Let $W(t)$ be a standardized Wiener process. In this paper we prove that $\lim \sup_{T\rightarrow\infty} \max_{a_T \leq t \leq T}\frac{|W(T) - W(T - t)|}{\{2t\lbrack\log(T/t) + \log \log t\rbrack\}^{1/2}} = 1 \text{a.s.}$ under suitable conditions on $a_T$. In addition we prove various other related results all of which are related to earlier work by Csorgo and Revesz. Let $\{X_k\}$ be an i.i.d. sequence of random variables and let $S_N = X_1 + \cdots + X_N$. Our original objective was to obtain results similar to the ones obtained for the Wiener process but with $N$ replacing $T$ and $S_N$ replacing $W(T)$. Using the work of Komlos, Major, and Tusnady on the invariance principle, we obtain the desired results for i.i.d. sequences as immediate corollaries to our work for the Wiener process.
Publié le : 1983-08-14
Classification:
Increments of a Wiener process,
Wiener process,
law of iterated logarithm,
lag sums,
sums of random variables,
60F15,
60G15,
60G17
@article{1176993505,
author = {Hanson, D. L. and Russo, Ralph P.},
title = {Some Results on Increments of the Wiener Process with Applications to Lag Sums of I.I.D. Random Variables},
journal = {Ann. Probab.},
volume = {11},
number = {4},
year = {1983},
pages = { 609-623},
language = {en},
url = {http://dml.mathdoc.fr/item/1176993505}
}
Hanson, D. L.; Russo, Ralph P. Some Results on Increments of the Wiener Process with Applications to Lag Sums of I.I.D. Random Variables. Ann. Probab., Tome 11 (1983) no. 4, pp. 609-623. http://gdmltest.u-ga.fr/item/1176993505/