Almost sure convergence of weighted sums of independent random variables Guy Cohen and Michael Lin Dedicated to Arkady Tempelman Abstract. 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 X 1 is in a certain class of integrability. In particular, our condition allows us to generalize the rate obtained by Marcinkiewicz and Zygmund when E[|X 1 | p ] < ∞ for some 1 <p< 2. When applied to weighted averages, our result strengthens the SLLN of Jamison, Orey, and Pruitt in the case X 1 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. 1. Introduction Let (Ω, F , P) be a probability space, and let {X n } be a sequence of integrable centered independent random variables. In this paper, which is largely expository, we consider what conditions should be imposed on a complex sequence {b n } with |b n |→∞, in order to obtain a.s. convergence of the series ∑ n Xn bn , whenever {X n } is in a certain class of integrability. Of particular interest is the case of weighted averages, when {w n } is a sequence of positive numbers (weights) with divergent sum and b n = ∑ n k=1 w k /w n ; a.s. convergence of the above series implies the strong law of large numbers (SLLN) for the weighted averages. Another case of interest is when {c n } is a sequence with ∑ |c k | 2 = ∞ and we take b n = ∑ n k=1 |c k | 2 /c n ; a.s. convergence of the series implies strong consistency of the least square estimator (LSE) in a linear regression model. 1991 Mathematics Subject Classification. Primary: 60F15, 60G50, 60G42; Secondary: 62J05, 37A05. Key words and phrases. independent random variables, martingale differences, strong law of large numbers, weighted sums, linear regression, LSE, counting function, dynamical systems. 1