Complete convergence and almost sure convergence of weighted sums of random variables

Let r>1. For each n≥1, let {X_nk, -∞<k<∞} be a sequence of independent real random variables. We provide some very relaxed conditions which will guarantee {Mathematical expression} for every ε>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 effetive way of combining a certain maximal inequality of Hoffmann-Jørgensen 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 iid random variables are also established.