Existence of moments of a counting process and convergence in multidimensional time

Abstract

AbstractStarting with independent, identically distributed random variables X1,X2... and their partial sums (Sn), together with a nondecreasing sequence (b(n)), we consider the counting variable N=∑n1(Sn>b(n)) and aim for necessary and sufficient conditions on X1 in order to obtain the existence of certain moments for N, as well as for generalized counting variables with weights, and multi-index random variables. The existence of the first moment of N when b(n)=εn, i.e. ∑n=1∞ℙ(|Sn|>εn)<∞, corresponds to the notion of complete convergence as introduced by Hsu and Robbins in 1947 as a complement to Kolmogorov's strong law.