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

Yingying Jiang,Baokun Li,Fuming Lin

doi:10.1186/s13660-016-1082-4

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.

Highlights

Let {Xi, i ≥ } be a standardized normal sequence with correlation coefficient rij = Cov(Xi, Xj) satisfying the conventional assumption that rij → and rij log(|i – j|) → γ as j – i → +∞
More recent results for maxima of stationary normal sequences can be found in Ho and Hsing [ ], Tan and Peng [ ], and Hashorva et al [ ], among others
Some literature was devoted to study the maxima of nonstationary normal sequences; see Horowitz [ ] and Leadbetter et al [ ] for the weakly dependence case and Zhang [ ], Lin et al [ ], and Tan and Yang [ ] for the strongly dependence case

Summary

Introduction

Leadbetter et al [ ] considered a stationary weakly dependent normal sequence {Xi, i ≥ } and obtained the asymptotic joint probability distribution of Mn( ) and Mn( ) and even that of Mn( ) and L(n ). We prove that the time-normalized point process Nn converges in distribution to the in plane Cox process defined in Lin et al [ ] and extend the results in Lin et al [ ] to the case of more general normal sequences. . .} be the points of a Cox process N(r) on Lr with (stochastic) intensity exp(–xr – γ + γ ζ ), where ζ is a standard normal random variable, xr is a constant corresponding to the N(r), and Lr is the in plane fixed horizontal line on which exceedances are represented as points. Let {Xi, i ≥ } be a normal sequence satisfying the conditions of Theorem.

It is convenient to neglect any set

The proof is completed since