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.
Publié le : 1974-08-14
Classification:
Maxima of Gaussian process,
asymptotic rates,
processes with stationary increments,
60G15,
60G17,
60E05
@article{1176996613,
author = {Marcus, M. B.},
title = {Asymptotic Maxima of Continuous Gaussian Processes},
journal = {Ann. Probab.},
volume = {2},
number = {6},
year = {1974},
pages = { 702-713},
language = {en},
url = {http://dml.mathdoc.fr/item/1176996613}
}
Marcus, M. B. Asymptotic Maxima of Continuous Gaussian Processes. Ann. Probab., Tome 2 (1974) no. 6, pp. 702-713. http://gdmltest.u-ga.fr/item/1176996613/