## The Annals of Probability

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

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

#### 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