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December, 1981 Improved Erdos-Renyi and Strong Approximation Laws for Increments of Partial Sums
M. Csorgo, J. Steinebach
Ann. Probab. 9(6): 988-996 (December, 1981). DOI: 10.1214/aop/1176994269


Let $X_1, X_2,\cdots$ be an i.i.d. sequence with $EX_1 = 0, EX^2_1 = 1, Ee^{tX_1} < \infty (|t| < t_0)$, and partial sums $S_n = X_1 + \cdots + X_n$. Starting from some analogous results for the Wiener process, this paper studies the almost sure limiting behaviour of $\max_{0 \leq n \leq N - a_N} a^{-1/2}_N (S_{n + a_N} - S_n)$ as $N \rightarrow \infty$ under various conditions on the integer sequence $a_N$. Improvements of the Erdos-Renyi law of large numbers for partial sums are obtained as well as strong invariance principle-type versions via the Komlos-Major-Tusnady approximation. An appearing gap between these two results is also going to be closed.


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M. Csorgo. J. Steinebach. "Improved Erdos-Renyi and Strong Approximation Laws for Increments of Partial Sums." Ann. Probab. 9 (6) 988 - 996, December, 1981.


Published: December, 1981
First available in Project Euclid: 19 April 2007

MathSciNet: MR632971
Digital Object Identifier: 10.1214/aop/1176994269

Primary: 60F15
Secondary: 60F10 , 60F17 , 60G15 , 60G17

Keywords: increments of partial sums , large deviations , strong approximations , Strong invariance principles

Rights: Copyright © 1981 Institute of Mathematical Statistics

Vol.9 • No. 6 • December, 1981
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