Open Access
Translator Disclaimer
October, 1976 A Representation Theorem on Stationary Gaussian Processes and Some Local Properties
Ruben Klein
Ann. Probab. 4(5): 844-849 (October, 1976). DOI: 10.1214/aop/1176995988

Abstract

Let $X(t, \omega, a \leqq t \leqq b, \omega \in \Omega$ be a real continuous stationary Gaussian process with mean 0 and covariance $R$. We prove that there exist analytic functions $f_n$ defined on $\lbrack a, b\rbrack$ and independent random variables $X_nN(0, 1), n = 0,1,2, \cdots$, such that the series $\sum^\infty_{n=0} f_n(t)X_n$ converges uniformly to $X(t)$ with probability 1. Among other applications of this representation theorem, we show that if the second spectral moment is infinite and $\int^\delta_0 (R(0) - R(t))^{-\frac{1}{2}} dt < \infty$ for some $0 < \delta \leqq b - a$, then for any given $u\in\mathbb{R}, P\{\omega\mid X_\omega^{-1}(u)$ is infinite$\} > 0$.

Citation

Download Citation

Ruben Klein. "A Representation Theorem on Stationary Gaussian Processes and Some Local Properties." Ann. Probab. 4 (5) 844 - 849, October, 1976. https://doi.org/10.1214/aop/1176995988

Information

Published: October, 1976
First available in Project Euclid: 19 April 2007

zbMATH: 0344.60028
MathSciNet: MR415749
Digital Object Identifier: 10.1214/aop/1176995988

Subjects:
Primary: 60G10
Secondary: 60G15, 60G17

Rights: Copyright © 1976 Institute of Mathematical Statistics

JOURNAL ARTICLE
6 PAGES


SHARE
Vol.4 • No. 5 • October, 1976
Back to Top