## The Annals of Probability

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

### Asymptotic Maxima of Continuous Gaussian Processes

#### 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**

links.jstor.org

**Subjects**

Primary: 60G15: Gaussian processes

Secondary: 60G17: Sample path properties 60E05: Distributions: general theory

**Keywords**

Maxima of Gaussian process asymptotic rates processes with stationary increments

#### Citation

Marcus, M. B. Asymptotic Maxima of Continuous Gaussian Processes. Ann. Probab. 2 (1974), no. 4, 702--713. doi:10.1214/aop/1176996613. https://projecteuclid.org/euclid.aop/1176996613