A Representation Theorem on Stationary Gaussian Processes and Some Local Properties

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$\} > 0$.