Convergence rates in the law of large numbers

Abstract

Introduction. Let {Xk : k ^ 1} denote a sequence of random variables, {a" : n 2 1} a sequence of real numbers, {bn: n ^ 1} a nondecreasing sequence of positive real numbers and let Sn = lTk = xXk. Many of the limit theorems of probability theory may then be formulated as theorems concerning the convergence of either the sequence {P( | (S" -an)jb" | > e) : n ^ 1} or {P(sup," | (Sk -ak)jbk \>s): n ^ 1}, fore > 0, to an appropriate limiting value. It is the purpose of this paper to study the rates of convergence of such sequences. The results of this paper will include those previously announced in