Weighted ergodic theorems and strong laws of large numbers

Michael Lin, Michel Weber

Research output: Contribution to journalArticlepeer-review

11 Scopus citations

Abstract

We investigate the convergence, in norm and almost everywhere (a.e.), of weighted ergodic averages as well as weighted sums of independent identically distributed (iid) random variables. The averages are true ones, normalized by the corresponding sums of weights, which are only assumed to be non-negative. The L2-norm convergence in the mixing case is shown to rely upon very simple conditions on the weights. We show that 'quasimonotone weights' with a simple additional condition yield a.e. convergence of weighted averages for all Dunford-Schwartz contractions of probability spaces and L1-functions. For independent random variables, we look at weighted averages of centered random variables with bounded variances (or bounded moments of some order greater than 1), in particular the iid case, and obtain several sufficient conditions on the weights for almost sure convergence (weighted SLLN). For example, in Theorem 4.14 we show that if a weight sequence {wk} with divergent partial sums Wn satisfies supn≥1 1/W nk=1n wk(log(w k+1))β < ∞ for some β > 1 then for any iid sequence in the class L(log+L)1+∈ the weighted averages converge almost surely to the expectation.

Original languageEnglish
Pages (from-to)511-543
Number of pages33
JournalErgodic Theory and Dynamical Systems
Volume27
Issue number2
DOIs
StatePublished - 1 Apr 2007

ASJC Scopus subject areas

  • Mathematics (all)
  • Applied Mathematics

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