Doubly Stochastic Compound Poisson Processes in Extreme Value Theory

Helena Ferreira

Abstract: For some linear models, chain-dependent sequences and doubly stochastic max-autoregressive processes, which do not satisfy the long range dependence condition $\Delta(u_{n})$ from Hsing {\sl et al.} ([7]), the sequence $\{S_{n}\}_{n\ge1}$, of point processes of exceedances of a real level $u_{n}$ by $X_{1},...,X_{n}$, $n\ge1$, converges in distribution to a compound Poisson process with stochastic intensity.
These examples illustrate the main result of this paper: for sequences $\{X_{n}\}_{n\ge1}$ that conditional on a random variable $X$ satisfy the usual dependence conditions in the extreme value theory, we obtain the convergence of $\{S_{n}\}$ to a point process whose distribution is a mixture of distributions of compound Poisson processes. Such result permits the identification of a class of sequences for which the extremal behaviour can be described by mixtures of extreme value distributions.

Keywords: Extreme value theory; point processes; mixtures of distributions.

Classification (MSC2000): 60F05, 60G55

Full text of the article:

• Compressed PostScript file (84 kilobytes)
• PDF file (164 kilobytes)