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October, 1977 The Law of the Iterated Logarithm on Arbitrary Sequences for Stationary Gaussian Processes and Brownian Motion
Clifford Qualls
Ann. Probab. 5(5): 724-739 (October, 1977). DOI: 10.1214/aop/1176995715

Abstract

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.

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Clifford Qualls. "The Law of the Iterated Logarithm on Arbitrary Sequences for Stationary Gaussian Processes and Brownian Motion." Ann. Probab. 5 (5) 724 - 739, October, 1977. https://doi.org/10.1214/aop/1176995715

Information

Published: October, 1977
First available in Project Euclid: 19 April 2007

zbMATH: 0375.60035
MathSciNet: MR451369
Digital Object Identifier: 10.1214/aop/1176995715

Subjects:
Primary: 60F10
Secondary: 60F20 , 60G10 , 60G17 , 60J65

Keywords: Brownian motion , Gaussian process , Law of iterated logarithm , stationary process , sure sequences

Rights: Copyright © 1977 Institute of Mathematical Statistics

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Vol.5 • No. 5 • October, 1977
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