Convergence Rates Related to the Strong Law of Large Numbers

Abstract

Let $X_1, X_2, \cdots$ be independent random variables with common distribution function $F$, zero mean, unit variance, and finite moment generating function, and with partial sums $S_n$. According to the strong law of large numbers, $p_m \equiv P\big\{\frac{S_n}{n} > c_n \text{for some} n \geq m\big\}$ decreases to 0 as $m$ increases to $\infty$ when $c_n \equiv c > 0$. For general $c_n$'s the Hewitt-Savage zero-one law implies that either $p_m = 1$ for every $m$ or else $p_m \downarrow 0$ as $m \uparrow \infty$. Assuming the latter case, we consider here the problem of determining $p_m$ up to asymptotic equivalence. For constant $c_n$'s the problem was solved by Siegmund (1975); in his case the rate of decrease depends heavily on $F$. In contrast, Strassen's (1967) solution for smoothly varying $c_n = o(n^{-2/5})$ is independent of $F$. We complete the solution to the convergence rate problem by considering $c_n$'s intermediate to those of Siegmund and Strassen. The rate (Theorem 1.1) in this case depends on an ever increasing number of terms in the Cramer series for $F$ the more slowly $c_n$ converges to zero.