Acta Mathematica Hungarica

, Volume 151, Issue 2, pp 510–530 | Cite as

Almost sure limit behavior of Cesàro sums with small order

Article

Abstract

Various methods of summation for divergent series have been extended to analogs for sums of i.i.d. random variables. The present paper deals with a special class of matrix weighted sums of i.i.d. random variables where the weights \({a_{n,k}}\) are defined as the weights from Cesàro summability, i.e., \({a_{n,k}=\binom{n-k+\alpha-1}{n-k}/\binom{n+\alpha}{n}}\), where \({\alpha > 0}\). A strong law of large numbers (SLLN) has been shown to hold in this setting iff \({E {|X|}^{1/\alpha}<\infty}\), but a law of the iterated logarithm (LIL) has been shown for the case \({\alpha \geqq 1}\) only. We will study the case \({0 < \alpha < 1}\) in more detail, giving an LIL for \({1/2 < \alpha < 1}\) and some additional strong limit theorems under appropriate moment conditions for \({1/2 \leqq \alpha < 1}\).

Key words and phrases

Cesàro summation strong law law of the iterated logarithm exponential bound 

Mathematics Subject Classification

primary 60F15 60G50 secondary 40G05 

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2016

Authors and Affiliations

  1. 1.Department of MathematicsUppsala UniversityUppsalaSweden
  2. 2.Department of Number Theory and ProbabilityUlm UniversityUlmGermany

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