mathrm{\infty }\right],j\ge 1,

where {\rho }_{n,j}=E\left({\xi }_{ni},{\xi }_{n,i+j}\right),{\delta }_0=0, and the following conclusion was drawn:

Proposition 1.1　 For the standard normal triangular array mentioned above, if condition (1.2) holds, there will exist two sequences of positive integers r_n,l_n satisfying when n\to \mathrm{\infty },

(1.3)\frac{l_n}{r_n}\to 0,\frac{r_n}{n}\to 0,

and

(1.4)\underset{n\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}} \frac{n^2}{r_n}\sum _{j=l_n}^n \frac{\left|{\rho }_{n,j}\right|}{{\left(1-{\rho }_{n,j}\right)}^{\frac{1}{2}}}\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{2\ln n-\mathrm{l}\mathrm{n}\ln n}{1+{\rho }_{n,j}}\right)=0,

and

(1.5)\underset{m\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}} \underset{n\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}\mathrm{u}\mathrm{p}} \sum _{j=m}^{r_n} n^{-\frac{1-{\rho }_{n,j}}{1+{\rho }_{n,j}}}\frac{\mathrm{l}\mathrm{n}{n}^{-\frac{{\rho }_{n,j}}{1+{\rho }_{n,j}}}}{{\left(1-{\rho }_{n,j}^2\right)}^{\frac{1}{2}}}=0.

Then

(1.6)\underset{n\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}} P\left(\underset{1\le i\le n}{\mathrm{m}\mathrm{a}\mathrm{x}} {\xi }_{ni}\le u_n\left(x\right)\right)=\mathrm{e}\mathrm{x}\mathrm{p}\left(-\vartheta {\mathrm{e}}^{-x}\right),

where u_n\left(x\right)=\frac{x}{a_n}+b_n. Normalizing constants a_n and b_n are defined as follows:

(1.7)a_n={\left(2\ln n\right)}^{\frac{1}{2}},b_n={\left(2\ln n\right)}^{\frac{1}{2}}-\frac{1}{2}{\left(2\ln n\right)}^{-\frac{1}{2}}\left(\ln \ln n+\mathrm{l}\mathrm{n}4\pi \right),

and

If for all k\ge 1, it satisfies {\delta }_k=\mathrm{\infty }, then \vartheta =1, where E represents a standard exponential distribution independent of W_k, and is a joint normal distribution with mean zero. There is

E\left(W_i{W}_j\right)=\frac{{\delta }_i+{\delta }_j-{\delta }_{\left|i-j\right|}}{2{\left({\delta }_i{\delta }_j\right)}^{\frac{1}{2}}}.

There have been numerous studies on the convergence rate of sequences of random variables. Recent researches include: [7] presents the convergence rate of the skewed normal distribution; [2] shows the convergence rate of the asymmetric normal distribution, and the like. In this paper, we will give the convergence rate where the triangular array in Proposition 1.1 converges to its limit distribution.

The structure of this paper is as follows: Section 2 introduces the main conclusions; Sections 3 and 4 present their proofs.

2 Main conclusions

Let \left\{{\xi }_n\right\} be a sequence of random variables. If for the subsets I = \left\{i_1,i_2,\dots ,i_p\right\} and J=\left\{j_1,j_2,\dots ,j_{p^{\text{'}}}\right\} of the set \{1,2,\dots ,n\}, {\mathrm{m}\mathrm{i}\mathrm{n}}_{i\in J}i-{\mathrm{m}\mathrm{a}\mathrm{x}}_{i\in I}i\ge l_n, then u_n is a sequence of real numbers. We say

(2.1){\alpha }_{n,l_n}=\mathrm{m}\mathrm{a}\mathrm{x}\left\{\left|P\left({\xi }_i\le u_n,i\in I\cup J\right)-P\left(X_i\le u_n,i\in I\right)P\left(X_i\le u_n,i\in J\right)\right|\right\}

is a mixing coefficient. If for a certain sequence l_n=o\left(n\right) satisfying {\alpha }_{n,l_n}\to 0 (n\to \mathrm{\infty } ), we say \left\{{\xi }_n\right\} satisfies the \mathrm{\Delta }\left(u_n\right) condition.

Theorem 2.1　 For the triangular array in Proposition 1.1, denote M_n={\mathrm{m}\mathrm{a}\mathrm{x}}_{1\le i\le n}{\xi }_{ni}, and under the assumption of Proposition 1.1, there exists a constant C, such that

(2.2)\underset{x\in R}{\mathrm{s}\mathrm{u}\mathrm{p}} \left|P\left(M_n\le u_n\right)-\mathrm{e}\mathrm{x}\mathrm{p}\left(-\vartheta {\mathrm{e}}^{-x}\right)\right|\le \left(\frac{l_n}{r_n}+\frac{n{\alpha }_{n,l_n}}{r_n}+\frac{(\mathrm{l}\mathrm{n}\ln n{)}^2}{\ln n}+\rho \left(n\right)\right)C,

where

Note 2.1　From Theorem 2.1, it can be seen that the convergence rate of the triangular array in Proposition 1.1 to its extreme value distribution is determined by the slower one between \frac{l_n}{r_n},\frac{{\left(\ln \ln n\right)}^2}{\ln n}, \frac{n{\alpha }_{n,l_n}}{r_n} and \rho \left(n\right), which is not faster than O\left(\frac{{\left(\ln \ln n\right)}^2}{\ln n}\right).

Example 2.1　Assume \left\{{\xi }_{n,j}\right\} is a standard normal triangular array, and for any n, \left\{{\xi }_{n,j}\right\} is an AR(1) process, that is

{\xi }_{n,j}=d_n{\xi }_{n,j-1}+{\left(1-d_n^2\right)}^{\frac{1}{2}}{Z}_j,j\ge 0,

where Z_j is a sequence which is independent and identically distributed on the standard normal distribution. Assume

d_n=\left(1-\frac{\zeta }{\ln n}\right),\zeta >0.

In this case, {\rho }_{n,j}=d_n^j={\left(1-\frac{\zeta }{\ln n}\right)}^j, and {\delta }_j=j\zeta in (1.2). The reasons are as follows.

(2.3)\left(1-{\rho }_{n,j}\right)\ln n=\left(\left(1-{\left(1-\frac{\zeta }{\ln n}\right)}^j\right)\ln n\right)=j\zeta -\frac{j\left(j-1\right){\zeta }^2}{\ln n}\left(1+o\left(1\right)\right).

Select l_{\mathrm{n}}=\ln n{\left(\ln \ln n\right)}^2, then

(2.4)\underset{j\ge l_n}{\mathrm{s}\mathrm{u}\mathrm{p}} \left|{\rho }_{n,j}\right|\ln n\le \left|{\rho }_{n,l_n}\right|\ln n\le \mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{\zeta l_n}{\ln n}+\mathrm{l}\mathrm{n}\ln n\right)\to 0\left(n\to \mathrm{\infty }\right).

Select r_n=\left[n{\left({\epsilon }_n\right)}^{\frac{1}{2}}\right]\vee \left[{\left(n{l}_n\right)}^{\frac{1}{2}}\right], where{\epsilon }_n={\mathrm{s}\mathrm{u}\mathrm{p}}_{j\ge l_n}\left|{\rho }_{n,j}\right|\ln n. It can be verified that the proposition holds. In the conclusion, \frac{l_n}{r_n}=\frac{l_n}{\left[n{\left({\epsilon }_n\right)}^{\frac{1}{2}}\right]\vee \left[{\left(n{l}_n\right)}^{\frac{1}{2}}\right]}\le {\left(\frac{\ln n}{n}\right)}^{\frac{1}{2}}\ln \ln n; it can be seen that \rho \left(n\right)=\frac{1}{\ln n} from (2.3); for \frac{n{\alpha }_n,l_n}{r_n}, it is known from (2.4) that there exists the constant A, satisfying

\frac{2\ln n-\ln \ln n}{1+{\rho }_{n,j}}\le 2\ln n-\ln \ln n+A.

Hence, it can be obtained from normal distribution lemma (refer to [8]) that

(2.5)\begin{array}{rl}\frac{n{\alpha }_{n,l_n}}{r_n}& \le \frac{n}{r_n}\frac{n}{2\pi }\sum _{j=l_n}^n\frac{\left|{\rho }_{n,j}\right|}{{\left(1-{\rho }_{n,j}\right)}^{\frac{1}{2}}}\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{2\ln n-\ln \ln n}{1+{\rho }_{n,j}}\right)\\ & \le A\frac{n^3}{r_n}\frac{{\epsilon }_n}{\ln n}\mathrm{e}\mathrm{x}\mathrm{p}\left(-2\ln n+\mathrm{l}\mathrm{n}\ln n\right)\\ & =A\frac{n}{r_n}{\epsilon }_n\le A{\left({\epsilon }_n\right)}^{\frac{1}{2}}\le A\frac{1}{{\left(\ln n\right)}^{\frac{1}{2}}}.\end{array}

Considering (2.5), the conclusion can be simplified as follows:

(2.6)\underset{x\in R}{\mathrm{s}\mathrm{u}\mathrm{p}} \left|P\left(M_n\le u_n\right)-\mathrm{e}\mathrm{x}\mathrm{p}\left(-\vartheta {\mathrm{e}}^{-x}\right)\right|\le \frac{(\mathrm{l}\mathrm{n}\ln n{)}^2}{\ln n}C.

Next, MATLAB R2016b is used for numerical simulation. Take \zeta =1, then \vartheta =0.641, denoted by

{\mathrm{\Delta }}_n=\underset{x\in B}{\mathrm{s}\mathrm{u}\mathrm{p}} \left|P\right(M_n\le u_n)-\mathrm{e}\mathrm{x}\mathrm{p}(-\vartheta {\mathrm{e}}^{-x}\left)\right|.

B is denoted as a certain known set, B^{\mathrm{♯}} represents the number of elements in the set B, and \overline{{\mathrm{\Delta }}_n} refers to the mean absolute error, i.e.,

\overline{{\mathrm{\Delta }}_n}=\frac{1}{B^{\mathrm{♯}}}\sum _{x\in B} \left|P\right(M_n\le u_n)-\exp (-\vartheta {\mathrm{e}}^{-x}\left)\right|.

Corresponding diagram is as follows (Fig.1).

For different n, in order to calculate the probability P\left(M_n\le u_n\right), 10000 random numbers following this distribution are generated. The kernel density estimation with a normal kernel is used to obtain its distribution function. Take C=0.4, and denote C_n=C\frac{{\left(\ln \left(\ln n\right)\right)}^2}{\ln n},B=\left\{x|x=-5+0.1k,0\le k\le 130,k\in \mathbb{Z}\right\}. The error table (Tab.1) is obtained through numerical simulation.

3 Relevant lemmas

Lemma 3.1　 Assume \left\{{\xi }_n,n\ge 1\right\} is a stationary sequence of random variables sharing the same distribution function F and satisfying the \mathrm{\Delta }\left(u_n\right) condition; r_n, l_n are sequences of positive integers satisfying r_n=o\left(n\right), l_n=o\left(r_n\right), {\alpha }_{n,l_n}=o\left(r_n\right). If , \mathrm{\infty }, x\in \mathbb{R}. Let k_n=\left[\frac{n}{r_n+l_n}\right]. Then

(3.1)\left|P\left(M_n\le u_n\right)-P^{k_n}\left(M_{\left[\frac{n}{k_n}\right]}\le u_n\right)\right|\le 3C_1\frac{l_n}{r_n}+\frac{n{\alpha }_{n,l_n}}{r_n}.

Proof　From Lemma 3.3.1 and Lemma 3.3.2 in [7], it can be obtained that if the condition \mathrm{\Delta }\left(u_n\right) is satisfied, then

where M\left(I_1\right)={\mathrm{m}\mathrm{a}\mathrm{x}}_{i\in I_1}{\xi }_i, M\left(I_1^{\ast }\right)={\mathrm{m}\mathrm{a}\mathrm{x}}_{i\in I_1^{\ast }}{\xi }_i, I_1=\left\{1,2,\dots ,\left[\frac{n}{k_n}\right]-l_n\right\}, I_1^{\ast }=\left\{\left[\frac{n}{k_n}\right]-l_n+1,\dots ,\left[\frac{n}{k_n}\right]\right\}. Since ,

so

(3.2)

Besides, given l_n=o\left(r_n\right),

(3.3)\left(k_n-1\right){\alpha }_{n,l_n}\le k_n{\alpha }_{n,l_n}=\frac{n{\alpha }_{n,l_n}}{r_n+l_n}\le \frac{n{\alpha }_{n,l_n}}{r_n}.

It can be seen from (3.2) and (3.3) the conclusion holds. Q.E.D. □

Lemma 3.2　 Under the assumption of Proposition 1.1, for any finite subset K of the set \left\{2,3,\dots ,n\right\}, \mathrm{\forall }i\in K, . Denote . Then there exists the constant C>0 (related to the number of elements in the set K), such that

(3.4)\begin{array}{rl}& \left|P\left({\xi }_{n,k}\le u_n,k\in K|{\xi }_{n,1}>u_n\right)-P\left(\frac{2}{E}+\sqrt{{\delta }_k}{W}_k\le {\delta }_k,k\in K\right)\right|\\ & \le C\left(\rho \left(n\right)+\frac{\ln \ln n}{\ln n}\right).\end{array}

Proof　Given

(3.5)\begin{array}{rl}P({\xi }_{n,k}& \le u_n,k\in K|{\xi }_{n,1}>u_n)\\ & ={\int }_0^{\mathrm{\infty }}P\left({\xi }_{n,k}\le u_n,k\in K|{\xi }_{n,1}=u_n+\frac{y}{u_n}\right)\frac{\varphi \left(u_n+\frac{y}{u_n}\right)}{u_n\left(1-\Phi \left(u_n\right)\right)}\mathrm{d}y,\end{array}

it can be obtained from Lemma 4.1 in [10] that

\begin{array}{rl}P({\xi }_{n,k}& \le u_n,k\in K|{\xi }_{n,1}=u_n+\frac{y}{u_n})\\ & =P\left(Z_{n,k}\le v_{n,k},k\in K\right)\to P\left(W_k\le v_k,k\in K\right),\end{array}

where both {\left(Z_{n,k},k\in K\right)}^{\mathrm{T}} and {\left(W_k,k\in K\right)}^{\mathrm{T}} follow the multi-dimensional positive distribution. The following covariance matrix is as follows:

\begin{array}{rl}{\left({\rho }_{n,i,j}\right)}_{i,j\in K}& ={\left(\frac{{\rho }_{n,\left|i-j\right|}-{\rho }_{n,i}{\rho }_{n,j}}{{\left(1-{\rho }_{n,i}^2\right)}^{\frac{1}{2}}{\left(1-{\rho }_{n,j}^2\right)}^{\frac{1}{2}}}\right)}_{i,j\in K},\\ {\left({\rho }_{i,j}\right)}_{i,j\in K}& ={\left(\frac{{\delta }_i+{\delta }_j-{\delta }_{\left|i-j\right|}}{2{\left({\delta }_i{\delta }_j\right)}^{\frac{1}{2}}}\right)}_{i,j\in K},\end{array}

v_{n,k}=u_n\sqrt{\frac{1-{\rho }_{n,k}}{1+{\rho }_{n,k}}}-\frac{y{\rho }_{n,k}}{u_n\sqrt{1-{\rho }_{n,k}^2}} \left(k\in K\right), v_k=\sqrt{{\delta }_k}-\frac{y}{2\sqrt{{\delta }_k}}, and when n\to \mathrm{\infty }, v_{n,k}\to v_k \left(k\in K\right).

Hence,

(3.6)\begin{array}{rl}& \left|P\left(Z_{n,k}\le v_{n,k},k\in K\right)-P\left(W_k\le v_k,k\in K\right)\right|\\ & \le \left|P\left(Z_{n,k}\le v_{n,k},k\in K\right)-P\left(W_k\le v_{n,k},k\in K\right)\right|\\ & +\left|P\left(W_k\le v_{n,k},k\in K\right)-P\left(W_k\le v_k,k\in K\right)\right|\\ & =J_{n,1}+J_{n,2}.\end{array}

For J_{n,1}, with normal comparison lemma (refer to [8]), there is

(3.7)

From (1.2), denote \left(1-{\rho }_{n,k}\right)\ln n={\delta }_k+g_k\left(n\right), , k\in K, where g_k\left(n\right)=o\left(1\right) n\to \mathrm{\infty }, such that

(3.8)\begin{array}{rl}& {\rho }_{n,i,j}=\frac{{\rho }_{n,\left|i-j\right|}-{\rho }_{n,i}{\rho }_{n,j}}{{\left(1-{\rho }_{n,i}^2\right)}^{\frac{1}{2}}{\left(1-{\rho }_{n,j}^2\right)}^{\frac{1}{2}}}\\ & =\frac{\left(1-{\rho }_{n,i}\right)+\left(1-{\rho }_{n,j}\right)-\left(1-{\rho }_{n,\left|i-j\right|}\right)-\left(1-{\rho }_{n,i}\right)\left(1-{\rho }_{n,j}\right)}{{\left(1-{\rho }_{n,i}\right)}^{\frac{1}{2}}{\left(1-{\rho }_{n,j}\right)}^{\frac{1}{2}}{\left(1+{\rho }_{n,i}\right)}^{\frac{1}{2}}{\left(1+{\rho }_{n,j}\right)}^{\frac{1}{2}}}\\ & =\frac{{\delta }_i+g_i\left(n\right)+{\delta }_j+g_j\left(n\right)-{\delta }_{\left|i-j\right|}-g_{\left|i-j\right|}\left(n\right)-{\left(\ln n\right)}^{-1}\left({\delta }_i+g_i\left(n\right)\right)\left({\delta }_j+g_j\left(n\right)\right)}{{\left({\delta }_i+g_i\left(n\right)\right)}^{\frac{1}{2}}{\left({\delta }_j+g_j\left(n\right)\right)}^{\frac{1}{2}}{\left(2-\left({\delta }_i+g_i\left(n\right)\right){\left(\ln n\right)}^{-1}\right)}^{\frac{1}{2}}{\left(2-\left({\delta }_j^{-1}+g_j\left(n\right)\right){\left(\ln n\right)}^{-1}\right)}^{\frac{1}{2}}}\\ & =\left(\left({\delta }_i+{\delta }_j-{\delta }_{\left|i-j\right|}\right)-\frac{{\delta }_i{\delta }_j}{\ln n}+g_i\left(n\right)\left(1+o\left(1\right)\right)+g_j\left(n\right)\left(1+o\left(1\right)\right)\right)\\ & \times \frac{1}{2}{\left({\delta }_i{\delta }_j\right)}^{-\frac{1}{2}}{\left(1+\frac{g_i\left(n\right)}{{\delta }_i}+\frac{g_j\left(n\right)}{{\delta }_j}+\frac{g_i\left(n\right)g_j\left(n\right)}{{\delta }_i{\delta }_j}-\frac{{\delta }_i+{\delta }_j}{2\ln n}\left(1+o\left(1\right)\right)\right)}^{-\frac{1}{2}}\\ & =\left(\left({\delta }_i+{\delta }_j-{\delta }_{\left|i-j\right|}\right)-\frac{{\delta }_i{\delta }_j}{\ln n}+g_i\left(n\right)\left(1+o\left(1\right)\right)+g_j\left(n\right)\left(1+o\left(1\right)\right)\right)\\ & \times \frac{1}{2}{\left({\delta }_i{\delta }_j\right)}^{-\frac{1}{2}}\left(1-\frac{1}{2}\left(\frac{g_i\left(n\right)}{{\delta }_i}\left(1+o\left(1\right)\right)+\frac{g_j\left(n\right)}{{\delta }_j}\left(1+o\left(1\right)\right)-\frac{{\delta }_i+{\delta }_j}{2\ln n}\left(1+o\left(1\right)\right)\right)\right)\\ & ={\rho }_{i,j}+\frac{1}{4}{\left({\delta }_i{\delta }_j\right)}^{-\frac{1}{2}}\left(2-\frac{{\delta }_i+{\delta }_j-{\delta }_{\left|i-j\right|}}{{\delta }_i}\right)g_i\left(n\right)\left(1+o\left(1\right)\right)-\frac{1}{2}{\left({\delta }_i{\delta }_j\right)}^{-\frac{1}{2}}{g}_{\left|i-j\right|}\left(n\right)\\ & +\frac{1}{4}{\left({\delta }_i{\delta }_i\right)}^{-\frac{1}{2}}\left(2-\frac{{\delta }_i+{\delta }_j-{\delta }_{\left|i-j\right|}}{{\delta }_j}\right)g_j\left(n\right)\left(1+o\left(1\right)\right)\\ & +\left(\frac{\left({\delta }_i+{\delta }_j\right)\left({\delta }_i+{\delta }_j-{\delta }_{\left|i-j\right|}\right)}{8{\left({\delta }_i{\delta }_j\right)}^{\frac{1}{2}}}-\frac{1}{2}{\left({\delta }_i{\delta }_j\right)}^{\frac{1}{2}}\right)\frac{1}{\ln n}\left(1+o\left(1\right)\right),\end{array}

i.e.,

(3.9)\begin{array}{rl}{\rho }_{n,i,j}-{\rho }_{i,j}=& \frac{1}{4}{\left({\delta }_i{\delta }_j\right)}^{-\frac{1}{2}}\left(2-\frac{{\delta }_i+{\delta }_j-{\delta }_{\left|i-j\right|}}{{\delta }_i}\right)g_i\left(n\right)\left(1+o\left(1\right)\right)-\frac{1}{2}{\left({\delta }_i{\delta }_j\right)}^{-\frac{1}{2}}{g}_{\left|i-j\right|}\left(n\right)\\ & +\frac{1}{4}{\left({\delta }_i{\delta }_j\right)}^{-\frac{1}{2}}\left(2-\frac{{\delta }_i+{\delta }_j-{\delta }_{\left|i-j\right|}}{{\delta }_j}\right)g_j\left(n\right)\left(1+o\left(1\right)\right)\\ & +\left(\frac{\left({\delta }_i+{\delta }_j\right)\left({\delta }_i+{\delta }_j-{\delta }_{\left|i-j\right|}\right)}{8{\left({\delta }_i{\delta }_j\right)}^{\frac{1}{2}}}-\frac{1}{2}{\left({\delta }_i{\delta }_j\right)}^{\frac{1}{2}}\right)\frac{1}{\ln n}\left(1+o\left(1\right)\right).\end{array}

From (3.7)−(3.9), there exists the constant C_1 (depending on K),

(3.10)J_{n,1}\le C_1\left(\sum _{k\in K} g_k\left(n\right)+\frac{1}{\ln n}\right).

For J_{n,2}, assume the density function of {\left(W_k,k\in K\right)}^{\mathrm{T}} is f\left(\mathrm{x}\right), where

f\left(\mathrm{x}\right)=\frac{1}{{\left(2\pi \right)}^{K^{\mathrm{♯}}}{\left|\rho \right|}^{\frac{1}{2}}}\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{1}{2}{\mathrm{x}}^{\mathrm{T}}{\rho }^{-1}\mathrm{x}\right).

From the positive definiteness of \rho, \mathrm{\forall }\mathrm{x}\in {\mathbb{R}}^{K^{\mathrm{♯}}} (K^{\mathrm{♯}} represents the number of elements in the set K), and there is always \mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{1}{2}{\mathrm{x}}^{\mathrm{T}}{\rho }^{-1}\mathrm{x}\right)\le 1, such that

(3.11)\begin{array}{rl}& |{\int }_{-}{\mathrm{\infty }}^{v_{n,k}}\int f\left(x_1,x_2,\dots ,x_{K^{\mathrm{♯}}}\right)\mathrm{d}{x}_1\mathrm{d}{x}_2\cdots \mathrm{d}{x}_{K^{\mathrm{♯}}}\\ & -{\int }_{-}{\mathrm{\infty }}^{v_k}\int f\left(x_1,x_2,\dots ,x_{K^{\mathrm{♯}}}\right)\mathrm{d}{x}_1\mathrm{d}{x}_2\cdots \mathrm{d}{x}_{K^{\mathrm{♯}}}|\\ & =\left|{\int }_{v_k}^{v_{n,k}}\int f\left(x_1,x_2,\dots ,x_{K^{\mathrm{♯}}}\right)\mathrm{d}{x}_1\mathrm{d}{x}_2\cdots \mathrm{d}{x}_{K^{\mathrm{♯}}}\right|\\ & \le \frac{1}{(2\pi {)}^{K^{\mathrm{♯}}}|\rho {|}^{\frac{1}{2}}}\prod _{k\in K}\left|v_{n,k}-v_k\right|.\end{array}

For v_{n,k}=u_n\sqrt{\frac{1-{\rho }_{n,k}}{1+{\rho }_{n,k}}}-\frac{y{\rho }_{n,k}}{u_n{\left(1-{\rho }_{n,k}^2\right)}^{\frac{1}{2}}},k\in K, and with the help of the assumption in Lemma 3.2 and u_n^2=2\ln n-\mathrm{l}\mathrm{n}\ln n+ O\left(1\right), there is

(3.12)\begin{array}{rl}{u}_n\sqrt{\frac{1-{\rho }_{n,k}}{1+{\rho }_{n,k}}}& ={\left(\frac{\left(2\ln n-\ln \ln n+O\left(1\right)\right){\left(\ln n\right)}^{-1}\left({\delta }_k+g_k\left(n\right)\right)}{2-{\delta }_k{\left(\ln n\right)}^{-1}+g_k\left(n\right){\left(\ln n\right)}^{-1}}\right)}^{\frac{1}{2}}\\ & ={\left({\delta }_k+g_k\left(n\right)-\frac{{\delta }_k}{2}\frac{\ln \ln n}{\ln n}+o\left(\frac{\mathrm{l}\mathrm{n}\ln n}{\ln n}\right)\right)}^{\frac{1}{2}}\\ & ={\delta }_k^{\frac{1}{2}}+\frac{1}{2}{\delta }_k^{-\frac{1}{2}}{g}_k\left(n\right)\left(1+o\left(1\right)\right)-\frac{1}{4}{\delta }_k^{\frac{1}{2}}\frac{\ln \ln n}{\ln n}\left(1+o\left(1\right)\right),\end{array}

(3.13)\begin{array}{rl}\frac{{\rho }_{n,k}}{u_n{\left(1-{\rho }_{n,k}^2\right)}^{\frac{1}{2}}}& =\left(1-{\delta }_k{\left(\ln n\right)}^{-1}+g_k\left(n\right){\left(\ln n\right)}^{-1}\right)\\ & {\left(4{\delta }_k-g_k\left(n\right)-2{\delta }_k\frac{\ln \ln n}{\ln n}\left(1+o\left(1\right)\right)\right)}^{-\frac{1}{2}}\\ & =\left(1-{\delta }_k{\left(\ln n\right)}^{-1}-g_k\left(n\right){\left(\ln n\right)}^{-1}\right)\\ & \times \left[\frac{1}{2}{\delta }_k^{-\frac{1}{2}}+\frac{1}{4}{\delta }_k^{-\frac{3}{2}}{g}_k\left(n\right)\left(1+o\left(1\right)\right)+\frac{1}{8}{\delta }_k^{-\frac{1}{2}}\frac{\ln \ln n}{\ln n}\left(1+o\left(1\right)\right)\right]\\ & =\frac{1}{2}{\delta }_k^{-\frac{1}{2}}+\frac{1}{4}{\delta }_k^{-\frac{3}{2}}{g}_k\left(n\right)\left(1+o\left(1\right)\right)+\frac{1}{8}{\delta }_k^{-\frac{1}{2}}\frac{\ln \ln n}{\ln n}\left(1+o\left(1\right)\right).\end{array}

From (3.12) and (3.13), there is

(3.14)\begin{array}{rl}& \prod _{k\in K} \left|\begin{array}{c}{v}_{n,k}-v_k\end{array}\right|\\ & =\prod _{k\in K} \left|u_n\sqrt{\frac{1-{\rho }_{n,k}}{1+{\rho }_{n,k}}}-\frac{y{\rho }_{n,k}}{u_n(1-{\rho }_{n,k}^2{)}^{\frac{1}{2}}}-\left({\delta }_k^{\frac{1}{2}}-\frac{y}{2}({\delta }_k{)}^{-\frac{1}{2}}\right)\right|\\ & =\prod _{k\in K} \left|\left(\frac{1}{2}{\delta }_k^{-\frac{1}{2}}-\frac{y}{4}{\delta }_k^{-\frac{3}{2}}\right)g_k\left(n\right)(1+o(1\left)\right)-\frac{1}{4}{\delta }_k^{\frac{1}{2}}\left(1+\frac{y}{2{\delta }_k}\right)\frac{\mathrm{l}\mathrm{n}\ln n}{\ln n}(1+o(1\left)\right)\right|.\end{array}

It is known from (3.10), (3.11), and (3.14) that there exists such constant C_2 that for any k\in K,

(3.15)J_{n,2}\le C_2\left(g_k\left(n\right)+\frac{\ln \ln n}{\ln n}\right).

It can be seen from (3.6), (3.10), and (3.15) that there exists the constant C_3, such that

(3.16)\left|P\left(Z_{n,k}\le v_{n,k},k\in K\right)-P\left(W_k\le v_k,k\in K\right)\right|\le C_3\left(\sum _{k\in K} g_k\left(n\right)+\frac{\mathrm{l}\mathrm{n}\ln n}{\ln n}\right).

For (3.5), based on Mills’s Inequality

(3.17)\varphi \left(x\right)\left(1-x^2\right)\le 1-\Phi \left(x\right)\le \frac{\varphi \left(x\right)}{x},x>0,

we can get

(3.18)\mathrm{e}\mathrm{x}\mathrm{p}\left(-y-\frac{y^2}{2u_n^2}\right)\le \frac{\varphi \left(u_n+\frac{y}{u_n}\right)}{u_n\left(1-\Phi \left(u_n\right)\right)}\le \frac{1}{1-u_n^{-2}}\mathrm{e}\mathrm{x}\mathrm{p}\left(-y-\frac{y^2}{2u_n^2}\right).

Let C=\mathrm{m}\mathrm{a}\mathrm{x}\left\{C_1,C_3\right\} . From (3.5) and (3.17)‒(3.18),

(3.19)\begin{array}{rl}& \left|P\left({\xi }_{n,k}\le u_n,k\in K|{\xi }_{n,1}>u_n\right)-P\left(\frac{E}{2}+\sqrt{{\delta }_k}{W}_k\le {\delta }_k,k\in K\right)\right|\\ & \le C\left(\sum _{k\in K} g_k\left(n\right)+\frac{\mathrm{l}\mathrm{n}\ln n}{\ln n}\right)\le C\left({\rho }_{\left(n\right)}+\frac{\mathrm{l}\mathrm{n}\ln n}{\ln n}\right).\end{array}

Q.E.D. □

4 Proofs of theorems

Proof of Theorem 2.1　According to triangular inequality, we get

(4.1)\begin{array}{rl}|P(M_n& \le u_n)-\mathrm{e}\mathrm{x}\mathrm{p}\left(-\vartheta {\mathrm{e}}^{-x}\right)|\\ & \le \left|P\left(M_n\le u_n\right)-\mathrm{e}\mathrm{x}\mathrm{p}\left(-nP\left(\bigcap _{j=2}^{r_n} \left\{{\xi }_{n,j}\le u_n\right\},{\xi }_{n,1}>u_n\right)\right)\right|\\ & +\left|\mathrm{e}\mathrm{x}\mathrm{p}\left(-nP\left(\bigcap _{j=2}^{r_n} \left\{{\xi }_{n,j}\le u_n\right\},{\xi }_{n,1}>u_n\right)\right)-\mathrm{e}\mathrm{x}\mathrm{p}\left(-\vartheta {\mathrm{e}}^{-x}\right)\right|\\ & =I_{n,1}+I_{n,2}.\end{array}

For I_{n,1}, denote M_{1,p}={\mathrm{m}\mathrm{a}\mathrm{x}}_{2\le i\le p}{\xi }_{n,i}, then M_n=M_{0,n}. Since the function y={\left(1+x\right)}^{\frac{1}{x}} monotonically decreases on (-1,0), and {\mathrm{l}\mathrm{i}\mathrm{m}}_{x\to 0}{-\left(1+x\right)}^{\frac{1}{x}}=e, considering [9] and Lemma 3.1, there exists the constant C_1^{\prime } (related with the selection of u_n ), such that

(4.2)\begin{array}{rl}P\left(M_n\le u_n\right)& \le P^{k_n}\left(M_{\left[\frac{n}{k_n}\right]}\le u_n\right)+3C_1^{\prime }\frac{l_n}{r_n}+\frac{n{\alpha }_{n,l_n}}{r_n}\\ & \le {\left(1-\left[\frac{n}{k_n}\right]P\left({\xi }_{n,1}>u_n,M_{1,\left[\frac{n}{kn}\right]}\le u_n\right)\right)}^{k_n}+3C_1^{\prime }\frac{l_n}{r_n}+\frac{n{\alpha }_{n,r_n}}{l_n}\\ & \le \mathrm{e}\mathrm{x}\mathrm{p}\left(-nP\left({\xi }_{n,1}>u_n,M_{1,\left[\frac{n}{k_n}\right]}\le u_n\right)\right)+3C_1^{\prime }\frac{l_n}{r_n}+\frac{n{\alpha }_{n,l_n}}{r_n}.\end{array}

For the reverse direction of Inequality (4.2), s_n is a sequence of positive integers, and r_n=o\left(s_n\right) . Let t_n=\frac{n}{s_n+l_n}. It is known from Lemma 3.1 that there exists constant C_1^{″}, such that

(4.3)\left|P\left(M_n\le u_n\right)-P^{t_n}\left(M_{\left[\frac{n}{t_n}\right]}\le u_n\right)\right|\le 3C_1^{″}\frac{l_n}{s_n}+\frac{n{\alpha }_{n,l_n}}{r_n},

and from r_n=o\left(s_n\right), (4.2), and (4.3), it can be obtained that

(4.4)\left|P^{k_n}\left(M_{\left[\frac{n}{k_n}\right]}\le u_n\right)-P^{t_n}\left(M_{\left[\frac{n}{t_n}\right]}\le u_n\right)\right|\le 3C_1^{″}\left(\frac{l_n}{r_n}+\frac{l_n}{s_n}\right)+2\frac{n{\alpha }_{n,l_n}}{r_n}.

Let f\left(n\right)=3C_1^{″}\left(\frac{l_{\mathrm{n}}}{r_{\mathrm{n}}}+\frac{l_{\mathrm{n}}}{s_{\mathrm{n}}}\right)+2\frac{n{\alpha }_{\mathrm{n},l_n}}{r_n}. From (4.4), we get

(4.5)\left|P^{k_n}\left(M_{\left[\frac{n}{k_n}\right]}\le u_n\right)-P^{t_n}\left(M_{\left[\frac{n}{t_n}\right]}\le u_n\right)\right|\le f\left(n\right),

i.e.,

\begin{array}{rl}& 1-{\left({\left(1-P\left(M_{\left[\frac{n}{t_n}\right]}>u_n\right)\right)}^{t_n}+f\left(n\right)\right)}^{\frac{1}{k_n}}\le P\left(M_{\left[\frac{n}{k_n}\right]}>u_n\right)\\ & \le 1-{\left({\left(1-P\left(M_{\left[\frac{n}{t_n}\right]}>u_n\right)\right)}^{t_n}-f\left(n\right)\right)}^{\frac{1}{k_n}},\end{array}

where

(4.6)\begin{array}{rl}& 1-{\left({\left(1-P\left(M_{[\frac{n}{t_n]}}>u_n\right)\right)}^{t_n}+f\left(n\right)\right)}^{\frac{1}{k_n}}\\ & =1-{\left(1-t_{n}P\left(M_{\left[\frac{n}{t_n}\right]}>u_n\right)+o\left(t_{n}P\left(M_{[\frac{n}{t_n]}}>u_n\right)\right)+f\left(n\right)\right)}^{\frac{1}{k_n}}\\ & =1-(1-\frac{1}{k_n}\left(t_{n}P\left(M_{\left[\frac{n}{t_n}\right]}>u_n\right)+o\left(t_{n}P\left(M_{\left[\frac{n}{t_n}\right]}>u_n\right)\right)-f\left(n\right)\right)\\ & +o\left(\frac{1}{k_n}\left(t_{n}P\left(M_{[\frac{n}{t_n]}}>u_n\right)+o\left(t_{n}P\left(M_{[\frac{n}{t_n]}}>u_n\right)\right)-f\left(n\right)\right)\right))\\ & =\frac{t_n}{k_n}P\left(M_{[\frac{n}{t_n]}}>u_n\right)(1+o(1\left)\right)-\frac{f\left(n\right)}{k_n}(1+o(1\left)\right).\end{array}

By the same token,

(4.7)\begin{array}{rl}& 1-{\left({\left(1-P\left(M_{\left[\frac{n}{t_n}\right]}>u_n\right)\right)}^{t_n}-f\left(n\right)\right)}^{\frac{1}{k_n}}\\ & =\frac{t_n}{k_n}P\left(M_{\left[\frac{n}{t_n}\right]}>u_n\right)\left(1+o\left(1\right)\right)+\frac{f\left(n\right)}{k_n}\left(1+o\left(1\right)\right).\end{array}

From (4.6) and (4.7), it is known that

(4.8)\begin{array}{rl}& \left|P\left(M_{\left[\frac{n}{k_n}\right]}>u_n\right)-\frac{t_n}{k_n}P\left(M_{\left[\frac{n}{t_n}\right]}>u_n\right)\right|\\ & \le o\left(\frac{t_n}{k_n}P\left(M_{\left[\frac{n}{t_n}\right]}>u_n\right)\right)+\frac{f\left(n\right)}{k_n}\left(1+o\left(1\right)\right).\end{array}

For the sake of convenience, letp=\left[\frac{n}{t_n}\right], q = \left[\frac{n}{k_n}\right]. Then, based on (3.8) and P\left(M_p>u_n\right) = P\left(M_{p-q}\right) > u_n,M_{p-q,p} \le u_n)+P(M_q>u_n), there is

\begin{array}{rl}P\left(M_p>u_n\right)& \le \frac{k_n}{k_n-t_n}P\left(M_{p-q}>u_n,M_{p-q,p}\le u_n\right)+o\left(\frac{t_n}{k_n-t_n}P\left(M_p>u_n\right)\right)\\ & +\frac{f\left(n\right)}{k_n-t_n}\left(1+o\left(1\right)\right)\\ & \le \frac{k_n}{k_n-t_n}\sum _{i=1}^{p-q}P\left({\xi }_{n,i}>u_n,M_{i,q+i-1}\le u_n\right)+o\left(\frac{t_n}{k_n-t_n}pP\left({\xi }_{n,1}>u_n\right)\right)\\ & +\frac{f\left(n\right)}{k_n-t_n}\left(1+o\left(1\right)\right)\\ & \le \frac{k_n}{k_n-t_n}pP\left({\xi }_{n,1}>u_n,M_{1,q}\le u_n\right)+o\left(\frac{t_n}{k_n-t_n}\times \frac{p}{n}\right)\\ & +\frac{f\left(n\right)}{k_n-t_n}\left(1+o\left(1\right)\right).\end{array}

Hence, from (4.3), we get

(4.9)\begin{array}{rl}P(M_n\le u_n)& \ge P^{t_n}\left(M_{\left[\frac{n}{t_n}\right]}\le u_n\right)-3C_1^{″}\frac{l_n}{r_n}-\frac{n{\alpha }_{n,l_n}}{r_n}\\ & \ge (1-(\frac{k_n}{k_n-t_n}pP({\xi }_{n,1}>u_n,M_{|1,q}\le u_n)+o\left(\frac{t_n}{k_n-t_n}\times \frac{p}{n}\right)\\ & +\frac{f\left(n\right)}{k_n-t_n}(1+o(1))){)}^{t_n}+3C_1^{″}\frac{l_n}{r_n}+\frac{n{\alpha }_{n,l_n}}{r_n}\\ & \ge \exp \left(-nP\left({\xi }_{n,1}>u_n,M_{1,\left[\frac{n}{k_n}\right]}\right)\right)+o\left(\frac{t_n}{k_n-t_n}\times \frac{p}{n}\right)\\ & +\frac{f\left(n\right)}{k_n-t_n}-3C_1^{″}\frac{l_n}{r_n}-\frac{n{\alpha }_{n,l_n}}{r_n}.\end{array}

Since \frac{t_n}{k_n}\sim \frac{r_n}{s_n}, it is known from (4.2)‒(4.8) that there exists the constant C_2, such that

(4.10)\left|I_{n,1}\right|\le \left(\frac{l_n}{r_n}+\frac{n{\alpha }_{n,l_n}}{r_n}\right)C_2.

For I_{n,2}, from [9, p.39], it is known that

(4.11)\begin{array}{rl}& nP\left(\bigcap _{j=2}^{r_n} \left\{{\xi }_{n,j}\le u_n,{\xi }_{n,1}>u_n\right\}\right)\\ & =nP\left({\xi }_{n,1}>u_n\right)P\left(\bigcap _{j=2}^{r_n} \left(\left\{{\xi }_{n,j}\le u_n\right\}|{\xi }_{n,1}>u_n\right)\right)\\ & ={\mathrm{e}}^{-x}\left(1-\frac{{\left(\ln \ln n\right)}^2}{16\ln n}\left(1+o\left(1\right)\right)\right)P\left(\bigcap _{j=2}^{r_n} \left(\left\{{\xi }_{n,j}\le u_n\right\}|{\xi }_{n,1}>u_n\right)\right).\end{array}

It can be seen from the proofs of relevant Proposition 1 in [8, p.682] that

(4.12)\underset{m\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\underset{n\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}P\left(\bigcap _{j=2}^m \{{\xi }_{n,j}\le u_n\}|{\xi }_{n,1}>u_n\right)\to \vartheta ,

(4.13)\underset{m\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\underset{n\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\mathrm{s}\mathrm{u}\mathrm{p}P\left(\bigcup _{j=m}^{r_n} \{{\xi }_{n,j}\le u_n\}|{\xi }_{n,1}>u_n\right)\to 0.

Based on (4.12), (4.13), and Lemma 3, there exists the constant C_3, such that

(4.14)\left|P\left(\bigcap _{j=2}^{r_n} \left\{{\xi }_{n,j}\le u_n\right\}\right|{\xi }_{n,1}>u_n\right)-\vartheta |\le C_3\left(\rho \left(n\right)+\frac{\mathrm{l}\mathrm{n}\ln n}{\ln n}\right).

From (4.11) and (4.14) it is known that

(4.15)\begin{array}{rl}& nP\left(\bigcap _{j=2}^{r_n} \left\{{\xi }_{n,j}\le u_n,{\xi }_{n,1}>u_n\right\}\right)\\ & \le {\mathrm{e}}^{-x}\left(1-\frac{{\left(\mathrm{l}\mathrm{n}\ln n\right)}^2}{16\ln n}\left(1+o\left(1\right)\right)\right)\left(\vartheta +C_3\left(\rho \left(n\right)+\frac{\ln \ln n}{\ln n}\right)\right)\\ & =\vartheta {\mathrm{e}}^{-x}+{\mathrm{e}}^{-x}{C}_3\rho \left(n\right)\left(1+o\left(1\right)\right)-{\mathrm{e}}^{-x}\vartheta \frac{{\left(\ln \ln n\right)}^2}{16\ln n}\left(1+o\left(1\right)\right).\end{array}

For any real numbers x, y satisfying, with Taylor’s Theorem, there exists , such that

(4.16){\mathrm{e}}^{-y}-{\mathrm{e}}^{-x}={\mathrm{e}}^{-x}\left({\mathrm{e}}^{x-y}-1\right)={\mathrm{e}}^{-x}\left(x-y+\frac{1}{2}{\left(x-y\right)}^2{\mathrm{e}}^{\theta \left(x-y\right)}\right).

Hence, from (4.14) and (4.15), it is known that

(4.17)I_{n,2}\le \left|{\mathrm{e}}^{-x}{C}_3\rho \left(n\right)\left(1+o\left(1\right)\right)-{\mathrm{e}}^{-x}\vartheta \frac{{\left(\mathrm{l}\mathrm{n}\ln n\right)}^2}{16\ln n}\left(1+o\left(1\right)\right)\right|,

and based on (4.1), (4.10), and (4.17), there exists the constant C such that (2.2) is established. Q.E.D. □

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