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Convergence of exceedance point processes of normal sequences with a seasonal component and its applications

Journal of Inequalities and Applications20162016:141

https://doi.org/10.1186/s13660-016-1082-4

Received: 22 April 2015

Accepted: 12 May 2016

Published: 26 May 2016

Abstract

In this paper, we prove that, under some mild conditions, a time-normalized point process of exceedances by a nonstationary and strongly dependent normal sequence with a seasonal component converges in distribution to the in plane Cox process. As an application of the convergence result, we deduce two important joint limit distributions for the order statistics.

Keywords

in plane Cox processexceedance point processnonstationary sequences with a seasonal componentstrongly dependent normal sequences kth maxima

MSC

60F0562E2062F1262M10

1 Introduction

Let \(\{X_{i}, i\geq1\}\) be a standardized normal sequence with correlation coefficient \(r_{ij}=\operatorname{Cov}(X_{i}, X_{j})\) satisfying the conventional assumption that \(r_{ij}\rightarrow0\) and \(r_{ij}\log(|i-j|)\rightarrow\gamma\) as \(j-i\rightarrow+\infty\). Normal sequences are weakly dependent if \(\gamma=0\), strongly dependent if \(0<\gamma<\infty\) and stationary if \(r_{ij}\) are related to \(|i-j|\) only, and nonstationary otherwise. Denote by \(M_{n}^{(k)}\) the kth maximum of \(\{X_{i},1\leq i\leq n\}\), whose location is denoted \(L_{n}^{(k)}\) and may vary among \(\{1,\ldots,n\}\). Leadbetter et al. [1] considered a stationary weakly dependent normal sequence \(\{X_{i}, i\geq1\}\) and obtained the asymptotic joint probability distribution of \(M_{n}^{(1)}\) and \(M_{n}^{(2)}\) and even that of \(M_{n}^{(2)}\) and \(L_{n}^{(2)}\). Mittal and Ylvisaker [2] proved that if \(\{X_{i}, i\geq1\}\) is stationary and strongly dependent, then \(M^{(1)}_{n}\) (also called the maximum of the sequence) after normalization converges in distribution to the convolution of \(\exp(-e^{-x})\) and a normal distribution function. More recent results for maxima of stationary normal sequences can be found in Ho and Hsing [3], Tan and Peng [4], and Hashorva et al. [5], among others. Meanwhile, some literature was devoted to study the maxima of nonstationary normal sequences; see Horowitz [6] and Leadbetter et al. [1] for the weakly dependence case and Zhang [7], Lin et al. [8], and Tan and Yang [9] for the strongly dependence case.

In particular, Leadbetter et al. [1] developed an important tool, the weak convergence of exceedance point processes, which is crucial to study the joint asymptotic distributions of some extremes of sequences. Due to its importance, many authors further studied the asymptotic behavior of exceedance point processes under different conditions; see Piterbarg [10], Hu et al. [11], Falk et al. [12], Peng et al. [13], Hashorva et al. [14] Wiśniewski [15], Lin et al. [8], Lin et al. [16], and the references therein.

In the paper, we consider \(\{X_{i}=Y_{i}+m_{i}, i\geq1\}\) where \(Y_{i}\) is a standardized nonstationary and strongly dependent normal sequence and \(m_{i}\) is a trend or seasonal component. Define \(\eta_{n}(t)\), \(t\in[0,1]\), as a continuous stochastic function such that \(\eta_{n}(t)\) is linear on \([(i-1)/n,i/n]\), \(i=1,2,\ldots,n\), and has the value \(X_{i}\) at the point \(i/n\) (\(\eta_{n}(0)=0\)). A similar definition can also be found in Leadbetter et al. [1]. A vector point process \(N_{n}\) formed by exceedances of the levels \(u^{(1)}_{n}, u^{(2)}_{n} ,\ldots, u^{(r)}_{n}\) by the stochastic function is called the time-normalized one since we use ‘\(j/n\)’ and set \(t\in[0,1]\) in the definition of \(\eta_{n}(t)\). In the sequel, for convenience, the expression ‘exceedances by \(\{X_{j},j\geq1\}\)’ stands for ‘exceedances by \(\eta(t)\).’ We prove that the time-normalized point process \(N_{n}\) converges in distribution to the in plane Cox process defined in Lin et al. [16] and extend the results in Lin et al. [16] to the case of more general normal sequences.

The remainder of the paper is structured as follows. In Section 2, we present the notation and main results. Proofs of the main results are postponed to Section 3. Throughout the paper, C stands for a constant that may vary from line to line, and ‘→’ stands for the convergence in distribution as \(n\rightarrow\infty\).

2 Notation and main results

Let \(\{X_{i}=Y_{i}+m_{i}, i\geq1\}\) be a standardized normal sequence plus a seasonal component with the correlation coefficient of \(\{Y_{i}, i\geq1\}\) and seasonal component satisfying the following:
$$\begin{aligned} &\sup\bigl\{ \vert r_{ij}\vert , i\neq j\bigr\} < 1 \quad \mbox{and}\quad r_{ij}\log(j-i)\rightarrow\gamma\in(0,\infty)\quad \mbox{as } j-i\rightarrow+\infty, \end{aligned}$$
(2.1)
$$\begin{aligned} &\beta_{n}=\max_{1\leq i\leq n}\vert m_{i}\vert =o\bigl((\log n)^{1/2}\bigr)\quad \mbox{as } n \rightarrow+\infty, \end{aligned}$$
(2.2)
$$\begin{aligned} &\frac{1}{n}\sum^{n}_{i=1} \exp\biggl(a_{n}^{*}\bigl(m_{i}-m_{n}^{*} \bigr)-\frac {1}{2}\bigl(m_{i}-m_{n}^{*} \bigr)^{2}\biggr)\rightarrow1 \quad \mbox{as } n\rightarrow\infty, \end{aligned}$$
(2.3)
where \(a_{n}^{*}=(2\log n)^{1/2}-\log\log n/2((2\log n)^{1/2})\), and \(m_{n}^{*}\) is a sequence of constants such that \(|m_{n}^{*}|\leq\beta_{n}\). Condition (2.3) is the same as condition (6.2.2) in Leadbetter et al. [1]. Throughout, the standardized constants \(a_{n}\) and \(b_{n}\) are defined by
$$\begin{aligned} a_{n}=(2\log n)^{1/2}, \qquad b_{n}=a_{n}-(2a_{n})^{-1}(\log\log n+ \log4\pi). \end{aligned}$$
(2.4)

Before presenting the main results, we first give the definition of the in plane Cox process.

Definition 2.1

Let \(\{\sigma_{1j}, j=1,2,\ldots\}\) be the points of a Cox process \(N^{(r)}\) on \(L_{r}\) with (stochastic) intensity \(\exp(-x_{r}-\gamma+\sqrt{2\gamma}\zeta)\), where ζ is a standard normal random variable, \(x_{r}\) is a constant corresponding to the \(N^{(r)}\), and \(L_{r}\) is the in plane fixed horizontal line on which exceedances are represented as points. \(N^{(r)}\) has the distribution characterized as follows:
$$ \begin{aligned}[b] P \Biggl(\bigcap^{I}_{i=1} \bigl\{ N^{(r)}(B_{i})=k_{i}\bigr\} \Biggr)={}& \int^{\infty }_{-\infty}\prod^{I}_{i=1} \biggl(\frac{(m(B_{i})\exp(-x_{r}-\gamma+\sqrt{2\gamma }z))^{k_{i}}}{k_{i}!} \\ & {}\cdot\exp\bigl(-m(B_{i})e^{-x_{r}-\gamma+\sqrt{2\gamma}z}\bigr) \biggr)\phi(z) \,dz, \end{aligned} $$
(2.5)
where \(B_{i}\) are Borel sets, and \(m(\cdot)\) is the Lebesgue measure. Let \(\beta_{j}, j=1,2,\dots\) , be independent and identically distributed (i.i.d.) random variables, independent also of the Cox process on \(L_{r}\), taking the values \(1,2,\ldots,r\) with conditional probabilities
$$P(\beta_{j}=s|\zeta=z)= \textstyle\begin{cases} (\tau_{r-s+1}-\tau_{r-s})/\tau_{r}& \mbox{for }s=1,2,\ldots,r-1,\\ \tau_{1}/\tau_{r}&\mbox{for }s=r, \end{cases} $$
that is, \(P(\beta_{j}\geq s|\zeta=z)=\tau_{r-s+1}/\tau_{r}\) for \(s=1,2,\ldots,r\), where \(\tau_{i}=e^{-x_{i}-\gamma+\sqrt{2\gamma}z}\), \(i=1, 2,\ldots,r\). For each j, placing points \(\sigma_{2j}, \sigma_{3j},\ldots, \sigma_{\beta_{j}j}\) on \(\beta_{j}-1\) lines \(L_{r-1}, L_{r-2},\ldots, L_{r-\beta_{j}+1}\), vertically above \(\sigma_{1j}\), we can obtain an in plane Cox process N. Specifically, the conditional probability that a point appears on \(L_{r-1}\) above \(\sigma_{1j}\) is \(P(\beta_{j}\geq 2|\zeta=z)=\tau_{r-1}/\tau_{r}\), and the deletions are conditionally independent, so that \(N^{(r-1)}\) is obtained as a conditionally independent thinning of the Cox process \(N^{(r)}\). Similarly, the other \(N^{(k)}\), \(1\leq k\leq r-2\), can be constructed.

In Theorem 2.1, we study a vector point process \(N_{n}=(N_{n}^{(1)},N_{n}^{(2)},\ldots,N_{n}^{(r)})\) that arises when \(\{X_{i},1\leq i\leq n\}\) exceeds the levels \(u_{n}^{(1)},u_{n}^{(2)},\ldots,u_{n}^{(r)}\), the structure of which is the same as that of the exceedance process on pp.111-112 in Leadbetter et al. [1]. We record the exceedance points corresponding to the levels \(u_{n}^{(1)},u_{n}^{(2)},\ldots,u_{n}^{(r)}\) on fixed horizontal lines \(L_{1}, L_{2}, \ldots, L_{r}\) in the plane.

Theorem 2.1

Suppose that \(\{X_{i}=Y_{i}+m_{i}, i\geq1\}\) satisfies conditions (2.1)-(2.3), and let \(u_{n}^{(k)}=x_{k}/a_{n}+b_{n}+m_{n}^{*}\) (\(1\leq k\leq r\)) satisfy \(u_{n}^{(1)}\geq u_{n}^{(2)} \geq\cdots\geq u_{n}^{(r)}\). Then the time-normalized exceedance point process \(N_{n}\) of levels \(u^{(1)}_{n}, u^{(2)}_{n}, \ldots,u^{(r)}_{n}\) by \(\{X_{i}, 1\leq i\leq n\}\) converges in distribution to the before-mentioned in plane Cox process.

Corollary 2.1

Let \(\{X_{i}, i\geq1\}\) satisfy the conditions of Theorem 2.1. Let \(B_{1}, B_{2}, \ldots, B_{s}\) be Borel subsets of the unit interval whose boundaries have zero Lebesgue measures. Then, for integers \(m_{j}^{(k)}\),
$$\begin{aligned} &P\bigl(N_{n}^{(k)}(B_{j})=m_{j}^{(k)}, j=1,2,\ldots, s; k=1,2,\ldots,r\bigr) \\ &\quad \rightarrow P\bigl(N^{(k)}(B_{j})=m_{j}^{(k)}, j=1,2,\ldots, s; k=1,2,\ldots,r\bigr). \end{aligned}$$

Theorem 2.2

Suppose that the levels \(u_{n}^{(k)}\) (\(1\leq k\leq r\)) satisfy
$$\begin{aligned} P\Bigl(\max_{1\leq i\leq n}X_{i}\leq u_{n}^{(k)} \Bigr)\rightarrow \int^{+\infty}_{-\infty}\exp\bigl(-e^{-x_{k}-\gamma +\sqrt{2\gamma}z}\bigr) \phi(z)\,dz \quad \textit{as } n\rightarrow\infty, \end{aligned}$$
with \(u_{n}^{(1)}\geq u_{n}^{(2)} \geq\cdots\geq u_{n}^{(r)}\). Let \(S_{n}^{(k)}\) be the numbers of exceedances of \(u_{n}^{(k)}\) by \(\{X_{i}, 1\leq i\leq n\}\) that satisfy the conditions of Theorem  2.1. Then, for \(k_{1}\geq0\), \(k_{2}\geq0,\ldots, k_{r}\geq0\),
$$\begin{aligned} \begin{aligned}[b] & P\bigl(S^{(1)}_{n}=k_{1},S^{(2)}_{n}=k_{1}+k_{2}, \ldots ,S^{(r)}_{n}=k_{1}+k_{2}+ \cdots+k_{r}\bigr) \\ & \quad \longrightarrow \frac{\tau_{1}^{k_{1}}(\tau_{2}-\tau_{1})^{k_{2}}\cdots(\tau_{r}-\tau _{r-1})^{k_{r}}}{k_{1}!k_{2}!\cdots k_{r}!} \\ & \qquad\quad {} \cdot \int^{+\infty}_{-\infty}\bigl(\exp(\sqrt{2\gamma}z-\gamma ) \bigr)^{k_{1}+k_{2}+\cdots+k_{r}}\cdot \exp\bigl(-e^{-x_{k}-\gamma+\sqrt{2\gamma}z}\bigr)\phi(z)\,dz. \end{aligned} \end{aligned}$$
(2.6)

Theorem 2.3

Let \(\{X_{i}, i\geq1\}\) be a normal sequence satisfying the conditions of Theorem  2.1. Let \(u^{(k)}_{n}=x_{k}/a_{n}+b_{n}+m_{n}^{*}\). Then, for \(x_{1}>x_{2}\), as \(n\rightarrow\infty\),
$$\begin{aligned} \begin{aligned}[b] & P\bigl(a_{n}\bigl(M_{n}^{(1)}-b_{n}-m_{n}^{*} \bigr)\leq x_{1},a_{n}\bigl(M_{n}^{(2)}-b_{n}-m_{n}^{*} \bigr)\leq x_{2}\bigr) \\ & \quad \longrightarrow \int^{+\infty}_{-\infty} \bigl(\exp(-x_{2}-\gamma+ \sqrt{2\gamma}z)-\exp(-x_{1}-\gamma+\sqrt{2\gamma }z)+1\bigr) \\ & \qquad\quad {} \times\exp\bigl(-e^{-x_{2}-\gamma+\sqrt{2\gamma}z}\bigr)\phi(z)\,dz \end{aligned} \end{aligned}$$
(2.7)
and
$$ P \biggl(\frac{1}{n}L_{n}^{(2)}\leq t,a_{n}\bigl(M_{n}^{(2)}-b_{n}-m_{n}^{*} \bigr)\leq x \biggr)\longrightarrow \int^{x}_{-\infty}H(y,t)\,dy, $$
(2.8)
where
$$\begin{aligned} H(y,t) =& \int^{+\infty}_{-\infty}(1-t)\exp(-y-\gamma+\sqrt{2\gamma}z) \exp\bigl(-(1-t)e^{-y-\gamma+\sqrt{2\gamma}z}\bigr)\phi(z)\,dz \\ & {}\cdot \int^{+\infty}_{-\infty}t\exp(-y-\gamma+\sqrt{2\gamma}z) \exp \bigl(-te^{-y-\gamma+\sqrt{2\gamma}z}\bigr)\phi(z)\,dz \\ &{} + \int^{+\infty}_{-\infty}\exp\bigl(-(1-t)e^{-y-\gamma+\sqrt{2\gamma}z}\bigr) \phi (z)\,dz \\ & {}\cdot \int^{+\infty}_{-\infty}t^{2}\exp(-2y-2\gamma+2 \sqrt{2\gamma}z) \exp\bigl(-te^{-y-\gamma+\sqrt{2\gamma}z}\bigr)\phi(z)\,dz. \end{aligned}$$

3 The proofs of main results

The proof of Theorem 2.1 will use the famous Berman inequality, which was first presented by Slepian [17] and Berman [18] and then polished up by Li and Shao [19]. For the latest results related to Berman’s inequality, we refer the reader to Hashorva and Weng [20] and Lu and Wang [21]. The upper bound of Berman’s inequality gives an estimate of the difference between two standardized n-dimensional distribution functions by a convenient function of their covariances. According to Hashorva and Weng [20], some results for normal sequences may be extended to nonnormal cases. The proof of Theorem 2.1 also depends on the following lemma of Zhang [7].

Lemma 3.1

Suppose that \(\{X_{i}, i\geq1\}\) is a standardized normal sequence with correlation coefficient \(r_{ij}\) satisfying (2.1). Define \(u_{n}=u_{n}(x)=x/a_{n}+b_{n}\) and \(\rho_{n}=\gamma/\log n\). Then
  1. (i)

    \(r_{ij}\rightarrow0\) as \(j-i\rightarrow+\infty\),

     
  2. (ii)

    \(\sum_{1\leq i< j\leq nb}|r_{ij}-\rho_{n}|\exp (-\frac{u_{n}^{2}}{1+w_{ij}} )\rightarrow 0\) as \(n\rightarrow+\infty\),

     
where \(0< b<+\infty\) and \(w_{ij}=\max\{|r_{ij}|,\rho_{n}\}\).

Proof of Theorem 2.1

It is sufficient to show that, as n goes to ∞,
  1. (a)

    \(E(N_{n}(B))\rightarrow E(N(B))\) for all sets B of the form \((c,d]\times(r,\delta]\), \(r<\delta\), \(0< c< d\), where \(d\leq1\), and \(E(\cdot)\) is the expectation, and

     
  2. (b)

    \(P(N_{n}(B)=0)\rightarrow P(N(B)=0)\) for all sets B that are finite unions of disjoint sets of this form.

     
First, consider (a). If \(B=(c,d]\times(r,\delta]\) intersects any of the lines, suppose that these are \(L_{s},L_{s+1},\ldots,L_{t}\) (\(1\leq s< t\leq r\)). Then
$$N_{n}(B)=\sum^{t}_{k=s}N_{n}^{(k)}\bigl((c,d]\bigr), \qquad N(B)=\sum^{t}_{k=s}N^{(k)}\bigl((c,d]\bigr), $$
and the number of points \(j/n\) in \((c,d]\) is \(([nd]-[nc])\). As in the proof Theorem 5.5.1 on p.113 in Leadbetter et al. [1], we have \(E(N_{n}(B))=([nd]-[nc])\sum^{t}_{k=s}(1-F(u_{n}^{(k)}))\) and
$$1-F\bigl(u_{n}^{(k)}\bigr)=1-\Phi\bigl(u_{n}^{(k)}-m_{j} \bigr),\quad 1\leq j\leq n. $$
Using conditions (2.2) and (2.3) yields
$$\begin{aligned} n\bigl(1-\Phi\bigl(u_{n}^{(k)}-m_{j} \bigr)\bigr)=n\bigl(1-\Phi (x_{k}/a_{n}+b_{n}+m_{n}-m_{j}) \bigr)\sim e^{-x_{k}}\quad \mbox{as } n\rightarrow\infty, \end{aligned}$$
(3.1)
where the last ‘’ attributes to the well-known fact that \(n(1-\Phi(x_{k}/a_{n}+b_{n}))\sim e^{-x}\) implies \(n(1-\Phi(x_{k}/\alpha_{n}+\beta_{n}))\sim e^{-x}\) if \(\alpha_{n}/a_{n}\rightarrow1\) and \((\beta_{n}-b_{n})/a_{n}\rightarrow0\). Thus, we have \(E(N_{n}(B))\sim n(d-c)\sum^{t}_{k=s}(\frac{e^{-x_{k}}}{n}+o(\frac{1}{n}))\rightarrow (d-c)\sum^{t}_{k=s}e^{-x_{k}}\). So, since
$$\begin{aligned} E\bigl(N(B)\bigr) =&\sum^{t}_{k=s}E \bigl((d-c)\exp(-x_{k}-\gamma+\sqrt{2\gamma}\zeta )\bigr) \\ =&\sum^{t}_{k=s}(d-c)e^{-x_{k}-\gamma} \cdot e^{\frac{(\sqrt{2\gamma})^{2}}{2}}=\sum^{t}_{k=s}(d-c)e^{-x_{k}}, \end{aligned}$$
the first result follows. In order to prove (b), we must prove that \(P(N_{n}(B)=0)\rightarrow P(N(B)=0)\), where \(B=\bigcup^{m}_{1}C_{k}\) with disjoint \(C_{k}=(c_{k}, d_{k}]\times(r_{k}, s_{k}]\). It is convenient to neglect any set \(C_{k}\) that does not intersect any of the lines \(L_{1}, L_{2}, \ldots, L_{r}\). Because there are intersections and differences of the intervals \((c_{k}, d_{k}]\), we may write B in the form \(\bigcup^{s}_{k=1}(c_{k}, d_{k}]\times E_{k}\), where \((c_{k}, d_{k}]\) are disjoint, and \(E_{k}\) is a finite union of semiclosed intervals. So we have
$$\begin{aligned} \bigl\{ N_{n}(B)=0\bigr\} =\bigcap ^{s}_{k=1}\bigl\{ N_{n}(F_{k})=0 \bigr\} , \end{aligned}$$
(3.2)
where \(F_{k}=(c_{k},d_{k}]\times E_{k}\). \(L_{l_{k}}\) stands for the lowest \(L_{j}\) intersecting \(F_{k}\). The aforementioned thinning property induces
$$\begin{aligned} \begin{aligned}[b] \bigl\{ N_{n}(F_{k})=0\bigr\} &=\bigl\{ N_{n}^{(l_{k})}\bigl((c_{k},d_{k}]\bigr)=0\bigr\} \\ &= \bigl\{ M_{n}(c_{k},d_{k})\leq u_{n}^{(l_{k})}\bigr\} , \end{aligned} \end{aligned}$$
(3.3)
where \(M_{n}(c_{k},d_{k})\) stands for the maximum of \(\{X_{k}\}\) with index k (\([cn]< k\leq[dn]\)). Calculating the probabilities of (3.2) and (3.3), we obtain
$$\begin{aligned} P\bigl(N_{n}(B)=0\bigr)=P \Biggl(\bigcap ^{s}_{k=1}\bigl\{ M_{n}(c_{k},d_{k}) \leq u_{n}^{(l_{k})}\bigr\} \Biggr). \end{aligned}$$
(3.4)
In order to get the limit of the right-hand side of (3.4), we first prove the following result. Define a sequence \(\{\bar{X}_{i}=\bar{Y}_{i}+m_{i}, i\geq1\}\), where \(\{\bar{Y}_{i}, i\geq1\}\) is a standardized normal sequence with correlation coefficient ρ, and \(\{m_{i}, i\geq1\}\) is the same as that in \(\{X_{i}, i\geq1\}\). \(M_{n}(c,d;\rho)\) stands for the maximum of \(\{\bar{X}_{k}\}\) with index k (\([cn]< k\leq[dn]\)). It is well known that \(M_{n}(c_{1}, d_{1}; \rho),\ldots,M_{n}(c_{k}, d_{k}; \rho)\) have the same distribution as \((1-\rho)^{1/2}M_{n}(c_{1}, d_{1}; 0)+\rho^{1/2}\zeta,\ldots, (1-\rho)^{1/2}M_{n}(c_{k}, d_{k}; 0)+\rho^{1/2}\zeta\), where \(c=c_{1}< d_{1}<\cdots<c_{k}<d_{k}=d\), and ζ is a standard normal variable. In the following, we estimate the bound of
$$\begin{aligned} \Biggl\vert P \Biggl(\bigcap^{s}_{k=1} \bigl\{ M_{n}(c_{k},d_{k})\leq u_{n}^{(l_{k})} \bigr\} \Biggr)-P \Biggl(\bigcap^{s}_{k=1} \bigl\{ M_{n}(c_{k},d_{k},\rho _{n})\leq u_{n}^{(l_{k})}\bigr\} \Biggr)\Biggr\vert , \end{aligned}$$
(3.5)
where \(\rho_{n}=\gamma/\log n\).
Using Berman’s inequality, the bound of (3.5) does not exceed
$$\begin{aligned} \begin{aligned}[b] & \frac{1}{2\pi}\sum \vert r_{ij}- \rho_{n}\vert \bigl(1-\rho_{n}^{2} \bigr)^{-1/2} \exp \biggl(-\frac{\frac{1}{2}((u_{n}^{i})^{2}+(u_{n}^{j})^{2})}{1+\omega _{ij}} \biggr) \\ & \quad \leq C \sum_{1\leq i< j\leq n}\vert r_{ij}-\rho_{n}\vert \exp \biggl(-\frac{\frac {1}{2}((x_{i}/a_{n}+b_{n})^{2}+(x_{j}/a_{n}+b_{n})^{2})}{1+\omega _{ij}} \biggr) \\ &\qquad {}\cdot \exp \bigl(\bigl( (x_{i}/a_{n}+b_{n})\bigl(m_{i}-m_{n}^{*}\bigr)+(x_{i}/a_{n}+b_{n})\bigl(m_{i}-m_{n}^{*}\bigr)\\ &\qquad {} -\frac{1}{2}\bigl(\bigl(m_{n}^{*}-m_{i}\bigr)^{2}+\bigl(m_{n}^{*}-m_{j}\bigr)^{2}\bigr)\bigr)/(1+\omega _{ij}) \bigr), \end{aligned} \end{aligned}$$
(3.6)
where the first sum is carried out over \(i,j\in\bigcup^{s}_{k=1}([c_{k}n],[d_{k}n]]\), \(i< j\), \(u_{n}^{i}\) or \(u_{n}^{j}\) stands for \(u_{n}^{(l_{k})}-m_{i}\) or \(u_{n}^{(l_{k})}-m_{j}\) when \(i\mbox{ or }j \in([c_{k}n],[d_{k}n]]\), and \(\omega_{ij}=\max\{|r_{ij}|,\rho_{n}\}\). Using the proof of Theorem 6.2.1 on p.129 in Leadbetter et al. [1], (2.3) implies that
$$\begin{aligned} \frac{1}{n}\sum^{n}_{i=1} \exp\biggl((x_{i}/a_{n}+b_{n}) \bigl(m_{i}-m_{n}^{*}\bigr)-\frac {1}{2} \bigl(m_{i}-m_{n}^{*}\bigr)^{2}\biggr) \rightarrow1. \end{aligned}$$
Since \(\omega_{ij}\) is bounded, we further get
$$\sup_{1 \leq i< j\leq n}\exp \biggl(\frac{(x_{i}/a_{n}+b_{n})(m_{i}-m_{n}^{*})-\frac {1}{2}(m_{i}-m_{n}^{*})^{2}}{1+\omega_{ij}} \biggr)< C. $$
So, (3.6) does not exceed
$$\begin{aligned} & C \sum_{1\leq i< j\leq n}\vert r_{ij}- \rho_{n}\vert \exp \biggl(-\frac{\frac {1}{2}((x_{i}/a_{n}+b_{n})^{2}+(x_{j}/a_{n}+b_{n})^{2})}{1+\omega _{ij}} \biggr) \\ & \quad < C\sum_{1\leq i< j\leq n}\vert r_{ij}- \rho_{n}\vert \exp \biggl(-\frac{((\min_{1\leq i\leq n}x_{i})/a_{n}+b_{n})^{2}}{1+\omega_{ij}} \biggr) \\ & \quad \rightarrow0. \end{aligned}$$
The last ‘→’ attributes to Lemma 3.1. In order to get the desired limit of (3.4), we only need to prove
$$P \Biggl(\bigcap^{s}_{k=1} \bigl\{ M_{n}(c_{k},d_{k},\rho_{n})\leq u_{n}^{(l_{k})}\bigr\} \Biggr)\rightarrow P\bigl(N(B)=0\bigr). $$
By the definition of \(M_{n}(c_{k}, d_{k}, \rho_{n})\) it clearly follows that
$$\begin{aligned} &P \Biggl(\bigcap^{s}_{k=1} \bigl\{ M_{n}(c_{k},d_{k},\rho_{n})\leq u_{n}^{(l_{k})}\bigr\} \Biggr)\\ &\quad =P \Biggl(\bigcap ^{s}_{k=1}\bigl\{ (1-\rho_{n})^{\frac {1}{2}}M_{n}(c_{k},d_{k},0) +\rho_{n}^{\frac{1}{2}}\zeta\leq u_{n}^{(l_{k})}\bigr\} \Biggr) \\ &\quad = \int^{+\infty}_{-\infty}P \Biggl(\bigcap ^{s}_{k=1}\bigl\{ M_{n}(c_{k},d_{k},0) \leq (1-\rho_{n})^{-\frac{1}{2}}\bigl(u_{n}^{(l_{k})}- \rho_{n}^{\frac{1}{2}}z\bigr)\bigr\} \Biggr)\phi(z)\,dz, \end{aligned}$$
where the proof of the last ‘=’ is the same as the argument on the first line from the bottom on p.136 in Leadbetter et al. [1]. Since \(a_{n}=(2\log n)^{\frac{1}{2}}\), \(b_{n}=a_{n}+O(a_{n}^{-1}\log\log n)\), and \(\rho_{n}=\gamma/\log n\), it is easy to show that
$$\begin{aligned} (1-\rho_{n})^{-\frac{1}{2}}\bigl(u_{n}^{(l_{k})}- \rho_{n}^{\frac{1}{2}}z\bigr) =\frac{x_{l_{k}}+\gamma-\sqrt{2\gamma}z}{a_{n}}+b_{n}+o \bigl(a_{n}^{-1}\bigr). \end{aligned}$$
Furthermore, we have
$$\begin{aligned} & P \Biggl(\bigcap^{s}_{k=1} \bigl\{ M_{n}(c_{k},d_{k},0) \leq (1- \rho_{n})^{-\frac{1}{2}}\bigl(u_{n}^{(l_{k})}- \rho_{n}^{\frac{1}{2}}z\bigr)\bigr\} \Biggr) \\ & \quad =P \Biggl(\bigcap^{s}_{k=1}\bigl\{ \tilde{\xi}_{[c_{k}n]+1}\leq (1-\rho_{n})^{-\frac{1}{2}} \bigl(u_{n}^{(l_{k})}-\rho_{n}^{\frac {1}{2}}z \bigr)-m_{[c_{k}n]+1}, \ldots, \tilde{\xi}_{[d_{k}n]}\\ & \qquad\leq(1- \rho_{n})^{-\frac {1}{2}}\bigl(u_{n}^{(l_{k})}- \rho_{n}^{\frac{1}{2}}z\bigr)-m_{[d_{k}n]}\bigr\} \Biggr) \\ & \quad \rightarrow\prod^{s}_{k=1}\exp \bigl(-(d_{k}-c_{k})e^{-x_{l_{k}}-\gamma+\sqrt {2\gamma}z}\bigr), \end{aligned}$$
where \(\tilde{\xi}_{k}\) stands for independent standard normal variables, and the proof of the last ‘→’ is the same as that of (3.1). Using the dominated convergence theorem yields that
$$\begin{aligned} & \int^{+\infty}_{-\infty}P \Biggl(\bigcap ^{s}_{k=1}\bigl\{ M_{n}(c_{k},d_{k},0) \leq (1-\rho_{n})^{-\frac{1}{2}}\bigl(u_{n}^{(l_{k})}- \rho_{n}^{\frac{1}{2}}z\bigr)\bigr\} \Biggr)\phi(z)\,dz \\ & \quad \rightarrow \int^{+\infty}_{-\infty}\prod^{s}_{k=1} \exp \bigl(-(d_{k}-c_{k})e^{-x_{l_{k}}-\gamma+\sqrt{2\gamma}z}\bigr)\phi(z)\,dz \\ & \quad =P\bigl(N(B)=0\bigr). \end{aligned}$$
The proof of (b) is completed. □

Proof of Corollary 2.1

Using Theorem 2.1, the proof is similar to that of Corollary 5.5.2 in Leadbetter et al. [1]. So we omit it. □

Proof of Theorem 2.2

By Corollary 2.1 the left-hand side of (2.6) converges to
$$\begin{aligned} P\bigl(S^{(1)}=k_{1},S^{(2)}=k_{1}+k_{2}, \ldots,S^{(r)}=k_{1}+k_{2}+\cdots+k_{r} \bigr), \end{aligned}$$
(3.7)
where \(S^{(i)}=N^{(i)}([0, 1])\) is the ith component of N. In our paper, the structure of the Cox process is similar to that of the Poisson process in plane in Leadbetter et al. [1]. So we can refer to the proof of Theorem 5.6.1 in Leadbetter et al. [1], and hence (3.7) equals
$$\begin{aligned} &\frac{(k_{1}+k_{2}+\cdots+k_{r})!}{k_{1}!k_{2}!\cdots k_{r}!} \biggl(\frac{\tau_{1}}{\tau_{r}} \biggr)^{k_{1}} \biggl(\frac{\tau _{2}-\tau_{1}}{\tau_{r}} \biggr)^{k_{2}}\biggl(\frac{\tau_{r}-\tau_{r-1}}{\tau_{r}} \biggr)^{k_{r}} \\ &\quad {}\cdot P\bigl(N^{(r)}\bigl((0,1]\bigr)=k_{1}+k_{2}+ \cdots+k_{r}\bigr). \end{aligned}$$
The proof is completed since
$$\begin{aligned} & P\bigl(N^{(r)}\bigl((0,1]\bigr)=k_{1}+k_{2}+ \cdots+k_{r}\bigr) \\ & \quad = \int^{+\infty}_{-\infty}\frac{(\exp(-x_{r}-\gamma+\sqrt{2\gamma }z))^{k_{1}+k_{2}+\cdots+k_{r}}}{ (k_{1}+k_{2}+\cdots+k_{r})!}\cdot\exp \bigl(-e^{-x_{r}-\gamma+\sqrt{2\gamma }z}\bigr)\phi(z)\,dz \\ & \quad =\frac{(\exp(-x_{r}))^{k_{1}+k_{2}+\cdots+k_{r}}}{ (k_{1}+k_{2}+\cdots+k_{r})!} \int^{+\infty}_{-\infty}\bigl(\exp(-\gamma+\sqrt {2\gamma}z) \bigr)^{k_{1}+k_{2}+\cdots+k_{r}} \\ & \qquad{}\times\exp\bigl(-e^{-x_{r}-\gamma+\sqrt{2\gamma}z}\bigr)\phi(z)\,dz. \end{aligned}$$
 □

Proof of Theorem 2.3

Clearly, the left-hand side of (2.7) is equal to
$$\begin{aligned} & P\bigl(a_{n}\bigl(M_{n}^{(1)}-b_{n}-m_{n}^{*} \bigr)\leq x_{1},a_{n}\bigl(M_{n}^{(2)}-b_{n}-m_{n}^{*} \bigr)\leq x_{2}\bigr) \\ & \quad =P\bigl(S_{n}^{(2)}=0\bigr)+ P\bigl(S_{n}^{(1)}=0, S_{n}^{(2)}=1\bigr), \end{aligned}$$
where \(S^{(i)}_{n}\) is the number of exceedances of \(u^{(i)}_{n}\) by \(X_{1}, X_{2}, \ldots, X_{n}\). Using Theorem 2.2 can complete the proof. In order to prove (2.8), write I, J for intervals \(\{1, 2,\ldots, [nt]\}\), \(\{[nt] + 1,\ldots, n\}\), respectively, and \(M^{(1)}(I)\), \(M^{(2)}(I)\), \(M^{(1)}(J)\), \(M^{(2)}(J)\) for the maxima and the second largest of \(\{X_{i},1\leq i\leq n\}\) in the intervals I, J. Let \(K_{n}(x_{1}, x_{2}, x_{3}, x_{4})\) be the joint d.f. of the normalized r.v.s
$$\begin{aligned} & X_{n}^{(1)}=a_{n}\bigl(M_{n}^{(1)}(I)-b_{n}-m_{n}^{*} \bigr),\qquad X_{n}^{(2)}=a_{n}\bigl(M_{n}^{(2)}(I)-b_{n}-m_{n}^{*} \bigr), \\ & Y_{n}^{(1)}=a_{n}\bigl(M_{n}^{(1)}(J)-b_{n}-m_{n}^{*} \bigr),\qquad Y_{n}^{(2)}=a_{n}\bigl(M_{n}^{(2)}(J)-b_{n}-m_{n}^{*} \bigr). \end{aligned}$$
Consider an interesting case of \(x_{1} > x_{2}\) and \(x_{3} > x_{4}\), that is,
$$\begin{aligned} &H_{n}(x_{1},x_{2},x_{3},x_{4})\\ &\quad = P\bigl(M_{n}^{(1)}(I)\leq u_{n}^{(1)},M_{n}^{(2)}(I) \leq u_{n}^{(2)},M_{n}^{(1)}(J)\leq u_{n}^{(3)},M_{n}^{(2)}(J)\leq u_{n}^{(4)}\bigr) \\ &\quad = P\bigl(N_{n}^{(1)}\bigl(I' \bigr)=0,N_{n}^{(2)}\bigl(I'\bigr) \leq1,N_{n}^{(3)}\bigl(J'\bigr)=0, N_{n}^{(4)}\bigl(J'\bigr)\leq1\bigr), \end{aligned}$$
where \(I'=(0,t]\) and \(J'=(t,1]\). By Corollary 2.1 with \(B_{1}=I'\) and \(B_{2} = J'\) we have
$$\begin{aligned} & \lim_{n\rightarrow\infty}H_{n}(x_{1},x_{2},x_{3},x_{4}) \\ & \quad =\lim_{n\rightarrow\infty}P\bigl(N_{n}^{(1)} \bigl(I'\bigr)=0,N_{n}^{(2)}\bigl(I' \bigr)\leq1\bigr)\cdot P\bigl(N_{n}^{(3)}\bigl(J' \bigr)=0, N_{n}^{(4)}\bigl(J'\bigr)\leq1\bigr) \\ & \quad = \int^{+\infty}_{-\infty}\bigl(\bigl(e^{-x_{2}}-e^{-x_{1}} \bigr)t\exp(\sqrt{2\gamma }-\gamma)+1\bigr) \exp\bigl(-te^{-x_{2}-\gamma+\sqrt{2\gamma}z}\bigr) \phi(z)\,dz \\ & \qquad{}\times \int^{+\infty}_{-\infty}\bigl(\bigl(e^{-x_{4}}-e^{-x_{3}} \bigr) (1-t)\exp(\sqrt{2\gamma }-\gamma)+1\bigr) \exp\bigl(-(1-t)e^{-x_{4}-\gamma+\sqrt{2\gamma}z} \bigr)\phi(z)\,dz \\ & \quad =H_{t}(x_{1},x_{2})H_{1-t}(x_{3},x_{4})=H(x_{1},x_{2},x_{3},x_{4}). \end{aligned}$$
Now the left-hand side of (2.8) is equal to
$$\begin{aligned} \begin{aligned}[b] & P\bigl(M^{(2)}_{n}(I)\leq u^{(2)}_{n},M^{(2)}_{n}(I)\geq M^{(1)}_{n}(J)\bigr) \\ & \quad {} +P\bigl(M^{(1)}_{n}(I)\leq u^{(2)}_{n}, M^{(1)}_{n}(J)>M^{(1)}_{n}(I)\geq M^{(2)}_{n}(J)\bigr). \end{aligned} \end{aligned}$$
(3.8)
Obviously, H is absolutely continuous, and the boundaries of sets in \(R^{4}\) such as \(\{(w_{1},w_{2},w_{3}, w_{4}): w_{2}\leq x_{2},w_{2}>w_{3}\}\) and \(\{(w_{1},w_{2},w_{3},w_{4}) : w_{1}\leq x_{2},w_{3} > w_{1}\geq w_{4}\}\) have zero Lebesgue measure. Thus, by Corollary 2.1, (3.8) converges to
$$P(X_{2}\leq x_{2},X_{2}\geq Y_{1})+P(X_{1}\leq x_{2},Y_{1}>X_{1} \geq Y_{2}). $$
Using the joint distribution \(H(x_{1}, x_{2}, x_{3}, x_{4})\) and a simple evaluation complete the proof. □

Declarations

Acknowledgements

The research of the authors was supported by the Scientific Research Fund of Sichuan Provincial Education Department under Grant 12ZB082, the Scientific research cultivation project of Sichuan University of Science & Engineering under Grant 2013PY07, and the Scientific Research Fund of Sichuan University of Science & Engineering under Grant 2013KY03.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Statistics, Southwestern University of Finance and Economics, Chengdu, China
(2)
School of Science, Sichuan University of Science & Engineering, Zigong, China

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