## The Annals of Probability

### Asymptotic Maxima of Continuous Gaussian Processes

M. B. Marcus

#### Abstract

Let $X(t)$ be a stationary Gaussian process with continuous sample paths. The behavior of $|X(t)|$ as $t \rightarrow \infty$ is considered. In particular, conditions on the spectrum of the process are given which determine whether $\lim \sup_{t\rightarrow\infty}|X(t)|/(\log t)^{\frac{1}{2}} = \operatorname{Const.} > 0$. These conditions are complete except when the spectrum of the process is continuous-singular. The main concern of this paper is to study the asymptotic behavior of some specific examples of $X(t)$ with continuous-singular spectra. Many examples are given showing the asymptotic behavior of stationary Gaussian processes with discrete spectra and their indefinite integrals.

#### Article information

Source
Ann. Probab., Volume 2, Number 4 (1974), 702-713.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176996613

Digital Object Identifier
doi:10.1214/aop/1176996613

Mathematical Reviews number (MathSciNet)
MR370726

Zentralblatt MATH identifier
0304.60024

JSTOR