Inequalities for the Law of Large Numbers

Abstract

Let $X_1, X_2, X_3, \cdots$ be independent random variables and $a_1, a_2, a_3, \cdots$ positive real numbers. Define $F(t) = \sup_k P\{|X_k| > t\}$ and $S_m = \sum^m_{k=1} a_k X_k.$ Inequalities of the form $P\{\sup_m|S_m| > \delta\} \leqq C \sum_k \int^1_0 \varphi'(u)F(u/a_k) du$ are given for a large class of functions $\varphi$, as well as inequalities of a somewhat different form that are appropriate for considering exponential convergence rates. Examples of how the inequalities can be used to prove rate theorems are also given.