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August, 1978 A Strong Invariance Theorem for the Strong Law of Large Numbers
Jon A. Wellner
Ann. Probab. 6(4): 673-679 (August, 1978). DOI: 10.1214/aop/1176995488


Let $X_1, X_2,\cdots$ be i.i.d. random variables with mean 0 and variance 1. Let $S_n = X_1 + \cdots + X_n$, and let $\{H_n\}$ be the standard partial sum processes on $\lbrack 0, \infty)$ defined in terms of the $S_n$'s and normalized as in Strassen. Each function of the "tail" behavior of the process $H_n$ is the dual of a function of the "initial" behavior of the process $H_n$, the duality being induced by the time inversion map $R$. The dual role of "initial" and "tail" functions is used to exploit an extension of Strassen's invariance theorem for the law of the iterated logarithm due to Wichura, and thereby obtain limit theorems for a variety of functions of the "tail" behavior of the sums $S_n$. For example, with probability one, $$\lim \sup_{n\rightarrow \infty} (n/2 \log \log n)^\frac{1}{2} \max_{n\leqq k < \infty} (k^{-1}S_k) = 1$$ and $$\lim \sup_{n\rightarrow \infty} n^{-1} \max \{k \geqq 1: k^{-1}S_k \geqq \theta(2 \log \log n/n)^\frac{1}{2}\} = \theta^{-2}.$$


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Jon A. Wellner. "A Strong Invariance Theorem for the Strong Law of Large Numbers." Ann. Probab. 6 (4) 673 - 679, August, 1978.


Published: August, 1978
First available in Project Euclid: 19 April 2007

zbMATH: 0377.60034
MathSciNet: MR482966
Digital Object Identifier: 10.1214/aop/1176995488

Primary: 60F15
Secondary: 60B10

Keywords: invariance theorems , Law of the iterated logarithm , Strong law of large numbers , time inversion

Rights: Copyright © 1978 Institute of Mathematical Statistics


Vol.6 • No. 4 • August, 1978
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