The Annals of Probability

Convergence Rates Related to the Strong Law of Large Numbers

James Allen Fill

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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.

Article information

Ann. Probab., Volume 11, Number 1 (1983), 123-142.

First available in Project Euclid: 19 April 2007

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Zentralblatt MATH identifier


Primary: 60F15: Strong theorems
Secondary: 60F10: Large deviations 60G50: Sums of independent random variables; random walks 60J15

Random walk laws of large numbers convergence rates boundary crossing probabilities invariance principles large deviations law of iterated logarithm Brownian motion


Fill, James Allen. Convergence Rates Related to the Strong Law of Large Numbers. Ann. Probab. 11 (1983), no. 1, 123--142. doi:10.1214/aop/1176993663.

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