SLLN for weighted independent identically distributed random variables

John Baxter, Roger Jones, Michael Lin, James Olsen

Research output: Contribution to journalArticlepeer-review

9 Scopus citations

Abstract

For any sequence {ak} with sup 1/n ∑k = 1 n |ak|q < ∞ for some q > 1, we prove that 1/n ∑k = 1n akXk converges to 0 a.s. for every {Xn} i.i.d. with E(|X1|) < ∞ and E(X1) = 0; the result is no longer true for q = 1, not even for the class of i.i.d. with X1 bounded. We also show that if {ak} is a typical output of a strictly stationary sequence with finite absolute first moment, then for every i.i.d. sequence {Xn} with finite absolute p th moment for some p > 1, 1/n ∑ k = 1n ak Xk converges a.s.

Original languageEnglish
Pages (from-to)165-181
Number of pages17
JournalJournal of Theoretical Probability
Volume17
Issue number1
DOIs
StatePublished - 1 Jan 2004

Keywords

  • Besicovitch sequences
  • Independent random variables
  • Law of large numbers
  • Weighted averages

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