Limit properties of exceedance point processes of strongly dependent normal sequences
- Fuming Lin^{1, 2}Email author,
- Daimin Shi^{1} and
- Yingying Jiang^{2}
https://doi.org/10.1186/s13660-015-0585-8
© Lin et al.; licensee Springer. 2015
Received: 28 September 2014
Accepted: 2 February 2015
Published: 20 February 2015
Abstract
In this paper, we define an in plane Cox process and prove the time-normalized point process of exceedances by a dependent normal sequence converging to the Cox process in distribution under some mild conditions. As some applications of the convergence result, two important joint asymptotic distributions for the order statistics are derived.
Keywords
MSC
1 Introduction
Throughout this paper, let \(\{\xi_{i}, i\geq1\}\) be a standardized strongly dependent stationary normal sequence with correlation coefficients \(r_{ij}=\operatorname{Cov}(\xi_{i},\xi_{j})\). C stands for a constant which may vary from line to line and ‘→’ for the convergence as \(n\rightarrow\infty\). The remainder of the paper is organized as follows. In Section 2, we define an in plane Cox process and prove that the time-normalized point process \(N_{n}\) of exceedances of levels \(u^{(1)}_{n}, u^{(2)}_{n} ,\ldots, u^{(r)}_{n}\) by \(\{\xi_{i},1\leq i\leq n\}\) converges in distribution to the in plane Cox process. In Section 3, as the applications of our main result, the asymptotic results of the probabilities \(P(a_{n}(M_{n}^{(2)}-b_{n})\leq x, L_{n}^{(2)}/n\leq t)\), and \(P(a_{n}(M_{n}^{(1)}-b_{n})\leq x_{1}, a_{n}(M_{n}^{(2)}-b_{n})\leq x_{2})\) are established.
2 Convergence of point processes of exceedances
Lemma 2.1
Proof
The proof can be found on p.134 in Leadbetter et al. [1]. □
Lemma 2.2
Suppose \(\{\xi_{i},i\geq1\}\) is a standard stationary normal sequence with covariances satisfying (2.1). Then the point process \(N_{n}\) of time-normalized exceedances of the level \(u_{n}(u_{n}=x/a_{n}+b_{n})\) converges in distribution to N on \((0,+\infty)\), where N is the Cox process defined by (2.2).
Proof
The proof can be found on p.136 in Leadbetter et al. [1]. □
In Theorem 2.1 below, we extend Lemma 2.2 to the case of exceedances of several levels and study a vector of point processes \(N_{n}=(N_{n}^{(1)},N_{n}^{(2)},\ldots,N_{n}^{(r)})\) which arises when \(\{\xi_{i},1\leq i\leq n\}\) exceeds the levels \(u_{n}^{(1)},u_{n}^{(2)},\ldots,u_{n}^{(r)}\), where \(u_{n}^{(k)}=x_{k}/a_{n}+b_{n}\), \(1\leq k\leq r\). For clarity, we record the locations of \(u_{n}^{(1)},u_{n}^{(2)},\ldots,u_{n}^{(r)}\) along fixed horizontal lines \(L_{1},L_{2},\ldots,L_{r}\) in the plane. The structure of the process vector is the same as that of the exceedances process on pp.111-112 in Leadbetter et al. [1] where the authors presented a detailed and visualized introduction. According to Lemma 2.2, each one-dimensional point process, on a given \(L_{k}\), i.e. \(N_{n}^{(k)}\) converges to a Cox process in distribution under appropriate conditions. Before presenting Theorem 2.1, we first give two definitions, one of which concerns a two-dimension Cox process, i.e. an in plane Cox process.
Definition 2.1
The locations of order statistics are the places where order statistics appear in the index set, for example the location of the maxima of \(\{\xi_{i},1\leq i\leq n\}\) varying among \(1,\ldots,n\).
Definition 2.2
The structure of the in plane Cox process N is very similar to that of the Poisson process on p.112 in Leadbetter et al. [1], but the independent thinning is replaced with the conditionally independent thinning here.
Theorem 2.1
Suppose \(\{\xi_{i},i\geq1\}\) is a standardized normal sequence satisfying the conditions in Lemma 2.2. Let \(u^{(k)}_{n}=x_{k}/a_{n}+b_{n}\) satisfy \(u^{(1)}_{n}\geq u^{(2)}_{n}\geq\cdots\geq u^{(r)}_{n}\) (\(1\leq k\leq r\)) where \(a_{n}\) and \(b_{n}\) are defined in (1.1). Then the time-normalized point process \(N_{n}\) of exceedances of levels \(u^{(1)}_{n}, u^{(2)}_{n} ,\ldots, u^{(r)}_{n}\) by \(\{\xi_{i},1\leq i\leq n\}\) converges in distribution to the above-mentioned in plane Cox process.
Proof
- (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,
- (b)
\(P(N_{n}(B)=0)\rightarrow P(N(B)=0)\) for all sets B which are finite unions of disjoint sets of this form.
Corollary 2.1
Proof
Combining Theorem 2.1 and the proof of Corollary 5.5.2 in Leadbetter et al. [1], we can complete the proof. □
Theorem 2.2
Proof
3 The joint distributions of some order statistics
This section contains two important results which concerns the joint distributions of order statistics of \(\{\xi_{i},i\geq1\}\).
Theorem 3.1
Proof
Remark 3.1
We may obtain the joint asymptotic distribution of \(M^{(1)}_{n} ,M^{(2)}_{n} , \ldots, M^{(k)}_{n}\) by using the same method as in Theorem 3.1.
Declarations
Acknowledgements
The research of the second author was supported by the National Natural Science Foundation of China, Grant No. 71171166. The research of other 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 and Engineering under Grant 2013PY07, the Scientific Research Fund of Sichuan University of Science and Engineering under Grant 2013KY03, and the Science Research Programs for Doctors in Southwestern University of Finance and Economics, Grant No. JBK1207085.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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