Abstract
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Reports of the Academy of Sciences of the USSR
- Volume 171, No. 6
MATHEMATICS
Yu. A. ROZANOV
ON GAUSSIAN LINEAR FUNCTIONALS ON A COUNTABLY HILBERT SPACE
(Presented by Academician A. N. Kolmogorov on 4 III 1966)
Let (U) be a real separable countably Hilbert space, whose topology is given by means of a countable number of scalar products
[
(u,v)_1,\quad (u,v)_2,\ldots,
\tag{1}
]
(X=U') is the set of continuous linear functionals (x=\langle u,x\rangle) on the space (U); (\mathfrak A) is the (\sigma)-algebra of Borel sets (with respect to the weak topology) of the space (X).
A probability measure (P(dx)) on the (\sigma)-algebra (\mathfrak A) is called Gaussian if, for every (u\in U), the random variable (u=\langle u,x\rangle), considered as a function of (x\in X), is Gaussian. This is equivalent to saying that the characteristic functional
[
\chi(u)=\int_X e^{i\langle u,x\rangle}P(dx)
]
of the measure (P(dx)) has the form
[
\chi(u)=e^{ia(u)-\frac12 b(u,u)},
]
where
[
a(u)=\int_X \langle u,x\rangle P(dx),
\tag{2}
]
[
b(u,v)=\int_X [\langle u,x\rangle-a(u)][\langle v,x\rangle-a(v)]P(dx).
\tag{3}
]
Measures in linear topological spaces and arbitrary Gaussian measures have been the subject of numerous and relatively recent investigations. The principal results are contained partly already in the monograph ((^1)), in survey articles ((^2,\ ^3)) (see also ((^4))). The facts given below, concerning Gaussian measures in the space of linear functionals on a countably Hilbert space, are a simple consequence of these results.
Let (P(dx)) be a Gaussian measure on the (\sigma)-algebra (\mathfrak A) of the space (X=U'), and let (a(u)) and (b(u,v)) be the functionals on the countably Hilbert space (U) defined by formulas (2) and (3); (a(u)) is called the mathematical expectation, and (b(u,v)) the correlation functional.
(a(u)), (b(u,v)) ((u,v\in U)) are the mathematical expectation and correlation functional of some Gaussian measure if and only if they are representable in the form
[
a(u)=(u,a)_n;\qquad b(u,v)=(Bu,v)_n,
\tag{4}
]
where ((u,v)_n) is some scalar product from (1); (a) is an element of the Hilbert space (\bar U=U_n) with this scalar product, obtained by completing (U); (B) is a positive nuclear operator in (\bar U).
Let us establish, for example, formula (4) for the correlation functional—
$b(u,v)$. As is known, this is a positive bilinear functional, which for some $n$ is bounded not only on the sphere $(u,u)_n \leqslant 1$, but also on a set of the form $(Su,u)_n \leqslant 1$, where $S$ is a positive nuclear operator in $\overline U = U_n$ (see (²), p. 414). This says that $b(u,v)$ is representable in the form (4), where $B$ is a linear positive operator such that $Bu=0$ when $Su=0$ and the product $BS^{-1}$ is bounded. If $u_1,u_2,\ldots$ is an orthonormal system of eigenvectors of the nuclear operator $S$, corresponding to positive eigenvalues $\lambda_1,\lambda_2,\ldots$, then $|BS^{-1}u_k|=\lambda_k^{-1}|Bu_k|\leq C$, whence it is seen that, together with the series $\sum \lambda_k$, the series $\sum |Bu_k|$ also converges. Consequently, $B$ is a nuclear operator (see (¹), p. 65).
Let us next consider two Gaussian measures $P(dx)$ and $P_1(dx)$. Like any Gaussian measures, they are either mutually absolutely continuous (that is, equivalent) or perpendicular. We indicate conditions for their equivalence and an expression for the corresponding density $p(x)=P_1(dx)/P(dx)$.
For the equivalence of $P(dx)$ and $P_1(dx)$ it is necessary and sufficient that, first, the corresponding mathematical expectations $a(u)$, $a_1(u)$ and correlation functionals $b(u,v)$, $b_1(u,v)$ have the form (4) for some common $n$; second, the corresponding nuclear operators $B$ and $B_1$ in the Hilbert space $\overline U=U_n$ have the same null subspace $\overline U^0$, and the difference $I-C$, $C\equiv B^{-1/2}B_1B^{-1/2}$, be a Hilbert–Schmidt operator in the subspace $\overline U^+$ orthogonal to $\overline U^0$; third, the difference of the corresponding elements $a_1-a$, defining by formula (4) the mathematical expectations $a(u)$ and $a_1(u)$, belong to the domain of definition of the operator $B^{-1/2}$.
This fact (cf. (⁵)) follows easily from the general condition for equivalence of Gaussian measures (see (³), p. 449), which in the case under consideration can be reformulated as follows: for a complete system in $\overline U^+$ of elements $v_1,v_2,\ldots$ such that $(Bv_k,v_k)_n=1$ and
[
(Bv_k,v_j)_n=0 \quad \text{for } k\ne j;\quad k,j=1,2,\ldots,
]
the series
[
\sum_{k,j}\left|(Bv_k,v_j)_n-(B_1v_k,v_j)_n\right|^2
]
and
[
\sum_k (v_k,a_1-a)^2
]
converge, with $a_1-a\in\overline U^+$. If one puts
[
u_k=B^{1/2}v_k,
]
then $u_1,u_2,\ldots$ will be a complete orthonormal system in $\overline U^+$. Moreover,
[
(Bv_k,v_j)_n-(B_1v_k,v_j)_n=(I-Cu_k,u_j)_n
]
and the series
[
\sum_{k,j}(I-Cu_k,u_j)_n^2,
]
[
\sum_k (B^{-1/2}u_k,a_1-a)_n^2
]
are convergent. This is equivalent to the fact that the difference $I-C$ is a Hilbert–Schmidt operator (see (¹), p. 50) and that the element $a_1-a$ belongs to the domain of definition of the operator $B^{-1/2}$.
In order to find an expression for the density $p(x)=P_1(dx)/P(dx)$ of the equivalent measures $P(dx)$ and $P_1(dx)$, we turn to the above-mentioned space $\overline U=U_n$. On it let us consider the new scalar product
[
(u,v)=(Bu,v)_n.
]
Let $\overline{\overline U}$ denote the corresponding completion of the space $\overline U$ with respect to this scalar product. To each element $u$ of the Hilbert space $\overline{\overline U}$ there corresponds a certain measurable function $u=u(x)$ of $x$. This correspondence is such that $u(x)=\langle u,x\rangle$ for $u\in U$, and to a convergent sequence of elements $u_1,u_2,\ldots$ there corresponds a sequence of functions $u_1=u_1(x), u_2=u_2(x),\ldots$ converging in mean square. The operator $B^{-1/2}$, considered in the space $\overline U$, is bounded on the ellipsoid of the form $(u,u)_n\leqslant 1$ and therefore can be uniquely defined on the whole subspace $\overline U^+$ of the space $\overline{\overline U}$.
Let $u_1,u_2,\ldots$ be a complete system of eigenvectors of the operator
[
C=B^{-1/2}B_1B^{-1/2}
]
in the Hilbert space $\overline U^+$, and let $\sigma_1^2,\sigma_2^2,\ldots$ be the system
corresponding eigenvalues. Without loss of generality, one may assume (a(u)\equiv 0). Put
[
m_k=(u_k,B^{-1/2}a_1),\qquad v_k=B^{-1/2}u_k,\qquad k=1,2,\ldots
\tag{5}
]
The density (p(x)=P_1(dx)/P(dx)) can be expressed by the formula
[
\log p(x)=-\frac{1}{2}\sum_k\left{\log\sigma_k^2+
\frac{[v_k(x)-m_k]^2}{\sigma_k^2}-v_k(x)^2\right},
\tag{6}
]
where (v_k=v_k(x)) are measurable functions of (x) corresponding to the elements (v_k\in\bar U) ((k=1,2,\ldots)).
This formula is easily obtained from general results, if one takes into account that the random variables (v_1,v_2,\ldots) are independent both with respect to the measure (P(dx)) and with respect to the measure (P_1(dx)).
Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR
Received
23 II 1966
CITED LITERATURE
- I. M. Gelfand, N. Ya. Vilenkin, Generalized Functions, vol. 4, Moscow, 1961.
- Yu. V. Prohorov, Proc. IV Berkeley Symposium on Math. Statistics and Probability, 2, 1960.
- Yu. A. Rozanov, Theory of Probability and Its Applications, 9, 3 (1964).
- Yu. A. Rozanov, DAN, 165, No. 5 (1965).
- C. R. Rao, V. S. Varadarajan, Sankhyā, A-25 (1963).