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

Part of: Limit theorems

Published online by Cambridge University Press:  25 July 2016

Oleg Klesov  and
Ulrich Stadtmüller
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
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Abstract

Starting with independent, identically distributed random variables X 1,X 2... and their partial sums (S n ), together with a nondecreasing sequence (b(n)), we consider the counting variable N=∑n 1(S n >b(n)) and aim for necessary and sufficient conditions on X 1 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(|S n |>εn)<∞, corresponds to the notion of complete convergence as introduced by Hsu and Robbins in 1947 as a complement to Kolmogorov's strong law.

Keywords

Baum‒Katz theoremcounting variables; momentsFuk‒Nagaev inequalitycomplete convergencelarge deviations

MSC classification

Primary: 60F15: Strong theorems
Secondary: 60F10: Large deviations
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue A: Probability, Analysis and Number Theory , July 2016 , pp. 181 - 201
Copyright
Copyright © Applied Probability Trust 2016 

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