ON THE UNIFORM INTEGRABILITY AND ALMOST SURE CONVERGENCE OF WEIGHTED SUMS

Abstract

Let X, X₁, … be independent and identically distributed random variables with zero and finite absolute p-th moment for some p greater than or equal to two. For each positive integer n, let a_n1, …, a_nn be constants. The boundedness of {(a²_n1+…+a²_nn)/n^2/p, n≥1} is established to be a necessary and sufficient condition for the uniform integrability of {|∑a_niX_i/n^1/p|^p, n≥1}. Conditions are also given for the central limit thorem and the convergence in probability of {∑a_niX_i/n^1/p, n≥1}. An improvement of Thrum's (1987) theorem on the almost sure convergence is obtained. As a result, the moment convergence and its limits are established.