Bounds on the distributions of extremal values of a scanning process
type: Item
instance of: scholarly article
title: Bounds on the distributions of extremal values of a scanning process
zbMATH Open document ID: 0549.60066
DOI: 10.1016/0304-4149(84)90303-X
author: Svante Janson
published in: Stochastic Processes and their Applications
publication date: 1984-01-01T00:00:00Z
review text: Let a Poisson process of intensity \(\lambda\) be defined on the real line and for fixed \(a>0\), let \(Y_ t\) be the number of events in [t-a,t). The author derives upper and lower bounds for the distributions of \(N_ T=\max_{0\leq t\leq T} Y_ t\) and \(M_ T=\min_{0\leq t\leq T} Y_ t\). This problem has received considerable attention in the literature, see \textit{N. D. Neff} and \textit{J. I. Naus}, The distribution of the size of the maximum cluster of points on a line. Selected Tables in Mathematical Statistics, Vol. VI (1980; Zbl 0445.62120). The author's emphasis is on easily computable bounds. For example, his simplest bound is \(P(N_ T\leq n)\leq e^{-\gamma}\) \((T>0)\) where \(\gamma =\lambda (p_ n\phi_ n\)-p\({}_{n+1}\phi_{n+1})\), \(p_ n=(a\lambda)^ ne^{-a\lambda}/n!\) and \(\phi_ n=\sum^{n}_{k=0}p_ k\). He obtains a number of asymptotic relations from his bounds. Numerical examples are given. The bounds are simple consequences of monotonicity properties and bounds satisfied by the absolutely continuous component of \(G(t)=P(\tau >t)=P(N_ t\leq n)\) where \(\tau =\inf\{t\geq 0:Y_ t\geq n\}\). Bounds are also derived for \(E\tau\).
Mathematics Subject Classification ID: 60J99
zbMATH DE Number: 3876341
zbMATH Keywords: Poisson process
zbMATH Keywords: upper and lower bounds
zbMATH Keywords: monotonicity properties
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3876341