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COMPLETE CONVERGENCE FOR ARRAYS AND THE LAW OF THE SINGLE LOGARITHM

Part of: Distribution theory - Probability Stochastic processes Limit theorems

Published online by Cambridge University Press:  29 August 2017

ALLAN GUT  and
ULRICH STADTMÜLLER Open the ORCID record for ULRICH STADTMÜLLER [Opens in a new window]
ALLAN GUT
Affiliation:
Department of Mathematics, Uppsala University, Box 480, SE-751 06 Uppsala, Sweden email allan.gut@math.uu.se
ULRICH STADTMÜLLER*
Affiliation:
Department of Number Theory and Probability Theory, Ulm University, D-89069 Ulm, Germany email ulrich.stadtmueller@uni-ulm.de
*
✉ulrich.stadtmueller@uni-ulm.de
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Abstract

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The present paper is devoted to complete convergence and the strong law of large numbers under moment conditions near those of the law of the single logarithm (LSL) for independent and identically distributed arrays. More precisely, we investigate limit theorems under moment conditions which are stronger than $2p$ for any $p<2$, in which case we know that there is almost sure convergence to 0, and weaker than $E\,X^{4}/(\log ^{+}|X|)^{2}<\infty$, in which case the LSL holds.

Keywords

arraysrow-wise independenceindependent and identically distributedthe law of the iterated logarithmcomplete convergencethe strong law of large numbersBorel–Cantelli lemmasexponential bounds

MSC classification

Primary: 60F15: Strong theorems
Secondary: 60E15: Inequalities; stochastic orderings 60G50: Sums of independent random variables; random walks
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 96 , Issue 2 , October 2017 , pp. 333 - 344
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

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