On the characterization of certain point processes. (English) Zbl 0645.60057

Hsing, Tailen

Stochastic Processes Appl. 26, 297-316 (1987).

This paper consists of two parts. First, a characterization is obtained for a class of infinitely divisible point processes on \({\mathbb{R}}\times {\mathbb{R}}_+'=(-\infty\), \(\infty)\times (0,\infty)\). Second, the result is applied to identify the weak limit of the point process \(N_ n\) with points (j/n, \(u_ n^{-1}(\xi_ j))\), \(j=0\), \(\pm 1\), \(\pm 2\),..., where \(\{\xi_ j\}\) is a stationary sequence satisfying a certain mixed condition \(\Delta\), and \(\{u_ n\}\) is a sequence of nonincreasing functions on (0,\(\infty)\) such that \[ \lim_{n\to \infty}P\{\max_{1\leq j\leq n}\xi_ j\leq u_ n(\tau)\}=e^{- \tau},\quad \tau >0. \] This application extends a result of T. Mori [Yokohama Math. J. 25, 155-168 (1977; Zbl 0374.60010)], which assumes that \(\{\xi_ j\}\) is \(\alpha\)-mixing, and that the distribution of \(\max_{1\leq j\leq n}\xi_ j\) can be linearly normalized to converge to a maximum stable distribution.

MSC:

60G55	Point processes (e.g., Poisson, Cox, Hawkes processes)
60F05	Central limit and other weak theorems
60G10	Stationary stochastic processes

Keywords:

infinitely divisible point processes; maximum stable distribution

Citations:

Zbl 0374.60010

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