TY - JOUR

T1 - Almost sure convergence of weighted sums of independent random variables

AU - Cohen, Guy

AU - Lin, Michael

PY - 2009/1/1

Y1 - 2009/1/1

N2 - 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.

AB - 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.

U2 - 10.1090/conm/485/09491

DO - 10.1090/conm/485/09491

M3 - Article

SN - 0271-4132

JO - Contemporary Mathematics

JF - Contemporary Mathematics

ER -