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Short Communications

On a Density of one Gaussian Distribution with Respect to Another

Yu. A. Rozanov

Moscow

Abstract: In this paper two arbitrary Gaussian measures $P_1(d\omega)$ and $P_2(d\omega)$ of a stochastic process $\{\xi_\alpha(\omega)\}$ with an abstract parameter $\alpha$ are considered. It is proved that they are equivalent if and only if the operator $B$ (in (12)) on the Hilbert space $H$ of random variables (10) has a pure point spectrum, and the eigen-vectors and the eigen-values of $B$ satisy conditions (15) and (16); the density $p(\omega)=P_1(d\omega)/P_2(d\omega)$ satisfies equation (17).

Received: 24.08.1960

 English version: Theory of Probability and its Applications, 1962, 7:1, 82–87