Almost sure convergence of weighted sums of independent random variables

Let (Ω,F ,P) be a probability space, and let {Xn} be a sequence of integrable centered i.i.d. random variables. In this paper we consider what conditions should be imposed on a complex sequence {bn} with |bn| → ∞, in order to obtain a.s. convergence of P n Xn bn , whenever X1 is in a certain class of integrability. In particular, our condition allows us to generalize the rate obtained by Marcinkiewicz and Zygmund when E[|X1|] < ∞ for some 1 < p < 2. When applied to weighted averages, our result strengthens the SLLN of Jamison, Orey, and Pruitt in the case X1 is symmetric. An analogous question is studied for {Xn} an Lp-bounded martingale difference sequence. An extension of Azuma’s SLLN for weighted averages of uniformly bounded martingale difference sequences is also presented. Applications are made also to modulated averages and to strong consistency of least squares estimators in a linear regression. The main tool for the general approach is (a generalization of) the counting function introduced by Jamison et al. for the SLLN for weighted averages.