## The Annals of Probability

- Ann. Probab.
- Volume 6, Number 4 (1978), 673-679.

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

#### Article information

**Source**

Ann. Probab., Volume 6, Number 4 (1978), 673-679.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176995488

**Digital Object Identifier**

doi:10.1214/aop/1176995488

**Mathematical Reviews number (MathSciNet)**

MR482966

**Zentralblatt MATH identifier**

0377.60034

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F15: Strong theorems

Secondary: 60B10: Convergence of probability measures

**Keywords**

Law of the iterated logarithm strong law of large numbers time inversion invariance theorems

#### Citation

Wellner, Jon A. A Strong Invariance Theorem for the Strong Law of Large Numbers. Ann. Probab. 6 (1978), no. 4, 673--679. doi:10.1214/aop/1176995488. https://projecteuclid.org/euclid.aop/1176995488