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

Deli Li, Yongcheng Qi, Andrew Rosalsky

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

Abstract

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
Volume24
Issue number4
DOIs
StatePublished - Dec 1 2011

Keywords

  • 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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