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February, 1983 Convergence Rates Related to the Strong Law of Large Numbers
James Allen Fill
Ann. Probab. 11(1): 123-142 (February, 1983). DOI: 10.1214/aop/1176993663


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.


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James Allen Fill. "Convergence Rates Related to the Strong Law of Large Numbers." Ann. Probab. 11 (1) 123 - 142, February, 1983.


Published: February, 1983
First available in Project Euclid: 19 April 2007

zbMATH: 0502.60021
MathSciNet: MR682804
Digital Object Identifier: 10.1214/aop/1176993663

Primary: 60F15
Secondary: 60F10 , 60G50 , 60J15

Keywords: boundary crossing probabilities , Brownian motion , Convergence rates , Invariance principles , large deviations , Law of iterated logarithm , laws of large numbers , Random walk

Rights: Copyright © 1983 Institute of Mathematical Statistics

Vol.11 • No. 1 • February, 1983
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