A Strong Invariance Theorem for the Strong Law of Large Numbers

Abstract

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 $$\lim \sup_{n\rightarrow \infty} n^{-1} \max \{k \geqq 1: k^{-1}S_k \geqq \theta(2 \log \log n/n)^\frac{1}{2}\} = \theta^{-2}.$$