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.
Publié le : 1981-12-14
Classification:
Increments of partial sums,
strong approximations,
strong invariance principles,
large deviations,
60F15,
60F10,
60F17,
60G15,
60G17
@article{1176994269,
author = {Csorgo, M. and Steinebach, J.},
title = {Improved Erdos-Renyi and Strong Approximation Laws for Increments of Partial Sums},
journal = {Ann. Probab.},
volume = {9},
number = {6},
year = {1981},
pages = { 988-996},
language = {en},
url = {http://dml.mathdoc.fr/item/1176994269}
}
Csorgo, M.; Steinebach, J. Improved Erdos-Renyi and Strong Approximation Laws for Increments of Partial Sums. Ann. Probab., Tome 9 (1981) no. 6, pp. 988-996. http://gdmltest.u-ga.fr/item/1176994269/