Weighted ergodic theorems and strong laws of large numbers

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 $L_2$-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 $L_1$-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 $\{w_k\}$ with divergent partial sums $W_n$ satisfies

\[ \sup_{n\ge 1} \frac1{W_n} \sum_{k=1}^n w_k ({\rm log}(w_k+1))^\beta < \infty\quad \text{for some }\beta > 1 \]

then for any iid sequence in the class $L({\rm log}^+L)^{1+\epsilon}$ the weighted averages converge almost surely to the expectation.