1 Introduction

The almost sure central limit theorem (ASCLT) has served as a basis for a large group of investigations of fundamental significance both in the theory of probability and in its numerous applications to statistics, natural sciences, engineering, and economics. Its methods and results continue to have great influence on other fields of probability theory, mathematical statistics, and their applications. In recent decades, there has been much work on the ASCLT. Cheng et al. [2], Fahrner and Stadtmüller [3], and Berkes and Csáki [4] considered the ASCLT for the maximum of i.i.d. random variables. For more related work on ASCLT, see [5–13]. An influential work is Csáki and Gonchigdanzan [14], which proved the almost sure limit theorem for the maximum of stationary weakly dependent sequence. Furthermore, Lin [15] considered the theorem which ASCLT version of the theorem proved by Leadbetter et al. [16]. Chen et al. [17] extended [14] to the multivariate stationary case. Lin et al. [1] partially extended [14] to the case of strongly dependent nonstationary Gaussian sequences and obtained the following theorem.

Theorem A

Let \(\{\xi_{n}:n\geq1\}\) be a sequence of nonstationary standard Gaussian random variables with covariances \(r_{ij}\) satisfying \(|r_{ij}-\frac{r}{\ln(j-i)}|\ln(j-i)(\ln\ln (j-i))^{1+\varepsilon}=O(1)\) for \(r>0\).

If

$$ a_{n}=(2\ln n)^{1/2},\quad b_{n}=(2\ln n)^{1/2}-\frac{1}{2}(2\ln n)^{-1/2}\bigl(\ln\ln n+\ln(4\pi) \bigr), $$
(1.1)

then

$$ \lim_{n\rightarrow\infty}\frac{1}{\ln n}\sum^{n}_{k=1} \frac{1}{k}I \bigl(a_{k}(M_{k}-b_{k})\leq x \bigr)= \int^{\infty}_{-\infty}\exp\bigl(-\mathrm{e}^{-x-r+\sqrt{2r}z} \bigr)\phi (z)\,\mathrm{d}z \quad \mathrm{a.s.}, $$
(1.2)

where I denotes an indicator function and ϕ is the standard normal density function.

The purpose of this paper is to give substantial improvements for both weight sequence and the range of random variables of Theorem A.

Throughout the paper, let \(\{\boldsymbol {\xi}_{i}=(\xi_{i}(1),\xi_{i}(2), \ldots,\xi_{i}(d)):i\geq1\}\) be a standardized nonstationary Gaussian vector sequence with

$$\begin{aligned} &\mathbb{E} \boldsymbol {\xi}_{n}=\bigl(\mathbb{E}\xi_{n}(1), \mathbb{E}\xi_{n}(2), \ldots, \mathbb{E}\xi_{n}(d)\bigr)=(0, 0, \ldots, 0), \\ &\operatorname{Var} \boldsymbol {\xi}_{n}=\bigl(\operatorname{Var} \xi_{n}(1), \operatorname{Var}\xi_{n}(2), \ldots , \operatorname{Var}\xi_{n}(d)\bigr)=(1, 1, \ldots, 1), \\ &r_{ij}(p)=\operatorname{Cov}\bigl(\xi_{i}(p), \xi_{j}(p)\bigr), \\ &r_{ij}(p,q)=\operatorname{Cov}\bigl(\xi_{i}(p), \xi_{j}(q)\bigr),\quad \mbox{for } 1\leq p\neq q\leq d. \end{aligned}$$

Let \(\{\boldsymbol {\eta}_{i}=(\eta_{i}(1),\eta_{i}(2), \ldots,\eta_{i}(d)):i\geq1\}\) be a d-dimensional vector sequence. For \(i\geq1\), we define

$$\boldsymbol {\xi}_{i}\boldsymbol {\eta}_{i}=\bigl(\xi_{i}(1) \eta_{i}(1),\xi_{i}(2)\eta_{i}(2), \ldots, \xi_{i}(d)\eta_{i}(d)\bigr). $$

Let \(\mathbf{u}_{ni}= (u_{ni}(1), u_{ni}(2), \ldots,u_{ni}(d))\) be a d-dimensional real vector, and \(\mathbf{u}_{ni}>\mathbf{u}_{ki}\) means \(u_{ni}(p)>u_{ki}(p)\) for \(p= 1, 2, \ldots, d\). Suppose

$$ r_{ij}(p)\ln(j-i)\rightarrow r,\qquad r_{ij}(p,q)\ln (j-i) \rightarrow r,\quad \mbox{as } i, j\rightarrow\infty, $$
(1.3)

where throughout \(r\geq0\) and \(i< j\).

\(\{\boldsymbol {\xi}_{n}:n\geq1\}\) is called weakly dependent for \(r =0\) and strongly dependent for \(r>0\).

In the paper, a very natural and mild assumption is

$$\begin{aligned} \begin{aligned} & \biggl|r_{ij}(p)-\frac{r}{\ln(j-i)} \biggr|\ln(j-i) (\ln D_{j-i})^{1+\varepsilon}=O(1),\\ & \biggl|r_{ij}(p, q)-\frac{r}{\ln(j-i)} \biggr|\ln(j-i) (\ln D_{j-i})^{1+\varepsilon}=O(1), \end{aligned} \end{aligned}$$
(1.4)

where

$$ d_{k}=\frac{\exp(\ln^{\alpha}k)}{k},\qquad D_{n}=\sum _{k=1} ^{n}d_{k},\quad \mbox{for } 0\leq\alpha< \frac{1}{2}. $$
(1.5)

Let \(\boldsymbol {\eta}_{i} = \boldsymbol {\xi}_{i} + \mathbf{m}_{i}\) where \(\mathbf{ m}_{i}=(m_{i}, m_{i}, \ldots, m_{i})\) is a real vector. The constant \(m_{i}\) satisfies

$$ \beta_{n}\triangleq\max_{1\leq i\leq n}| m_{i}|=o \bigl((\ln n)^{\frac{1}{2}}\bigr),\quad \mbox{as } n\rightarrow\infty. $$
(1.6)

\(m_{n}^{*}\) is defined so that \(|m_{n}^{*}|\leq\beta_{n}\) and

$$ \frac{1}{n}\sum^{n}_{i=1}\exp \biggl(a_{n}^{*}\bigl(m_{i}-m_{i}^{*} \bigr)-\frac {1}{2(m_{i}-m_{n}^{*})^{2}}\biggr)\rightarrow1, \quad \mbox{as } n\rightarrow\infty, $$
(1.7)

where \(a_{n}^{*}=a_{n}-\ln\ln\frac{n}{2a_{n}}\).

2 Results and proofs

We mainly consider the ASCLT of the maximum of nonstationary Gaussian vector sequence satisfying (1.4), which is crucial to consider other versions of the ASCLT such as that of the maximum of stationary strongly dependent sequence and the function of the maximum. In the sequel, \(a_{n}\ll b_{n}\) denotes the existence of a constant \(c>0\) such that \(a_{n}\ll cb_{n}\) for sufficiently large n. We also define the normalized real vector \(\mathbf{a}_{k}= (a_{k}, a_{k},\ldots, a_{k})\), \(\mathbf{b}_{k}= (b_{k}, b_{k},\ldots, b_{k})\), where \(a_{k}\) and \(b_{k}\) are defined by (1.1). The main results are as follows.

Theorem 1

Let \(\{\boldsymbol {\eta}_{i}:i\geq1\}\) be defined by \(\boldsymbol {\eta}_{i} = \boldsymbol {\xi}_{i} + \mathbf{m}_{i}\) where \(\{\boldsymbol {\xi}_{i}:i\geq1\}\) is the standard nonstationary Gaussian vector sequence with covariances satisfying (1.4). Suppose that \(\{m_{i}\}\) and \(m_{n}^{*}\) satisfy (1.6) and (1.7), respectively. Then

$$\begin{aligned} &\lim_{n\rightarrow\infty}\frac{1}{D_{n}}\sum_{k=1} ^{n}d_{k} I \Bigl(\mathbf{a}_{k}\Bigl(\max _{1\leq i\leq k}\boldsymbol {\eta}_{i}-\mathbf {b}_{k}- \mathbf{m}_{k}^{*}\Bigr)\leq \mathbf{x} \Bigr) \\ &\quad=\prod ^{d}_{p=1} \int_{\mathbb{R}}\exp\bigl(-\mathrm{e}^{-x(p)-r+\sqrt{2r}z}\bigr)\,\mathrm{d} \Phi(z) \quad \mathrm{a.s.}, \end{aligned}$$
(2.1)

for \(\mathbf{m}_{k}^{*}=(m_{k}^{*}, m_{k}^{*}, \ldots, m_{k}^{*})\) and \(\mathbf{x}=(x(1), x(2), \ldots, x(d))\in\mathbb{R}^{d}\), where \(\Phi(z)\) denotes the distribution function of a standard normal random variable.

Theorem 2

Let \(\{\boldsymbol {\xi}_{i}:i\geq1\}\) is the standard nonstationary Gaussian vector sequence with covariances satisfying (1.4), we have

$$\begin{aligned} &\lim_{n\rightarrow\infty}\frac{1}{D_{n}}\sum_{k=1} ^{n}d_{k} I \Bigl(\mathbf{a}_{k}\Bigl(\max _{1\leq i\leq t_{k}}\boldsymbol {\xi}_{i}-\mathbf {b}_{k}\Bigr) \leq \mathbf{x} \Bigr) \\ &\quad=\prod^{d}_{p=1} \int_{\mathbb{R}}\exp\bigl(-t\mathrm{e}^{-x(p)-r+\sqrt{2r}z}\bigr)\, \mathrm{d}\Phi(z)\quad \mathrm{a.s.}, \end{aligned}$$
(2.2)

for \(\mathbf{x}=(x(1), x(2), \ldots, x(d))\in\mathbb{R}^{d}\), where \({t_{n}}\) is an increasing sequence of positive integers such that \(\lim_{n\rightarrow\infty} \frac{t_{n}}{n}=t\) (\(t>0\)).

In the terminology of summation procedures, we have the following corollary.

Corollary 1

Equations (2.1) and (2.2) remain valid if we replace the weight sequence \(\{d_{k}:k\geq1\}\) by \(\{d_{k}^{\ast}:k\geq1\}\) such that \(0\leq d_{k}^{\ast}\leq d_{k}\), \(\sum_{k=1}^{\infty}d_{k}^{\ast}=\infty\).

Remark 1

Our results give substantial improvements for the weight sequence in Theorem A.

Remark 2

If \(\{\boldsymbol {\xi}_{i}:i\geq1\}\) is a standardized stationary Gaussian sequence, \(t=1\) and \(\alpha=0\), then (2.2) becomes (1.2). Thus Theorem A is a special case of Theorem 2.

Remark 3

Essentially, the problem whether Theorem 1 holds also for some \(1/2\leq\alpha<1\) remains open.

The following lemmas play important roles in the proofs of our theorems. The proofs are given in the Appendix.

Lemma 1

Let \(\{\boldsymbol {\xi}_{n}:n\geq1\}\) and \(\{ \boldsymbol {\xi}^{\prime}_{n}:n\geq1\}\) be two d-dimensional independent standardized nonstationary Gaussian sequences with

$$r_{ij}^{0}(p)=\operatorname{Cov}\bigl(\xi_{i}(p), \xi_{j}(p)\bigr), \qquad r_{ij}^{0}(p,q)= \operatorname{Cov}\bigl(\xi_{i}(p),\xi_{j}(q)\bigr) $$

and

$$r_{ij}^{\prime}(p)=\operatorname{Cov}\bigl(\xi^{\prime}_{i}(p), \xi^{\prime}_{j}(p)\bigr), \qquad r_{ij}^{\prime}(p,q)= \operatorname{Cov}\bigl(\xi^{\prime}_{i}(p),\xi^{\prime}_{j}(q) \bigr). $$

Write

$$\begin{aligned} &\rho_{ij}(p)=\max\bigl(\bigl|r_{ij}^{0}(p)\bigr|,\bigl|r_{ij}^{\prime}(p)\bigr| \bigr),\\ & \rho_{ij}(p,q)=\max \bigl(\bigl|r_{ij}^{0}(p,q)\bigr|,\bigl|r_{ij}^{\prime}(p,q)\bigr| \bigr). \end{aligned}$$

Assume that (1.4) holds. Let \(\mathbf{ u}_{ni}=(u_{ni}(1),u_{ni}(2),\ldots,u_{ni}(d))\) for \(i\geq1\) be real vectors such that \(n(1-\Phi(u_{ni}(p)))\) is bounded where Φ is the standard normal distribution function. There exist absolute constants \(K_{1}\), \(K_{2}\), if

$$\mathop{\max_{1\leq i< j\leq t_{n} }}_{ 1\leq p\leq d} \rho_{ij}(p)< 1 \quad\textit{and}\quad \mathop{\max _{1\leq i< j\leq t_{n} }}_{ 1\leq p\neq q\leq d}\rho_{ij}(p,q)< 1,\quad \textit{for }t>0, $$

then

$$\begin{aligned} &\bigl|\mathbb{P}(\boldsymbol {\xi}_{j}\leq\mathbf{u}_{nj}, j=1,2,\ldots,t_{n})-\mathbb{P}\bigl(\boldsymbol {\xi}^{\prime}_{j} \leq\mathbf{ u}_{nj}, j=1,2,\ldots,t_{n}\bigr)\bigr| \\ &\quad\leq K_{1}\sum_{p=1}^{d} \sum_{1\leq i< j\leq t_{n}}\bigl|r_{ij}^{0}(p)-r_{ij}^{\prime}(p) \bigr|\exp \biggl(-\frac{u_{ni}^{2}(p)+u^{2}_{nj}(p)}{2(1+\rho_{ij}(p))} \biggr) \\ & \qquad{}+K_{2}\sum_{1\leq p\neq q\leq d}\sum _{1\leq i< j\leq t_{n}}\bigl|r_{ij}^{0}(p,q)-r_{ij}^{\prime}(p,q) \bigr|\exp \biggl(-\frac{u_{ni}^{2}(p)+u^{2}_{nj}(q)}{2(1+\rho_{ij}(p,q))} \biggr), \end{aligned}$$

where \({t_{n}}\) is an increasing sequence of positive integers such that \(\lim_{n\rightarrow\infty} \frac{t_{n}}{n}=t\) (\(t>0\)).

Lemma 2

Let \(\{\xi_{n}:n\geq1\}\) be a standardized nonstationary Gaussian vector sequence such that conditions (1.4) holds, and further suppose that \(n(1 -\Phi (u_{ni}(p)))\) is bounded for \(p=1, 2, \ldots, d\) and \(\max_{p\neq q} (\sup_{n\geq0}|r_{n}(p,q)| )<1\). Let \(\rho_{n}=\frac{r}{\ln n}\), r defined in (1.3), \(\omega_{ij}=\max\{|r_{ij}(p)|,\rho_{n}\}\), \(\omega_{ij}^{\prime}=\max\{|r_{ij}(p,q)|,\rho_{n}\}\). For some \(\varepsilon>0\), then

$$ \sum_{p=1}^{d}\sum _{1\leq i< j\leq t_{n}}\bigl|r_{ij}(p)-\rho_{n} \bigr|\exp \biggl(- \frac{u^{2}_{ni}(p)}{2(1+|\omega_{ij}|)} \biggr)\ll(\ln D_{n})^{-(1+\varepsilon)} $$
(2.3)

and

$$ \sum_{1\leq p\neq q\leq d}\sum_{1\leq i< j\leq t_{n}}\bigl|r_{ij}(p,q)- \rho_{n} \bigr|\exp \biggl(-\frac{u_{ni}^{2}(p)+u^{2}_{nj}(q)}{2(1+|\omega_{ij}^{\prime }|)} \biggr)\ll(\ln D_{n})^{-(1+\varepsilon)}, $$
(2.4)

where \({t_{n}}\) is an increasing sequence of positive integers such that \(\lim_{n\rightarrow\infty} \frac{t_{n}}{n}=t\) (\(t>0\)).

Lemma 3

Let \(\{\tilde{\boldsymbol {\xi}}_{n}:n\geq1\}\) be a standard nonstationary Gaussian vector sequence with constant covariance \(\rho_{n}(p)=n/\ln n\) for \(p=1, 2, \ldots, d\) and \(\{\boldsymbol { \xi}_{n}:n\geq1\}\) satisfy the conditions of Theorem  1. Assume \(n(1 -\Phi(u_{ni}(p)))\) is bounded for \(p=1, 2, \ldots, d\) and (1.4) is satisfied. For \(p=1,2,\ldots,d\), then

$$\begin{aligned} &\bigl|\mathbb{E} \bigl(I\bigl(\tilde{\xi}_{1}(p)\leq u_{n1}(p), \ldots, \tilde{\xi}_{n}(p)\leq u_{nn}(p) \bigr)-I\bigl(\xi_{1}(p)\leq u_{n1}(p), \ldots, \xi_{n}(p)\leq u_{nn}(p)\bigr) \bigr) \bigr| \\ &\quad\ll(\ln D_{n})^{-(1+\varepsilon)}, \quad\textit{for some } \varepsilon>0. \end{aligned}$$
(2.5)

Lemma 4

Let \(\{\boldsymbol {\xi}_{n}:n\geq1\}\) be a standardized nonstationary Gaussian d-dimensional vector sequence with covariances satisfying (1.4). Suppose that the assumptions of Lemma  1 hold, then

$$ \lim_{n\rightarrow\infty} \mathbb{P} \Bigl(\mathbf{a}_{n}\Bigl(\max _{1\leq i\leq n}\boldsymbol {\eta}_{i}-\mathbf{b}_{n}- \mathbf{m}_{n}^{*}\Bigr)\leq \mathbf{x} \Bigr)=\prod ^{d}_{p=1} \int_{\mathbb{R}}\exp\bigl(-\mathrm{e}^{-x(p)-r+\sqrt{2r}z}\bigr)\,\mathrm{d} \Phi(z), $$
(2.6)

where \(\mathbf{x}=(x(1), x(2), \ldots, x(d))\in\mathbb{R}^{d}\).

Lemma 5

Let \(\zeta_{1}, \zeta_{2},\ldots,\zeta _{n},\ldots\) , be a sequence of bounded random variables. If

$$ \operatorname{Var} \Biggl(\sum^{n}_{k=1}d_{k} \zeta_{k} \Biggr)=O \biggl( \frac{D_{n}^{2}}{(\ln D_{n})^{1+\varepsilon}} \biggr),\quad \textit{for some } \varepsilon>0, $$
(2.7)

then

$$ \lim_{n\rightarrow\infty}\frac{1}{D_{n}}\sum_{k=1} ^{n}d_{k}(\zeta _{k}-\mathbb{E} \zeta_{k})=0\quad \mathrm{a.s.} $$
(2.8)

Proof of Theorem 1

By Lemma 4 and the Toeplitz lemma, note that (2.1) is equivalent to

$$\begin{aligned} &\lim_{n\rightarrow\infty}\frac{1}{D_{n}}\sum_{k=1} ^{n}d_{k} \Bigl(I \Bigl(\mathbf{a}_{k}\Bigl(\max _{1\leq i\leq n}\boldsymbol {\eta}_{i}-\mathbf {b}_{k}- \mathbf{ m}_{k}^{*}\Bigr)\leq \mathbf{x} \Bigr) \\ &\quad{}-\mathbb{P} \Bigl(\mathbf{a}_{k}\Bigl(\max_{1\leq i\leq n}\boldsymbol { \eta}_{i}-\mathbf{b}_{k}-\mathbf{ m}_{k}^{*} \Bigr)\leq \mathbf{x} \Bigr) \Bigr)=0 \quad\mathrm{a.s.} \end{aligned}$$
(2.9)

Let \(u_{ki}(p)=\frac{x(p)}{a_{k}}+b_{k}+m_{k}^{*}-m_{i}\), by (2.3) in [1], we have \(n(1-\Phi(u_{ki}(p)))\rightarrow\tau_{p}\) for \(x(p)\in\mathbb{R}\), \(0\leq\tau_{p}<\infty\). From Lemma 5, in order to prove (2.9), for \(p=1,2,\ldots,d\), it suffices to prove

$$\begin{aligned} &\operatorname{Var} \Biggl(\sum^{n}_{k=1}d_{k}I \biggl(\xi_{1}(p)\leq \frac{x(p)}{a_{k}}+b_{k}+m_{k}^{*}-m_{1}, \ldots, \xi_{k}(p)\leq \frac{x(p)}{a_{k}}+b_{k}+m_{k}^{*}-m_{k} \biggr) \Biggr) \\ &\quad =O \biggl( \frac{D_{n}^{2}}{(\ln D_{n})^{1+\varepsilon}} \biggr) \quad\mbox{for some } \varepsilon>0. \end{aligned}$$
(2.10)

Let \(\boldsymbol {\zeta}, \boldsymbol {\zeta}_{1}, \boldsymbol {\zeta}_{2},\ldots\) be d-dimensional independent standardized nonstationary Gaussian sequences, where \(\boldsymbol {\zeta}=(\zeta,\zeta_{,} \ldots,\zeta)\), \(\{\boldsymbol {\zeta}_{i}=(\zeta_{i}(1),\zeta_{i}(2), \ldots,\zeta_{i}(d)),i\geq1\}\). It can be shown that \(\{\lambda_{i}(p)=(1-\rho_{k})^{1/2}\zeta_{i}(p)+ \rho_{k}^{1/2}\zeta,i\geq1,p=1, 2, \ldots, d\}\) have constant covariance \(\rho_{k}=r/\ln k \). For \(p=1,2,\ldots,d\) using the well-known \(c_{2}\)-inequality, the left-hand side of (2.10) can be written as

$$\begin{aligned} &\operatorname{Var} \Biggl(\sum^{n}_{k=1}d_{k}I \biggl(\xi_{1}(p)\leq \frac{x(p)}{a_{k}}+b_{k}+m_{k}^{*}-m_{1}, \ldots, \xi_{k}(p)\leq \frac{x(p)}{a_{k}}+b_{k}+m_{k}^{*}-m_{k} \biggr) \\ &\qquad{}-\sum^{n}_{k=1}d_{k}I \bigl((1-\rho_{k})^{1/2}\zeta_{1}(p)+ \rho_{k}^{1/2}\zeta\leq u_{k1}(p), \ldots, (1- \rho_{k})^{1/2}\zeta_{k}(p)+ \rho_{k}^{1/2} \zeta\leq u_{kk}(p) \bigr) \\ &\qquad{}+\sum^{n}_{k=1}d_{k}I \bigl((1-\rho_{k})^{1/2}\zeta_{1}(p)+ \rho_{k}^{1/2}\zeta\leq u_{k1}(p), \ldots, (1- \rho_{k})^{1/2}\zeta_{k}(p)+ \rho_{k}^{1/2} \zeta\leq u_{kk}(p) \bigr) \Biggr) \\ &\quad\ll\operatorname{Var} \Biggl(\sum^{n}_{k=1}d_{k}I \bigl((1-\rho_{k})^{1/2}\zeta_{1}(p)+ \rho_{k}^{1/2}\zeta\leq u_{k1}(p), \ldots, \\ &\qquad(1- \rho_{k})^{1/2}\zeta_{k}(p)+ \rho_{k}^{1/2} \zeta\leq u_{kk}(p) \bigr) \Biggr) \\ &\qquad{}+\operatorname{Var} \Biggl(\sum^{n}_{k=1}d_{k}I \bigl(\xi_{1}(p)\leq u_{k1}(p), \ldots, \xi_{k}(p)\leq u_{kk}(p) \bigr) \\ &\qquad{}-\sum^{n}_{k=1}d_{k}I \bigl((1-\rho_{k})^{1/2}\zeta_{1}(p)+ \rho_{k}^{1/2}\zeta\leq u_{k1}(p), \ldots, (1- \rho_{k})^{1/2}\zeta_{k}(p)+ \rho_{k}^{1/2} \zeta\leq u_{kk}(p) \bigr) \Biggr) \\ &\quad=:L_{1}+L_{2}. \end{aligned}$$
(2.11)

We will show \(L_{i}\ll \frac{D_{n}^{2}}{(\ln D_{n})^{1+\varepsilon}}\), \(i=1,2\). For \(p=1,2,\ldots,d\), clearly

$$\begin{aligned} L_{1}={}&\mathbb{E} \Biggl(\sum^{n}_{k=1}d_{k}I \bigl(\zeta_{1}(p)\leq(1-\rho _{k})^{-1/2} \bigl(u_{k1}(p)-\rho_{k}^{1/2}\zeta\bigr), \ldots, \\ &\zeta_{k}(p)\leq(1-\rho_{k})^{-1/2} \bigl(u_{kk}(p)-\rho_{k}^{1/2}\zeta \bigr)\bigr)\Biggr) \\ &{}-\mathbb{P} \Biggl(\sum^{n}_{k=1}d_{k}I \bigl(\zeta_{1}(p)\leq (1-\rho_{k})^{-1/2} \bigl(u_{k1}(p)-\rho_{k}^{1/2}\zeta\bigr), \ldots, \\ &\zeta_{k}(p)\leq(1-\rho_{k})^{-1/2} \bigl(u_{kk}(p)-\rho _{k}^{1/2}\zeta\bigr)\bigr) \Biggr)^{2} \\ ={}& \int_{\mathbb{R}}\mathbb{E} \Biggl(\sum^{n}_{k=1}d_{k}I \bigl(\zeta _{1}(p)\leq(1-\rho_{k})^{-1/2} \bigl(u_{k1}(p)-\rho_{k}^{1/2}z\bigr), \ldots, \\ &\zeta_{k}(p)\leq(1-\rho_{k})^{-1/2} \bigl(u_{kk}(p)-\rho_{k}^{1/2}z\bigr) \bigr) \\ &{}-\mathbb{P} \bigl(\zeta_{1}(p)\leq(1-\rho _{k})^{-1/2} \bigl(u_{k1}(p)-\rho_{k}^{1/2}z\bigr), \ldots, \\ &\zeta_{k}(p)\leq(1-\rho_{k})^{-1/2} \bigl(u_{kk}(p)-\rho_{k}^{1/2}z\bigr)\bigr) \Biggr)^{2}\,\mathrm{d}\Phi(z), \end{aligned}$$
(2.12)

where

$$\begin{aligned} \eta _{k} =&I \bigl(\zeta_{1}(p)\leq(1- \rho_{k})^{-1/2}\bigl(u_{k1}(p)-\rho_{k}^{1/2}z \bigr), \ldots, \zeta_{k}(p)\leq(1-\rho_{k})^{-1/2} \bigl(u_{kk}(p)-\rho_{k}^{1/2}z\bigr) \bigr) \\ &{}-\mathbb{P} \bigl(\zeta_{1}(p)\leq(1-\rho _{k})^{-1/2} \bigl(u_{k1}(p)-\rho_{k}^{1/2}z\bigr), \ldots, \\ &\zeta_{k}(p)\leq(1-\rho_{k})^{-1/2} \bigl(u_{kk}(p)-\rho_{k}^{1/2}z\bigr) \bigr),\quad \mbox{for } p=1,2,\ldots,d. \end{aligned}$$

Write the expectation in (2.12) as

$$\begin{aligned} \mathbb{E} \Biggl(\sum^{n}_{k=1}d_{k} \eta_{k} \Biggr)^{2} =&\sum^{n}_{k=1}d_{k}^{2} \mathbb{E}|\eta_{k}|^{2}+2\sum_{1\leq k< l\leq n}d_{k}d_{l}\bigl| \mathbb{E}(\eta_{k}\eta_{l})\bigr| \\ =:&H_{1}+H_{2}. \end{aligned}$$
(2.13)

Noting that \(|\eta_{k}|\leq1\), \(\exp(\ln^{\alpha}x)=\exp (\int^{x}_{1}\frac{\alpha(\ln u)^{\alpha-1}}{u}\,\mathrm{d}u )\), we see that \(\exp(\ln^{\alpha}x)\) (\(\alpha<1/2\)) is a slowly varying function at infinity. Hence,

$$ H_{1 }\leq\sum^{n}_{k=1}d_{k}^{2}= \sum^{n}_{k=1}\frac{\exp(2\ln^{\alpha} k)}{k^{2}}\leq\sum ^{\infty}_{k=1}\frac{\exp(2\ln^{\alpha} k)}{k^{2}}< \infty. $$
(2.14)

For \(H_{2}\), similarly to the proof of the main result in [1], we have

$$\begin{aligned} H_{2} \ll&\sum_{1\leq k< l\leq n}d_{k}d_{l} \Biggl(\prod^{l}_{i=k+1}\Phi \bigl((1- \rho_{l})^{-1/2}\bigl(u_{li}(p)-\rho_{l}^{1/2}z \bigr)\bigr) \\ &{}-\prod^{l}_{i=1}\Phi \bigl((1- \rho_{l})^{-1/2}\bigl(u_{li}(p)-\rho_{l}^{1/2}z \bigr)\bigr) \Biggr) \\ \ll&\sum_{1\leq k< l\leq n}d_{k}d_{l} \frac{k}{l} \\ =&\mathop{\sum_{1\leq k< l\leq n }}_{\frac{l}{k}\geq \ln^{2}D_{n}}d_{k}d_{l} \frac{k}{l}+\mathop{\sum_{1\leq k< l\leq n}}_{ \frac{l}{k}< \ln^{2}D_{n}}d_{k}d_{l} \frac{k}{l} \\ =:&T_{1}+T_{2}. \end{aligned}$$
(2.15)

For \(T_{1}\), we have

$$ T_{1}\leq\sum_{1\leq k< l\leq n}\frac{d_{k}d_{l}}{\ln^{2}D_{n}}\leq \frac{D_{n}^{2}}{\ln^{2}D_{n}}. $$
(2.16)

According to Wu [18], for sufficiently large n, \(0<\alpha<\frac{1}{2}\), we have

$$ D_{n}\sim\frac{1}{\alpha} \bigl(\ln^{1-\alpha}n\exp\bigl( \ln^{\alpha}n\bigr) \bigr),\qquad \ln D_{n}\sim\ln^{\alpha}n,\qquad \exp \bigl(\ln^{\alpha}n\bigr)\sim\frac{\alpha D_{n}}{(\ln D_{n})^{\frac{1-\alpha}{\alpha}}}. $$
(2.17)

Since \(\alpha<1/2\) implies \((1-\alpha)/\alpha>1\), letting \(0<\varepsilon<(1-\alpha)/\alpha-1\), for sufficiently large n, we get

$$\begin{aligned} T_{2} \leq& \sum^{n}_{k=1}d_{k} \sum_{l=k}^{k\ln^{2}D_{n}}\frac{\exp(\ln^{\alpha} l)}{l} \\ \leq&\exp\bigl(\ln^{\alpha}n\bigr)\sum^{n}_{k=1}d_{k} \sum_{l=k}^{k\ln ^{2}D_{n}}\frac{1}{l} \\ \ll&\exp\bigl(\ln^{\alpha}n\bigr)D_{n}\ln\ln D_{n} \ll\frac{D_{n}^{2}\ln\ln D_{n}}{(\ln D_{n})^{\frac{1-\alpha}{\alpha}}} \\ \leq&\frac{D_{n}^{2}}{(\ln D_{n})^{1+\varepsilon}}. \end{aligned}$$
(2.18)

Combining (2.15)-(2.18), we can get

$$ H_{2}\ll\frac{D_{n}^{2}}{(\ln D_{n})^{1+\varepsilon}}. $$
(2.19)

By (2.13), (2.14), and (2.19), we have

$$ L_{1}\ll\frac{D_{n}^{2}}{(\ln D_{n})^{1+\varepsilon}}. $$
(2.20)

Clearly,

$$\begin{aligned} L_{2} =&\operatorname{Var} \Biggl(\sum^{n}_{k=1}d_{k}I \bigl(\xi_{1}(p)\leq u_{k1}(p), \ldots, \xi_{k}(p)\leq u_{kk}(p) \bigr) \\ &{}-I \bigl(\lambda_{1}(p)\leq u_{k1}(p), \ldots, \lambda_{k}(p)\leq u_{kk}(p) \bigr) \Biggr) \\ \leq&\sum^{n}_{k=1}d^{2}_{k} \operatorname{Var} \bigl(I \bigl(\xi_{1}(p)\leq u_{k1}(p), \ldots, \xi_{k}(p)\leq u_{kk}(p) \bigr) \\ &{}-I \bigl(\lambda_{1}(p)\leq u_{k1}(p), \ldots, \lambda_{k}(p)\leq u_{kk}(p) \bigr) \bigr) \\ &{}+2 \biggl|\sum_{1\leq i< j\leq n}d_{i}d_{j} \operatorname{Cov} \bigl(I \bigl(\xi_{1}(p)\leq u_{i1}(p), \ldots, \xi_{i}(p)\leq u_{ii}(p) \bigr) \\ &{}-I \bigl(\lambda_{1}(p)\leq u_{i1}(p), \ldots, \lambda_{i}(p)\leq u_{ii}(p) \bigr),I \bigl( \xi_{1}(p)\leq u_{j1}(p), \ldots, \xi_{j}(p)\leq u_{jj}(p) \bigr) \\ &{}-I \bigl(\lambda_{1}(p)\leq u_{j1}(p), \ldots, \lambda_{j}(p)\leq u_{jj}(p) \bigr) \bigr)\biggr| \\ =:&J_{1}+2J_{2}. \end{aligned}$$
(2.21)

Similarly to (2.14), we find that \(J_{1}\leq\sum^{\infty}_{k=1}d_{k}^{2}<\infty\). Note that

$$\begin{aligned} J_{2} \leq& \biggl|\sum_{1\leq i< j\leq n}d_{i}d_{j} \operatorname{Cov}\bigl(I \bigl(\xi_{1}(p)\leq u_{i1}(p), \ldots, \xi_{i}(p)\leq u_{ii}(p) \bigr) \\ &{}-I \bigl(\lambda_{1}(p)\leq u_{i1}(p), \ldots, \lambda_{i}(p)\leq u_{ii}(p) \bigr),I \bigl( \xi_{1}(p)\leq u_{j1}(p), \ldots, \xi_{j}(p)\leq u_{jj}(p) \bigr) \\ &{}-I \bigl(\lambda_{1}(p)\leq u_{j1}(p), \ldots, \lambda_{j}(p)\leq u_{jj}(p) \bigr)- \bigl(I \bigl( \xi_{1}(p)\leq u_{j1}(p), \ldots, \xi_{j}(p)\leq u_{jj}(p) \bigr) \\ &{}-I \bigl(\lambda_{i}(p)\leq u_{ji}(p), \ldots, \lambda_{j}(p)\leq u_{jj}(p) \bigr) \bigr) \bigr)\biggr| \\ &{}+ \biggl|\sum_{1\leq i< j\leq n}d_{i}d_{j} \operatorname{Cov} \bigl(I \bigl(\xi_{1}(p)\leq u_{i1}(p), \ldots, \xi_{i}(p)\leq u_{ii}(p) \bigr) \\ &{}-I \bigl(\lambda_{1}(p)\leq u_{i1}(p), \ldots, \lambda_{i}(p)\leq u_{ii}(p) \bigr),I \bigl( \xi_{i}(p)\leq u_{ji}(p), \ldots, \xi_{j}(p)\leq u_{jj}(p) \bigr) \\ &{}-I \bigl(\lambda_{i}(p)\leq u_{ji}(p), \ldots, \lambda_{j}(p)\leq u_{jj}(p) \bigr) \bigr) \biggr| \\ =:&J_{21}+J_{22}. \end{aligned}$$
(2.22)

For \(J_{21}\), we can get

$$\begin{aligned} J_{21} \leq&\sum_{1\leq i< j\leq n}d_{i}d_{j} \bigl\{ \bigl|\operatorname{Cov} \bigl(I \bigl(\xi_{1}(p)\leq u_{i1}(p), \ldots, \xi_{i}(p)\leq u_{ii}(p) \bigr) \\ &{}-I \bigl(\lambda_{1}(p)\leq u_{i1}(p), \ldots, \lambda_{i}(p)\leq u_{ii}(p) \bigr),I \bigl( \xi_{1}(p)\leq u_{j1}(p), \ldots, \xi_{j}(p)\leq u_{jj}(p) \bigr) \\ &{}-I \bigl(\xi_{i}(p)\leq u_{ji}(p), \ldots, \xi_{j}(p)\leq u_{jj}(p) \bigr) \bigr)\bigr| \\ &{}+ \bigl|\operatorname{Cov} \bigl(I \bigl(\xi_{1}(p)\leq u_{i1}(p), \ldots, \xi _{i}(p)\leq u_{ii}(p) \bigr) \\ &{}-I \bigl(\lambda_{1}(p)\leq u_{i1}(p), \ldots, \lambda_{i}(p)\leq u_{ii}(p) \bigr), I \bigl( \lambda_{1}(p)\leq u_{j1}(p), \ldots, \lambda_{j}(p) \leq u_{jj}(p) \bigr) \\ &{}-I \bigl(\lambda_{i}(p)\leq u_{ji}(p), \ldots, \lambda_{j}(p)\leq u_{jj}(p) \bigr) \bigr) \bigr| \bigr\} \\ \leq&2\sum_{1\leq i< j\leq n}d_{i}d_{j} \bigl\{ \mathbb{E}\bigl|I \bigl(\xi _{1}(p)\leq u_{j1}(p), \ldots, \xi_{j}(p)\leq u_{jj}(p) \bigr) \\ &{}-I \bigl(\xi_{i}(p)\leq u_{ji}(p), \ldots, \xi_{j}(p)\leq u_{jj}(p) \bigr) \bigr| \\ &{} +\mathbb{E} \bigl|I \bigl(\lambda_{1}(p)\leq u_{j1}(p), \ldots, \lambda _{j}(p)\leq u_{jj}(p) \bigr) \\ &{}-I \bigl(\lambda_{i}(p)\leq u_{ji}(p), \ldots, \lambda_{j}(p)\leq u_{jj}(p) \bigr) \bigr| \bigr\} \\ =&2\sum_{1\leq i< j\leq n}d_{i}d_{j} \bigl\{ \mathbb{P} \bigl(\xi _{i}(p)\leq u_{ji}(p), \ldots, \xi_{j}(p)\leq u_{jj}(p) \bigr) \\ &{}-\mathbb{P} \bigl(\xi_{1}(p)\leq u_{j1}(p), \ldots, \xi_{j}(p)\leq u_{jj}(p) \bigr) +\mathbb{P} \bigl( \lambda_{i}(p)\leq u_{ji}(p), \ldots, \lambda_{j}(p) \leq u_{jj}(p) \bigr) \\ &{} -\mathbb{P} \bigl(\lambda_{1}(p)\leq u_{j1}(p), \ldots, \lambda_{j}(p)\leq u_{jj}(p) \bigr) \bigr\} \\ \leq&2\sum_{1\leq i< j\leq n}d_{i}d_{j} \bigl\{ \bigl|\mathbb {P} \bigl(\xi_{i}(p)\leq u_{ji}(p), \ldots, \xi_{j}(p)\leq u_{jj}(p) \bigr) \\ &{}-\mathbb{P} \bigl(\lambda_{i}(p)\leq u_{ji}(p), \ldots, \lambda_{j}(p)\leq u_{jj}(p) \bigr) \bigr| \\ &{}+ \bigl|\mathbb{P} \bigl(\xi_{1}(p)\leq u_{j1}(p), \ldots, \xi_{j}(p)\leq u_{jj}(p) \bigr) \\ &{}-\mathbb{P} \bigl( \lambda_{1}(p)\leq u_{j1}(p), \ldots, \lambda_{j}(p) \leq u_{jj}(p) \bigr) \bigr| \\ &{}+2 \bigl|\mathbb{P} \bigl(\lambda_{i}(p)\leq u_{ji}(p), \ldots, \lambda_{j}(p)\leq u_{jj}(p) \bigr) \\ &{}-\mathbb{P} \bigl(\lambda_{1}(p)\leq u_{j1}(p), \ldots, \lambda_{j}(p)\leq u_{jj}(p) \bigr)\bigr| \bigr\} . \end{aligned}$$
(2.23)

By Lemma 3 and (2.17), for \(\alpha>0\), we have

$$\begin{aligned} &2\sum_{1\leq i< j\leq n}d_{i}d_{j} \bigl\{ \bigl|\mathbb{P} \bigl(\xi_{i}(p)\leq u_{ji}(p), \ldots, \xi_{j}(p)\leq u_{jj}(p) \bigr) \\ &\qquad{}-\mathbb{P} \bigl( \lambda_{i}(p)\leq u_{ji}(p), \ldots, \lambda_{j}(p) \leq u_{jj}(p) \bigr) \bigr| \\ & \qquad{}+ \bigl|\mathbb{P} \bigl(\xi_{1}(p)\leq u_{j1}(p), \ldots, \xi_{j}(p)\leq u_{jj}(p) \bigr)-\mathbb{P} \bigl( \lambda_{1}(p)\leq u_{j1}(p), \ldots, \lambda_{j}(p) \leq u_{jj}(p) \bigr)\bigr| \bigr\} \\ &\quad\ll\sum_{1\leq i< j\leq n}d_{i}d_{j}(\ln D_{j})^{-(1+\varepsilon)}=\sum_{j=1}^{n} \frac{\exp(\ln^{\alpha}j)}{j(\ln D_{j})^{1+\varepsilon}}\sum_{i=1}^{j}d_{i} \\ &\quad=\sum_{j=1}^{n}\frac{\exp(\ln^{\alpha}j)}{j(\ln D_{j} )^{1+\varepsilon}}D_{j} \ll\sum_{j=1}^{n}\frac{\exp(2\ln^{\alpha}j)(\ln j)^{1-\alpha}}{j(\ln j)^{(1+\varepsilon)\alpha}} \\ &\quad\sim \int^{n}_{e}\frac{\exp(2\ln^{\alpha}x)(\ln x)^{1-\alpha}}{(\ln x)^{\alpha+\alpha\varepsilon}}\,\mathrm{d}\ln x \\ &\quad= \int^{\ln n}_{1}\exp\bigl(2y^{\alpha} \bigr)y^{1-2\alpha-\alpha\varepsilon}\,\mathrm{d}y \\ &\quad\sim \int^{\ln n}_{1} \biggl(\exp\bigl(2y^{\alpha} \bigr)y^{1-2\alpha-\alpha \varepsilon}+\frac{2-3\alpha-\alpha\varepsilon}{2\alpha}\exp\bigl(2y^{\alpha } \bigr)y^{1-3\alpha-\alpha\varepsilon} \biggr)\,\mathrm{d}y \\ &\quad=\frac{1}{2\alpha}\exp\bigl(2y^{\alpha}\bigr)y^{2-3\alpha-\alpha\varepsilon} \Big|^{\ln n}_{1} \\ &\quad\ll\exp\bigl(2\ln^{\alpha}n\bigr) (\ln n)^{2-3\alpha-\alpha\varepsilon}\ll \frac{D_{n}^{2}}{(\ln D_{n})^{\frac{\alpha+\alpha\varepsilon}{\alpha}}} =\frac{D_{n}^{2}}{(\ln D_{n})^{1+\varepsilon}}. \end{aligned}$$
(2.24)

By (2.11)-(2.15), for \(p=1,2,\ldots,d\), we obtain

$$\begin{aligned} &\sum_{1\leq i< j\leq n}d_{i}d_{j} \bigl| \mathbb{P} \bigl(\lambda_{i}(p)\leq u_{ji}(p), \ldots, \lambda_{j}(p)\leq u_{jj}(p) \bigr) \\ &\qquad{}-\mathbb{P} \bigl(\lambda_{1}(p)\leq u_{j1}(p), \ldots, \lambda_{j}(p)\leq u_{jj}(p) \bigr) \bigr| \\ &\quad=\sum_{1\leq i< j\leq n}d_{i}d_{j}\bigl| \mathbb{P} \bigl(\zeta_{i}(p)\leq(1-\rho _{j})^{-1/2} \bigl(u_{ji}(p)-\rho_{j}^{1/2}\zeta\bigr), \ldots, \\ &\qquad\zeta_{j}(p)\leq(1-\rho_{j})^{-1/2} \bigl(u_{jj}(p)-\rho_{j}^{1/2}\zeta\bigr) \bigr) \\ &\qquad{}-\mathbb{P} \bigl({\zeta_{1}(p)\leq(1-\rho _{j})^{-1/2} \bigl(u_{j1}(p)-\rho_{j}^{1/2}\zeta\bigr),} \ldots, \\ & \qquad\zeta_{j}(p)\leq(1-\rho_{j})^{-1/2} \bigl(u_{jj}(p)-\rho_{j}^{1/2}\zeta \bigr) \bigr)\bigr| \\ &\quad=\sum_{1\leq i< j\leq n}d_{i}d_{j} \int_{\mathbb{R}} \bigl(\mathbb{P} \bigl(\zeta_{i}(p)\leq(1- \rho _{j})^{-1/2}\bigl(u_{ji}(p)-\rho_{j}^{1/2}z \bigr), \ldots, \\ &\qquad\zeta_{j}(p)\leq(1-\rho_{j})^{-1/2} \bigl(u_{jj}(p)-\rho_{j}^{1/2}z\bigr) \bigr) \\ &\qquad{}-\mathbb{P} \bigl(\zeta_{1}(p)\leq(1-\rho _{j})^{-1/2} \bigl(u_{j1}(p)-\rho_{j}^{1/2}z\bigr), \ldots, \\ & \qquad\zeta_{j}(p)\leq(1-\rho_{j})^{-1/2} \bigl(u_{jj}(p)-\rho_{j}^{1/2}z\bigr) \bigr) \bigr)\, \mathrm{d}\Phi(z) \\ &\quad=\sum_{1\leq i< j\leq n}d_{i}d_{j} \biggl( \int_{\mathbb{R}} \bigl(\Phi^{j-i} \bigl((1-\rho _{j})^{-1/2}\bigl({ \mathbf{u}}_{ji}(p)- \rho_{j}^{1/2}z\bigr) \bigr) \\ &\qquad{}-\Phi^{j} \bigl((1- \rho_{j})^{-1/2}\bigl({\mathbf {u}}_{j1}(p)- \rho_{j}^{1/2}z\bigr) \bigr) \bigr)\,\mathrm{d}\Phi(z) \biggr) \\ &\quad\ll\sum_{1\leq i< j\leq n}d_{i}d_{j} \int_{\mathbb{R}}\frac{i}{j}\,\mathrm{d}\Phi(z)=\sum _{1\leq i< j\leq n}d_{i}d_{j}\frac{i}{j} \\ &\quad =\frac{D_{n}^{2}}{(\ln D_{n})^{1+\varepsilon}}. \end{aligned}$$
(2.25)

By (2.23)-(2.25), we have

$$ J_{21}\ll\frac{D_{n}^{2}}{(\ln D_{n})^{1+\varepsilon}}. $$
(2.26)

For \(J_{22}\), noting that \(\{\xi_{i}(p):i\geq1\}\) and \(\{\lambda_{i}(p):i\geq1\}\) are independent, by Lemma 3 and (2.24), we get

$$\begin{aligned} J_{22} =&\biggl|\sum_{1\leq i< j\leq n}d_{i}d_{j} \bigl\{ \operatorname{Cov} \bigl(I \bigl(\xi_{1}(p)\leq u_{i1}(p), \ldots, \xi_{i}(p)\leq u_{ii}(p) \bigr), \\ & I \bigl(\xi_{i}(p)\leq u_{ji}(p), \ldots, \xi_{j}(p)\leq u_{jj}(p) \bigr) \bigr) \\ &{}+\operatorname{Cov} \bigl(I \bigl(\lambda_{1}(p)\leq u_{i1}(p), \ldots, \lambda_{i}(p)\leq u_{ii}(p) \bigr), \\ &I \bigl(\lambda_{i}(p)\leq u_{ji}(p), \ldots, \lambda_{j}(p)\leq u_{jj}(p) \bigr) \bigr) \bigr\} \biggr| \\ =& \biggl|\sum_{1\leq i< j\leq n}d_{i}d_{j} \bigl\{ \mathbb{P} \bigl(\xi_{1}(p)\leq u_{i1}(p), \ldots, \xi_{i}(p)\leq u_{ii}(p),\xi_{i}(p)\leq u_{ji}(p), \ldots, \xi_{j}(p)\leq u_{jj}(p) \bigr) \\ &{} -\mathbb{P} \bigl(\lambda_{1}(p)\leq u_{i1}(p), \ldots, \lambda_{i}(p)\leq u_{ii}(p),\lambda_{i}(p)\leq u_{ji}(p), \ldots, \lambda_{j}(p)\leq u_{jj}(p) \bigr) \\ &{}-\mathbb{P} \bigl(\xi_{1}(p)\leq u_{i1}(p), \ldots, \xi_{i}(p)\leq u_{ii}(p) \bigr)\mathbb{P} \bigl( \xi_{i}(p)\leq u_{ji}(p), \ldots, \xi_{j}(p)\leq u_{jj}(p) \bigr) \\ &{}-\mathbb{P} \bigl(\lambda_{1}(p)\leq u_{i1}(p), \ldots, \lambda_{i}(p)\leq u_{ii}(p) \bigr)\mathbb{P} \bigl( \lambda_{i}(p)\leq u_{ji}(p), \ldots, \lambda_{j}(p) \leq u_{jj}(p) \bigr) \bigr\} \biggr| \\ \leq& \sum_{1\leq i< j\leq n}d_{i}d_{j} \bigl\{ \bigl|\mathbb{P} \bigl(\xi_{1}(p)\leq u_{i1}(p), \ldots, \xi_{i}(p)\leq u_{ii}(p),\xi_{i}(p)\leq u_{ji}(p), \ldots, \xi_{j}(p)\leq u_{jj}(p) \bigr) \\ &{} -\mathbb{P} \bigl(\lambda_{1}(p)\leq u_{i1}(p), \ldots, \lambda_{i}(p)\leq u_{ii}(p),\lambda_{i}(p)\leq u_{ji}(p), \ldots, \lambda_{j}(p)\leq u_{jj}(p) \bigr) \bigr| \\ &{}+ \bigl|\mathbb{P} \bigl(\xi_{1}(p)\leq u_{i1}(p), \ldots, \xi_{i}(p)\leq u_{ii}(p) \bigr)-\mathbb{P} \bigl( \lambda_{1}(p)\leq u_{i1}(p), \ldots, \lambda_{i}(p) \leq u_{ii}(p) \bigr)\bigr| \\ &{} +\bigl|\mathbb{P} \bigl(\xi_{i}(p)\leq u_{ji}(p), \ldots, \xi_{j}(p)\leq u_{jj}(p) \bigr)-\mathbb{P} \bigl( \lambda_{i}(p)\leq u_{ji}(p), \ldots, \lambda_{j}(p) \leq u_{jj}(p) \bigr) \bigr| \bigr\} \\ &{}+2 \biggl|\sum_{1\leq i< j\leq n}d_{i}d_{j} \operatorname{Cov} \bigl(I \bigl(\lambda_{1}(p)\leq u_{i1}(p), \ldots, \lambda_{i}(p)\leq u_{ii}(p) \bigr), \\ &I \bigl(\lambda_{i}(p)\leq u_{ji}(p), \ldots, \lambda_{j}(p)\leq u_{jj}(p) \bigr) \bigr) \biggr| \\ \ll&\sum_{1\leq i< j\leq n}d_{i}d_{j}(\ln D_{j})^{-(1+\varepsilon)} \\ &{} + \biggl|\sum_{1\leq i< j\leq n}d_{i}d_{j} \operatorname{Cov} \bigl(I \bigl(\lambda_{1}(p)\leq u_{i1}(p), \ldots, \lambda_{i}(p)\leq u_{ii}(p) \bigr), \\ &I \bigl(\lambda_{i}(p)\leq u_{ji}(p), \ldots, \lambda_{j}(p)\leq u_{jj}(p) \bigr) \bigr)\biggr| \\ \ll&\frac{D_{n}^{2}}{(\ln D_{n})^{1+\varepsilon}}+ \biggl|\sum_{1\leq i< j\leq n}d_{i}d_{j} \operatorname{Cov} \bigl(I \bigl(\lambda_{1}(p)\leq u_{i1}(p), \ldots, \lambda_{i}(p)\leq u_{ii}(p) \bigr), \\ &I \bigl(\lambda_{i}(p)\leq u_{ji}(p), \ldots, \lambda_{j}(p)\leq u_{jj}(p) \bigr) \bigr) \biggr|. \end{aligned}$$
(2.27)

By (2.25), we have

$$\begin{aligned} & \biggl|\sum_{1\leq i< j\leq n}d_{i}d_{j} \operatorname{Cov} \bigl(I \bigl(\lambda_{1}(p)\leq u_{i1}(p), \ldots, \lambda_{i}(p)\leq u_{ii}(p) \bigr), \\ &\qquad I \bigl(\lambda_{i}(p)\leq u_{ji}(p), \ldots, \lambda_{j}(p)\leq u_{jj}(p) \bigr) \bigr)\biggr| \\ &\quad= \biggl|\sum_{1\leq i< j\leq n}d_{i}d_{j} \bigl\{ \operatorname{Cov} \bigl(I \bigl(\lambda_{1}(p)\leq u_{i1}(p), \ldots, \lambda_{i}(p)\leq u_{ii}(p) \bigr), \\ &\qquad I \bigl(\lambda_{i}(p)\leq u_{ji}(p), \ldots, \lambda_{j}(p)\leq u_{jj}(p) \bigr)-I \bigl( \lambda_{1}(p)\leq u_{j1}(p), \ldots, \lambda_{j}(p) \leq u_{jj}(p) \bigr) \bigr) \\ &\qquad{} +\operatorname{Cov} \bigl(I \bigl(\lambda_{1}(p)\leq u_{i1}(p), \ldots, \lambda_{i}(p)\leq u_{ii}(p) \bigr), \\ &\qquad I \bigl(\lambda_{1}(p)\leq u_{j1}(p), \ldots, \lambda_{j}(p)\leq u_{jj}(p) \bigr) \bigr) \bigr\} \biggr| \\ &\quad\leq\sum_{1\leq i< j\leq n}d_{i}d_{j} \mathbb{E} \bigl|I \bigl(\lambda_{1}(p)\leq u_{j1}(p), \ldots, \lambda_{j}(p)\leq u_{jj}(p) \bigr) \\ &\qquad{}-I \bigl(\lambda_{i}(p)\leq u_{ji}(p), \ldots, \lambda_{j}(p)\leq u_{jj}(p) \bigr) \bigr| \\ &\qquad{} + \biggl|\sum_{1\leq i< j\leq n}d_{i}d_{j} \operatorname{Cov} \bigl(I \bigl(\lambda_{1}(p)\leq u_{i1}(p), \ldots, \lambda_{i}(p)\leq u_{ii}(p) \bigr), \\ &\qquad I \bigl(\lambda_{1}(p)\leq u_{j1}(p), \ldots, \lambda_{j}(p)\leq u_{jj}(p) \bigr) \bigr) \biggr| \\ &\leq\sum_{1\leq i< j\leq n}d_{i}d_{j} \bigl(\mathbb{P} \bigl(\lambda_{i}(p)\leq u_{ji}(p), \ldots, \lambda_{j}(p)\leq u_{jj}(p) \bigr) \\ &\qquad{}-\mathbb{P} \bigl(\lambda_{1}(p)\leq u_{j1}(p), \ldots, \lambda_{j}(p)\leq u_{jj}(p) \bigr) \bigr) \\ &\qquad{} +\operatorname{Var} \Biggl(\sum_{i=1}^{n}d_{i}I \bigl(\lambda_{1}(p)\leq u_{i1}(p), \ldots, \lambda_{i}(p)\leq u_{ii}(p) \bigr) \Biggr) \\ &\qquad{}+\sum_{i=1}^{n}d_{i}^{2} \operatorname{Var} \bigl(I \bigl(\lambda_{1}(p)\leq u_{i1}(p), \ldots, \lambda_{i}(p)\leq u_{ii}(p) \bigr) \bigr) \\ &\quad\ll\frac{D_{n}^{2}}{(\ln D_{n})^{1+\varepsilon}}+\operatorname{Var} \Biggl(\sum _{i=1}^{n}d_{i}I \bigl( \lambda_{1}(p)\leq u_{i1}(p), \ldots, \lambda_{i}(p) \leq u_{ii}(p) \bigr) \Biggr). \end{aligned}$$
(2.28)

By (2.12)-(2.20), we have

$$ \operatorname{Var} \Biggl(\sum_{i=1}^{n}d_{i}I \bigl(\lambda_{1}(p)\leq u_{i1}(p), \ldots, \lambda_{i}(p)\leq u_{ii}(p) \bigr) \Biggr)\ll \frac{D_{n}^{2}}{(\ln D_{n})^{1+\varepsilon}}. $$
(2.29)

Together with (2.28) and (2.29), we obtain

$$\begin{aligned} & \biggl|\sum_{1\leq i< j\leq n}d_{i}d_{j} \operatorname{Cov} \bigl(I \bigl(\lambda_{1}(p)\leq u_{i1}(p), \ldots, \lambda_{i}(p)\leq u_{ii}(p) \bigr), \\ &\quad I \bigl(\lambda_{i}(p)\leq u_{ji}(p), \ldots, \lambda_{j}(p)\leq u_{jj}(p) \bigr) \bigr) \biggr| \ll \frac{D_{n}^{2}}{(\ln D_{n})^{1+\varepsilon}}. \end{aligned}$$
(2.30)

Hence by (2.27) and (2.30), we have

$$ J_{22}\ll\frac{D_{n}^{2}}{(\ln D_{n})^{1+\varepsilon}}. $$
(2.31)

By (2.21), (2.22), (2.26), and (2.31)), for \(\alpha>0\), we get

$$ L_{2}\ll\frac{D_{n}^{2}}{(\ln D_{n})^{1+\varepsilon}}. $$
(2.32)

Thus (2.10)-(2.32) together establish (2.9). The proof is completed. □

Proof of Theorem 2

According to Lin et al. [1], we have

$$\begin{aligned} &\lim_{n\rightarrow\infty}\mathbb{P} \Bigl(a_{k}\Bigl(\max _{1\leq i\leq t_{k}} \xi_{i}(p)-b_{k}\Bigr)\leq {\mathbf {x}} \Bigr) \\ &\quad= \int_{\mathbb{R}}\exp\bigl(-t\mathrm{e}^{-x(p)-r+\sqrt{2r}z}\bigr)\, \mathrm{d}\Phi(z),\quad \mbox{for } p=1,2,\ldots,d. \end{aligned}$$
(2.33)

By similar methods to the ones used to prove Lemma 4, we can prove

$$ \lim_{n\rightarrow\infty}\mathbb{P} \Bigl(\mathbf{a} _{k}\Bigl( \max_{1\leq i\leq t_{k}}\boldsymbol {\xi}_{i}-\mathbf{b}_{k} \Bigr)\leq \mathbf{x} \Bigr)=\prod^{d}_{p=1} \int_{\mathbb{R}}\exp\bigl(-t\mathrm{e}^{-x(p)-r+\sqrt{2r}z}\bigr)\, \mathrm{d}\Phi(z). $$
(2.34)

Note \(\lim_{n\rightarrow\infty} \frac{t_{n}}{n}=t\) (\(t>0\)) and we have Lemma 2, so the remainder of the proof is similar to that of Theorem 1. We thus omit it. □