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.