Let $W(T)$ for $0 \leq T < \infty$ be a standard Weiner process and suppose that $c_k$ and $b_k$ are fixed sequences of real numbers satisfying $0 \leq c_k < b_k < \infty$. Let $K(\omega)$ be the set of limit points (as $T \rightarrow \infty$) of $\frac{W(b_k;\omega) - W(c_k;\omega)}{\{2(b_k - c_k)\lbrack\log(b_k/(b_k - c_k)) + \log\log b_k\rbrack\}^{1/2}}$ where $\omega$ is a point in the probability space on which $W(T)$ is defined. We give necessary conditions on $b_k$ and $c_k$ to have $K(\omega) = \lbrack -1, 1\rbrack$ a.s. We also give some related results and discuss sharpness.
Publié le : 1983-11-14
Classification:
Increments of a Wiener process,
Wiener process,
law of iterated logarithm,
lag sums,
sums of random variables,
delayed sums,
60F15,
60G15,
60G17
@article{1176993449,
author = {Hanson, D. L. and Russo, Ralph P.},
title = {Some More Results on Increments of the Wiener Process},
journal = {Ann. Probab.},
volume = {11},
number = {4},
year = {1983},
pages = { 1009-1015},
language = {en},
url = {http://dml.mathdoc.fr/item/1176993449}
}
Hanson, D. L.; Russo, Ralph P. Some More Results on Increments of the Wiener Process. Ann. Probab., Tome 11 (1983) no. 4, pp. 1009-1015. http://gdmltest.u-ga.fr/item/1176993449/