Acta Mathematica Hungarica

, Volume 151, Issue 2, pp 510–530

Almost sure limit behavior of Cesàro sums with small order


DOI: 10.1007/s10474-016-0685-z

Cite this article as:
Gut, A. & Stadtmüller, U. Acta Math. Hungar. (2017) 151: 510. doi:10.1007/s10474-016-0685-z


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 

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