A Note on the Rate of Convergence and Its Applications

Abstract

Let $S_n$ denote the partial sums of i.i.d. random variables with mean zero and moment generating function existing in some neighborhood of the origin. We give explicit upper bounds for $P_m^+ = P(S_n \geqq a + bn$ for some $n \geqq m)$ and $P_m = P(|S_n| \geqq a + bn$ for some $n \geqq m), a \geqq 0, b > 0$. These bounds immediately give the rate of convergence for the strong law of large numbers. An application is also made to a sequential selection procedure.