UDC 519.21

MATHEMATICS

Yu. A. ROZANOV

ON THE DENSITY OF GAUSSIAN DISTRIBUTIONS AND WIENER–HOPF INTEGRAL EQUATIONS

(Presented by Academician A. N. Kolmogorov, 11 IX 1965)

Let, on some measurable space \((\Omega,\mathfrak A)\) (the space of elementary events \(\omega\)), there be given a system of real measurable functions \(\xi(\omega,t)\) (random variables \(\xi(t)\)), depending on a parameter \(t\). Suppose that the \(\sigma\)-algebra \(\mathfrak A\) is generated by the variables \(\xi(t)\), i.e., coincides with the minimal \(\sigma\)-algebra containing all possible sets of the form \(\{\xi(\omega,t)\le x\}\). Suppose that on the \(\sigma\)-algebra \(\mathfrak A\) two Gaussian distributions \(P(d\omega)\) and \(P_1(d\omega)\) are given, i.e., such probability measures with respect to which the joint probability distributions of the random variables \(\xi(t)\) are Gaussian. Gaussian distributions are completely determined by two numerical characteristics—the mean \(A(t)\) and the correlation function \(B(s,t)\) of the given random variables:

\[ A(t)=\int \xi(\omega,t)P(d\omega), \]

\[ B(s,t)=\int[\xi(\omega,s)-A(s)][\xi(\omega,t)-A(t)]P(d\omega). \]

The question is asked under what conditions on the corresponding functions \(A(t)\), \(B(s,t)\) and \(A_1(t)\), \(B_1(s,t)\) the Gaussian distributions \(P(d\omega)\) and \(P_1(d\omega)\) will be mutually absolutely continuous, or, as one also says, equivalent. How can one find the density \(p(\omega)=P_1(d\omega)/P(d\omega)\)? These questions are of great interest in mathematical statistics, information theory, and other areas of probability theory; they are closely connected with a number of interesting problems in functional analysis and the theory of functions of a complex variable, and in recent years have attracted the attention of many mathematicians (the main results can be found, for example, in the survey article \((^1)\)).

Suppose the Gaussian measures \(P(d\omega)\) and \(P_1(d\omega)\) define probability distributions of a stationary process \(\xi(t)\); then the parameter \(t\) runs over the interval \([0,T]\), and the correlation functions \(B(s,t)=B(s-t)\) and \(B_1(s,t)=B_1(s-t)\) depend only on the difference \(s-t\) and are represented in the form

\[ B(t)=\int e^{i\lambda t}F(d\lambda),\qquad B_1(t)=\int e^{i\lambda t}F_1(d\lambda), \]

where \(F(d\lambda)\) and \(F_1(d\lambda)\) are some positive and bounded measures on the line \(-\infty<\lambda<\infty\). The solution of the question of the equivalence of \(P(d\omega)\) and \(P_1(d\omega)\) in the most general setting is easily reduced to the two cases considered, when either \(B(t)\equiv B_1(t)\), while the functions \(A(t)\) and \(A_1(t)\) are arbitrary, or \(A(t)\equiv A_1(t)\equiv 0\), while the correlation functions \(B(t)\) and \(B_1(t)\) are arbitrary.

In the relatively simple case when \(B(t)\equiv B_1(t)\), the solution of the question of equivalence was given in many works and consists in the following...

Denote by \(L_T^2(F)\) the class of all complex functions \(\varphi(\lambda)\) on the line \(-\infty<\lambda<\infty\) such that

\[ \int |\varphi(\lambda)|^2 F(d\lambda)<\infty,\qquad \inf_{c_k,\;0\leqslant t_k\leqslant T} \int\left|\varphi(\lambda)-\sum_k c_k e^{i\lambda t_k}\right|^2 F(d\lambda)=0. \]

For the equivalence of \(P(d\omega)\) and \(P_1(d\omega)\) it is necessary and sufficient that the difference \(a(t)=A(t)-A_1(t)\) admit a representation of the form

\[ a(t)=\int e^{i\lambda t}\varphi(\lambda)F(d\lambda),\qquad 0\leqslant t\leqslant T, \tag{1} \]

where \(\varphi(\lambda)\) is some function of the class \(L_T^2(F)\). It is convenient to assume \(A(t)\equiv 0\), which in no way restricts generality. Then, with respect to the distribution \(P(d\omega)\), the process \(\xi(t)\) admits a spectral expansion of the form

\[ \xi(t)=\int e^{i\lambda t}\Phi(d\lambda) \]

(here \(\Phi(d\lambda)\) is a measure with orthogonal values in \(L^2(\Omega)\)), and the density \(p(\omega)=P_1(d\omega)/P(d\omega)\) is described by the formula

\[ p(\omega)=D\exp\left\{\int\varphi(\lambda)\Phi(d\lambda)\right\}, \]

where \(D\) is a normalizing factor determined from the condition

\[ \int p(\omega)P(d\omega)=1, \]

and the function \(\varphi(\lambda)\) is the same as in expression (1).

Let \(A(t)\equiv A_1(t)\equiv 0\). Here, for some special cases, quite definitive results have been obtained, but there has as yet been no complete solution of the question. Such a solution, in our opinion, is proposed below. In doing so we shall assume that

\[ 0<\underline{\lim}_{\lambda\to\infty}\frac{f_1(\lambda)}{f(\lambda)} \leqslant \overline{\lim}_{\lambda\to\infty}\frac{f_1(\lambda)}{f(\lambda)}<\infty, \tag{2} \]

where \(f(\lambda)\) and \(f_1(\lambda)\) denote the densities of the spectral measures \(F(d\lambda)\) and \(F_1(d\lambda)\) with respect to some positive measure \(G(d\lambda)\) (for absolutely continuous spectral measures, the functions \(f(\lambda)\) and \(f_1(\lambda)\) are simply the spectral densities of the stationary process \(\xi(t)\)). We note that deviations from condition (2) hardly deserve serious attention. Instead of condition (2), without restricting generality, one may assume that for all \(\lambda\)

\[ c\leqslant f_1(\lambda)/f(\lambda)\leqslant C \]

for some positive constants \(c\) and \(C\).

Denote by \(L_T^2(F,F_1)\) the class of all functions \(\varphi(\lambda,\mu)\) in the plane \(-\infty<\lambda,\mu<\infty\) such that

\[ \iint |\varphi(\lambda,\mu)|^2 F(d\lambda)F_1(d\mu)<\infty, \]

\[ \inf_{c_{kj},\;0\leqslant t_k,t_j\leqslant T} \iint\left|\varphi(\lambda,\mu)-\sum_{k,j} c_{kj}e^{i(\lambda t_k-\mu t_j)}\right|^2 F(d\lambda)F_1(d\mu)=0. \]

Theorem 1. For the equivalence of \(P(d\omega)\) and \(P_1(d\omega)\) it is necessary and sufficient that the difference \(b(s,t)=B(s-t)-B_1(s-t)\) of the corresponding correlation functions admit a representation of the form

\[ b(s,t)=\iint e^{-i(\lambda s-\mu t)}\varphi(\lambda,\mu)F(d\lambda)F_1(d\mu), \qquad 0\leqslant s,t\leqslant T, \tag{3} \]

where \(\varphi(\lambda,\mu)\) is some function of the class \(L_T^2(F,F_1)\).
The density \(p(\omega)\) is described by the formula:

\[ p(\omega)=D\exp\left\{-\frac12\operatorname*{l.i.m.}_{n\to\infty}\left[ \iint_{|\varphi|\leqslant n}\varphi(\lambda,\mu)\Phi(d\lambda)\Phi(d\mu)- \int_{|\varphi|\leqslant n}\varphi(\lambda,\lambda)F(d\lambda)\right]\right\}, \tag{4} \]

where \(D\) is the normalizing factor determined from the condition

\[ \int p(\omega)P(d\omega)=1, \]

and the function \(\varphi(\lambda,\mu)\) is the same as in expression (3).

Formula (4) is obviously simplified if the integral exists

\[ \int_{-\infty}^{\infty}\varphi(\lambda,\lambda)F(d\lambda). \]

Expression (3), as well as (1), is a Wiener–Hopf integral equation with respect to the unknown function \(\varphi\in L_T^2\). To emphasize the importance of the study of such equations (with arbitrary functions \(a(t)\) and \(b(s,t)\)), we note that many extremal problems of great interest for applications are reduced to their solution.

It is clear that, as an element of the corresponding space \(L_T^2\), the solution \(\varphi\) of equations (1) or (3) is always unique. It is known that, for equation (1), the condition for the existence of a solution is the following: there exists a constant \(C\) such that

\[ \left|\sum c(t)a(t)\right|^2 \leq C \int\left|\sum c(t)e^{i\lambda t}\right|^2 F(d\lambda) \]

for all \(c(t)\), \(0\leq t\leq T\). A completely analogous existence condition for equation (3) is:

\[ \left|\sum c(s,t)b(s,t)\right|^2 \leq C \iint\left|\sum c(s,t)e^{i(\lambda s-\mu t)}\right|^2 F(d\lambda)F_1(d\mu). \]

We formulate a result that opens one possible direction for the further study of equations (1) and (3), when the measure \(F(d\lambda)\) is absolutely continuous and the density \(f(\lambda)=F(d\lambda)/d\lambda\) has a prescribed asymptotic behavior as \(\lambda\to\infty\).

Theorem 2. Suppose that for some integer \(n\)

\[ 0<\underline{\lim}_{\lambda\to\infty}\lambda^{2n}f(\lambda) \leq \overline{\lim}_{\lambda\to\infty}\lambda^{2n}f(\lambda)<\infty. \]

For a solution of equation (1) to exist, it is necessary and sufficient that the function \(a(t)\) have an \((n-1)\)-st absolutely continuous derivative \(a^{(n-1)}(t)\) and

\[ \int_0^T |a^{(n)}(t)|^2\,dt<\infty. \]

For a solution of equation (3) to exist, it is necessary and sufficient that the function \(b(s,t)\) have absolutely continuous partial derivatives
\(\partial^{2n-1}b(s,t)/\partial s^{\,n-1}\partial t^n\),
\(\partial^{2n-1}b(s,t)/\partial s^n\partial t^{\,n-1}\), and

\[ \int_0^T\int_0^T \left|\frac{\partial^{2n}b(s,t)}{\partial s^n\partial t^n}\right|^2\,ds\,dt<\infty. \]

Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
10 IX 1965

REFERENCES

1. Yu. A. Rozanov, Theory of Probability and Its Applications, 9, 3 (1964).