## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 43, Number 2 (1972), 580-596.

### Asymptotic Properties of Gaussian Processes

Clifford Qualls and Hisao Watanabe

#### Abstract

We study separable mean zero Gaussian processes $X(t)$ with correlation $\rho (t, s)$ for which $1 - \rho (t, s)$ is asymptotic to a regularly varying (at zero) function of $|t - s|$ with exponent $0 < \alpha \leqq 2$. For such processes, we obtain the asymptotic distribution of the maximum of $X(t)$. This result is used to obtain a result for $X(t)$ as $t \rightarrow \infty$ similar to the so-called law of the iterated logarithm.

#### Article information

**Source**

Ann. Math. Statist., Volume 43, Number 2 (1972), 580-596.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177692638

**Digital Object Identifier**

doi:10.1214/aoms/1177692638

**Mathematical Reviews number (MathSciNet)**

MR307318

**Zentralblatt MATH identifier**

0247.60031

**JSTOR**

links.jstor.org

#### Citation

Qualls, Clifford; Watanabe, Hisao. Asymptotic Properties of Gaussian Processes. Ann. Math. Statist. 43 (1972), no. 2, 580--596. doi:10.1214/aoms/1177692638. https://projecteuclid.org/euclid.aoms/1177692638