Bounds on the distributions of extremal values of a scanning process
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 \(E\tau\).
scientific article; zbMATH DE number 3876341
scientific article