Let $\{W(t), 0 \leq t < +\infty\}$ be a standard Wiener process and $0 < b_t \leq t$ be a nondecreasing function of $t$. The properties of the process $Y_1(t) = b^{-1/2}_t \sup_{0\leq s \leq t - b_t}(W(s + b_t) - W(s))$ are investigated. One of the results says that $\lim_{t\rightarrow\infty}(Y_1(t) - (2 \log tb^{-1}_t)^{1/2}) = 0$ a.s. if $b_t$ is "much less" than $t$. Analogous properties of similar processes are studied.
Publié le : 1982-08-14
Classification:
Law of the iterated logarithm,
Wiener process,
empirical density function,
Kiefer process,
60F15,
60G17
@article{1176993771,
author = {Revesz, P.},
title = {On the Increments of Wiener and Related Processes},
journal = {Ann. Probab.},
volume = {10},
number = {4},
year = {1982},
pages = { 613-622},
language = {en},
url = {http://dml.mathdoc.fr/item/1176993771}
}
Revesz, P. On the Increments of Wiener and Related Processes. Ann. Probab., Tome 10 (1982) no. 4, pp. 613-622. http://gdmltest.u-ga.fr/item/1176993771/