## The Annals of Probability

- Ann. Probab.
- Volume 7, Number 1 (1979), 55-61.

### An Improvement of Strassen's Invariance Principle

#### Abstract

Let a distribution function $F(x), \int xdF(x) = 0, \int x^2dF(x) = 1$ be given. Strassen constructed two sequences $X_1, X_2, \ldots$ and $Y_1, Y_2, \ldots$ of independent, identically distributed random variables, the $X_i$ with distribution function $F(x)$ and the $Y_i$ with standard normal distribution, in such a way that the partial sums $S_n = \sum^n_{i = 1}X_i$ and $T_n = \sum^n_{i = 1} Y_i$ satisfy the relation $|S_n - T_n| = O((n \log \log n)^\frac{1}{2})$ with probability 1. Earlier we proved that this result cannot be improved. Now we show however that an approximation $|S_n - T_n| = O(n^\frac{1}{2})$ can be achieved, if the $Y_i$ are independent normal variables whose variances are appropriately chosen.

#### Article information

**Source**

Ann. Probab., Volume 7, Number 1 (1979), 55-61.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176995147

**Digital Object Identifier**

doi:10.1214/aop/1176995147

**Mathematical Reviews number (MathSciNet)**

MR515812

**Zentralblatt MATH identifier**

0392.60034

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G50: Sums of independent random variables; random walks

Secondary: 60B10: Convergence of probability measures

**Keywords**

Invariance principle sums of independent random variables

#### Citation

Major, P. An Improvement of Strassen's Invariance Principle. Ann. Probab. 7 (1979), no. 1, 55--61. doi:10.1214/aop/1176995147. https://projecteuclid.org/euclid.aop/1176995147