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

Published online by Cambridge University Press:  25 July 2016

Oleg Klesov*
Affiliation:
National Technical University of Ukraine `KPI'
Ulrich Stadtmüller*
Affiliation:
Ulm University
*
Department of Mathematical Analysis and Probability Theory, National Technical University of Ukraine `KPI', Peremogy Avenue 56, 03056 Kyiv, Ukraine. Email address: klesov@matan.kpi.ua
Department of Number and Probability Theory, Ulm University, 89069 Ulm, Germany. Email address: ulrich.stadtmueller@uni-ulm.de

Abstract

Starting 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=1ℙ(|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.

MSC classification

Type
Research Article
Copyright
Copyright © Applied Probability Trust 2016 

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