The Annals of Probability

The Law of the Iterated Logarithm on Arbitrary Sequences for Stationary Gaussian Processes and Brownian Motion

Clifford Qualls

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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.

Article information

Source
Ann. Probab., Volume 5, Number 5 (1977), 724-739.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176995715

Digital Object Identifier
doi:10.1214/aop/1176995715

Mathematical Reviews number (MathSciNet)
MR451369

Zentralblatt MATH identifier
0375.60035

JSTOR
links.jstor.org

Subjects
Primary: 60F10: Large deviations
Secondary: 60F20: Zero-one laws 60G10: Stationary processes 60G17: Sample path properties 60J65: Brownian motion [See also 58J65]

Keywords
Stationary process Gaussian process law of iterated logarithm Brownian motion sure sequences

Citation

Qualls, Clifford. The Law of the Iterated Logarithm on Arbitrary Sequences for Stationary Gaussian Processes and Brownian Motion. Ann. Probab. 5 (1977), no. 5, 724--739. doi:10.1214/aop/1176995715. https://projecteuclid.org/euclid.aop/1176995715


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