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February, 1979 An Improvement of Strassen's Invariance Principle
P. Major
Ann. Probab. 7(1): 55-61 (February, 1979). DOI: 10.1214/aop/1176995147


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.


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P. Major. "An Improvement of Strassen's Invariance Principle." Ann. Probab. 7 (1) 55 - 61, February, 1979.


Published: February, 1979
First available in Project Euclid: 19 April 2007

zbMATH: 0392.60034
MathSciNet: MR515812
Digital Object Identifier: 10.1214/aop/1176995147

Primary: 60G50
Secondary: 60B10

Keywords: invariance principle , Sums of independent random variables

Rights: Copyright © 1979 Institute of Mathematical Statistics

Vol.7 • No. 1 • February, 1979
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