A Refinement of the Kolmogorov-Marcinkiewicz-Zygmund Strong Law of Large Numbers

Deli Li, Yongcheng Qi, Andrew Rosalsky

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Let {Xn; n≥1} be a sequence of independent copies of a real-valued random variable X and set Sn=X1+{dot operator}{dot operator}{dot operator}+Xn, n≥1. This paper is devoted to a refinement of the classical Kolmogorov-Marcinkiewicz-Zygmund strong law of large numbers. We show that for 0 < p < 2, n≥1. Versions of the above result in a Banach space setting are also presented. To establish these results, we invoke the remarkable Hoffmann-Jørgensen (Stud. Math. 52:159-186, 1974) inequality to obtain some general results for sums of the form, (where {Vn; n≥1} is a sequence of independent Banach-space-valued random variables, and an≥0, n≥1), which may be of independent interest, but which we apply to.

Original languageEnglish (US)
Pages (from-to)1130-1156
Number of pages27
JournalJournal of Theoretical Probability
Issue number4
StatePublished - Dec 2011

Bibliographical note

Funding Information:
Acknowledgements The authors are extremely grateful to the referee for very carefully reading the manuscript and for offering numerous comments and suggestions which enabled us to substantially improve the paper. The referee did not only provide comments to help us to improve the presentation, but more significantly, the referee presented improved versions of some of the main results complete with their proofs. Specifically, the referee significantly improved our original version of Theorem 3.1, and this improved version led to more elegant proofs of many other results in the paper as was pointed out to us by the referee. The authors also thank Professor Andrzej Korzeniowski for his interest in our work and for pointing out to us the relevance of his paper [9] to ours. The research of Deli Li was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada, and the research of Yongcheng Qi was partially supported by NSF Grant DMS-0604176.


  • Kolmogorov-Marcinkiewicz-Zygmund strong law of large numbers
  • Rademacher type p Banach space
  • Real separable Banach space
  • Stable type p Banach space
  • Sums of i.i.d. random variables


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