Extremes of moving averages of random variables from the domain of attraction of the double exponential distribution
type: Item
instance of: scholarly article
title: Extremes of moving averages of random variables from the domain of attraction of the double exponential distribution
zbMATH Open document ID: 0657.60028
DOI: 10.1016/0304-4149(88)90075-0
published in: Stochastic Processes and their Applications
publication date: 1988-01-01T00:00:00Z
review text: Let \(Z_ k\), \(-\infty <k<\infty\) be a sequence of iid r.v's with d.f. \(F\in D(\Lambda)\), i.e. \[ \lim_{n\to \infty}F^ n(a_ nx+b_ n)=\Lambda (x),\quad x\in {\mathbb{R}} \] for some \(a_ n>0\), \(b_ n\in {\mathbb{R}}\) (n\(\geq 1)\) and \(\Lambda (x)=\exp \{-e^{-x}\}.\) The main object of this paper is the sequence of point processes based on the moving average process \(X_ n=\sum^{\infty}_{j=-\infty}C_ jZ_{n-j}\), \(n=0,\pm 1,..\). where \(\sum^{\infty}_{j=-\infty}| C_ j|^{1-\delta}<\infty\), \(\delta >0\) and \(\max \{| C_ i|:\) \(i=0,\pm 1,...\}=1\). For a fixed integer m define the vectors \[ X_{n,k}=a_ n^{-1}(Z_ k-b_ n,\quad -Z_ k-b_ n,\quad Z_{k- i},\quad 0<| i| \leq 2m) \] and let \(\epsilon_ x(F)=1\) if \(x\in F\), 0 otherwise; \(x\in E\), \(F\in {\mathcal E}\), E is a subset of compactified Euclidean space, \({\mathcal E}\) is the \(\sigma\)-algebra generated by open sets. Then (under certain conditions) \(\sum^{\infty}_{k=1}\epsilon_{(k_ n^{-1},X_{n,k})}\Rightarrow N\) where N is the sum of 2 independent Poisson random measures with specified mean measures. Numerous applications comprise the convergence of maxima and exceedances. Convergent sequences of point processes based on the max-moving average process \(\{\) \(\bigvee^{\infty}_{j=-\infty}C_ jZ_{n-j}\}\) are also investigated.
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60G55
zbMATH DE Number: 4074067
zbMATH Keywords: sequence of point processes
zbMATH Keywords: moving average process
zbMATH Keywords: Poisson random measures
zbMATH Keywords: convergence of maxima and exceedances
zbMATH Keywords: max-moving average process
author: Richard A. Davis
author: Sidney I. Resnick
MaRDI profile type: Publication
full work available at URL: 0304-4149(88)90075-0
OpenAlex ID: W2004444437
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description: scientific article; zbMATH DE number 4074067