Randon-Nikodym Derivatives of Stationary Gaussian Measures

Abstract

A problem of considerable importance in time series analysis is that of determining whether two Gaussian processes are equivalent, i.e., mutually absolutely continuous with respect to each other. If we are given $\{\Omega, \mathscr{a}, P_k\}, k = 0, 1$, where $\Omega$ is a set of real valued functions on some interval $\lbrack a, b\rbrack, \mathscr{a}$ is a Borel field of subsets of $\Omega$ and $P_k$ is a Gaussian probability measure on $\mathscr{a}$, then the problem is to determine the conditions under which $P_0$ is equivalent to $P_1$. Feldman (1958) has shown that a certain dichotomy exists in this problem in the following sense. If $\Omega$ is a linear space, then either $P_0$ and $P_1$ are equivalent or they are perpendicular, i.e., mutually singular. In addition, Feldman (1958) has shown, using some results of Segal (1958), that if $\Lambda$ is the linear span of $\Omega$ and the real constants, then $P_0$ and $P_1$ are equivalent iff the $P_0$-equivalence classes of $\Lambda$ are the same as the $P_1$-equivalence classes of $\Lambda$ and the identity correspondence between the $L_2(P_0)$ closure of $\Lambda$ and the $L_2(P_1)$ closure of $\Lambda$ is a bounded invertible operator $T$ such that $(T^\ast T)^{\frac{1}{2}} - I$ is a Hilbert-Schmidt operator. It is well known that a Gaussian process, and hence its probability measure, is uniquely determined by its covariance function, $R(s, t)$, assuming, of course, that the mean value functions is zero. It should, therefore, be possible to determine when $P_0$ and $P_1$ are equivalent in terms of the covariance functions of the two processes, $R_0(s, t), R_1(s, t)$. Indeed, it is desirable to have a theorem which states the conditions for equivalence of $P_0$ and $P_1$ in terms of the covariance functions, since in most applications the only information concerning the Gaussian processes consists of the covariance functions. In this sense, Feldman's (1958) theorem is in a form which is not too suitable for practical applications. This situation has been corrected by Feldman (1960) who, using his earlier results, has given the necessary and sufficient conditions for equivalence of $P_0$ and $P_1$ in terms of the covariance functions for a rather wide class of stationary Gaussian processes with zero mean value functions. However, Feldman (1960) has not computed the Radon-Nikodym derivative (RND) $dP_1/dP_0$ for pairs of equivalent Gaussian processes. The RND, or likelihood ratio, is of considerable importance in statistical inference for stochastic processes, as is well known. Recently, Parzen (1961), (1962) has described a new approach to time series analysis which is based on the notion of a reproducing kernel Hilbert space (RKHS). The theory of RKHS has been given by Aronszajn (1950). Using some results due to Aronszajn (1950) and Hajek (1958a), (1958b) Parzen (1962) has derived a necessary and sufficient condition for equivalence of pairs of Gaussian processes which have the same covariance function but have different mean value functions. In addition, he gives an explicit expression for the RND associated with pairs of equivalent Gaussian processes. The present work stems from a desire to derive the necessary and sufficient conditions for equivalence of pairs of stationary Gaussian processes with different covariance functions and zero mean value functions and to compute the RND for pairs of certain equivalent Gaussian processes. The method of proof is based on the RKHS approach used by Parzen. Thus, part of the results obtained will serve as an alternate derivation of Feldman's (1960) result. However, Feldman's proof involves a large number of complicated manipulations, while the present proof has the advantage of requiring only a few elementary manipulations. This simplification is achieved by employing convergence theorems of Aronszajn (1950), Hajek (1958a), (1958b) and Hormander (1963) and an expression for an inner product due to Parzen (1961).