Strong theorems on the extreme values of stationary Poisson processes
A number of strong limit theorems are derived for \(\max (Y_ 1, \dots, Y_ n)\), where \(Y_ 1, Y_ 2,\dots\) is an i.i.d. sequence of random variables with a Poisson-type tail behaviour. These results are used to study strong extreme value asymptotics for a sequence \(X_ i(t) = X_ i + W_ i(t)\), \(t \geq 0\), \(i = 1, 2, \dots, \) of stochastic processes, where \(X_ 1, X_ 2, \dots\) denote the points of a Poisson process in \(\mathbb{R}^ d\), and \(W_ i (t)\) be an i.i.d. sequence of \(\mathbb{R}^ d\)- valued Wiener processes. Almost sure upper and lower bounds are derived for various extreme value processes based on \(X_ i (t)\), e.g., for \(S(T) = \sup_{0 \leq t \leq T} s(t)\) with \(s(t)\) denoting the number of points \(X_ i(t)\) in the unit ball at time \(t\).
scientific article; zbMATH DE number 758087