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


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 (US)
Pages (from-to)165-181
Number of pages17
JournalJournal of Theoretical Probability
Issue number1
StatePublished - Jan 2004

Bibliographical note

Funding Information:
We are grateful for the hospitality and support offered by North Dakota State University to the third author, and by the University of Minnesota and Ben-Gurion University to the last author. The last author’s travel to Israel was partially supported by an NDSU grant from the VP of Academic Affairs. Research of J. Olsen partially supported by ND-EPSCoR through NSF Grant EPS-9874802.


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


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