Let $X(t)$ be a stationary Gaussian process with continuous sample paths, mean zero, and a covariance function satisfying (a) $r(t) \sim 1 - C|t|^\alpha$ as $t \rightarrow 0, 0 < \alpha \leqq 2$ and $C > 0$; and (b) $r(t) \log t = o(1)$ as $t \rightarrow \infty$. Let $\{t_n\}$ be any sequence of times with $t_n \uparrow \infty$. Then, for any nondecreasing function $f$, one obtains $P\{X(t_n) > f(t_n) \mathrm{i.o.}\} = 0$ or 1 according to a certain integral test. This result both combines and generalizes the law of iterated logarithm results for discrete and continuous time processes. In particular, it is shown that any sequence $t_n$ satisfying $\lim \sup_{n\rightarrow\infty} (t_n - t_{n-1})(\log n)^{1/\alpha} < \infty$ captures continuous time in the sense that the upper and lower class functions for the law of the iterated logarithm of $X(t_n)$ are exactly the same as those for the continuous time $X(t)$. Analogous results are obtained for Brownian motion.
Publié le : 1977-10-14
Classification:
Stationary process,
Gaussian process,
law of iterated logarithm,
Brownian motion,
sure sequences,
60F10,
60F20,
60G10,
60G17,
60J65
@article{1176995715,
author = {Qualls, Clifford},
title = {The Law of the Iterated Logarithm on Arbitrary Sequences for Stationary Gaussian Processes and Brownian Motion},
journal = {Ann. Probab.},
volume = {5},
number = {6},
year = {1977},
pages = { 724-739},
language = {en},
url = {http://dml.mathdoc.fr/item/1176995715}
}
Qualls, Clifford. The Law of the Iterated Logarithm on Arbitrary Sequences for Stationary Gaussian Processes and Brownian Motion. Ann. Probab., Tome 5 (1977) no. 6, pp. 724-739. http://gdmltest.u-ga.fr/item/1176995715/