This article is cited in 6 papers

Short Communications

On the Structure of the Infinitesimal $\sigma$-Algebra of a Gaussian Process

V. N. Tutubalin, M. I. Freidlin

Moscow

Abstract: Let $x(t)$ be a Gaussian stationary process $\mathfrak{M}_{+0}=\bigcap _{t>0}\mathfrak{M}_t$, where $\mathfrak{M}_t$ is the $\sigma $-algebra generated by $x(s),0\leq s\leq t$. It is proved that if the spectral density $f(\lambda)$ of the process satisfies the condition $f(\lambda)\geq{1}/{\lambda^p}$ for all $|\lambda|>\lambda_0$ and some $p>0$, the $\sigma $-algebra $\mathfrak{M}_{+0}$ is generated by $x(0),{dx(0)}/{dt},\dots,{dx^{(k)}{(0)}}/{dt^k}$, where $k$ is the order of the derivative the sample functions admit.

Received: 25.04.1960

 English version: Theory of Probability and its Applications, 1962, 7:2, 196–199