An intermediate Baum–Katz theorem

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• 作者：Allan Gut^a ; ^{allan.gut@math.uu.se"" rel=""nofollow} ; ; Ulrich Stadtmü ; ller^b ; ^{ulrich.stadtmueller@uni-ulm.de"" rel=""nofollow} ;
• 关键词：Sums of i.i.d.  ; random variables ; Convergence rates ; Law of large numbers ; Almost exponential moments ; Random fields
• 刊名：Statistics and Probability Letters
• 出版年：2011
• 出版时间：October 2011
• 年：2011
• 卷：81
• 期：10
• 页码：1486-1492
• 全文大小：221 K

文摘

We extend the classical Hsu–Robbins–Erdős theorem to the case when all moments exist, but the moment generating function does not, viz., we assume that Eexp{(log⁺X)^α}<∞ for some α>1. We also present multi-index versions of the same and of a related result due to Lanzinger in which the assumption is that Eexp{X^α}<∞ for some α(0,1).