COMPLETE CONVERGENCE AND ALMOST SURE CONVERGENCE OF WEIGHTED SUMS OF RANDOM-VARIABLES

Let r > 1. For each n greater than or equal to 1, let {X(nk), - infinity < k < infinity} be a sequence of independent real random variables. We provide some very relaxed conditions which will guarantee Sigma(n greater than or equal to 1)n(r-2)P{\Sigma(k=-infinity)(infinity) X(nk) \ greater than or equal to epsilon} < infinity for every epsilon > 0. This result is used to establish some results on complete convergence for weighted sums of independent random variables. The main idea is that we devise an effective way of combining a certain maximal inequality of Hoffmann-Jorgensen and rates of convergence in the Weak Law of Large Numbers to establish results on complete convergence of weighted sums of independent-random variables. New results as well as simple new proofs of known ones illustrate the usefulness of our method in this context. We show further that this approach can be used in the study of almost sure convergence for weighted sums of independent random variables. Convergence rates in the almost sure convergence of some summability methods of lid random variables are also established.