UDC 519.21

MATHEMATICS

Yu. I. Golyosov, A. A. Tempelman

ON THE EQUIVALENCE OF MEASURES CORRESPONDING TO GAUSSIAN VECTOR-VALUED FUNCTIONS

(Presented by Academician A. N. Kolmogorov on 14 VI 1968)

In what follows (\mathcal H) is a Hilbert space; ((\cdot,\cdot)) and (|\cdot|) are the scalar product and norm in (\mathcal H); (\mathcal G(\mathcal H)) is the Hilbert space of Hilbert–Schmidt operators with scalar product
[
(A,B){\mathcal G(\mathcal H)}=\operatorname{Sp}(AB^*);
]
(K(\mathcal H)) is the set of nuclear operators; (\mathcal G+(\mathcal H)) and (K_+(\mathcal H)) are the corresponding subsets of positive definite operators.

Let ((\Omega,\mathfrak B)) be a measurable space; (\mathcal P) a probability measure on (\mathfrak B). An (\mathcal H)-valued random Gaussian function (\xi(t)), (t\in T), determines the mathematical expectation (m(t)) and the correlation function (R(s,t)):
[
M(x,\xi(t))=(x,m(t)),\qquad
M(x,\xi(s)-m(s))(y,\xi(t)-m(t))=(x,R(s,t)y),
]
(x,y\in\mathcal H); (m(t)) is an (\mathcal H)-valued function; (R(s,t)) is a (K(\mathcal H))-valued positive definite function; (\widetilde R(s,t)=R(s,t)+m(s)\otimes m(t)).

1. Let (R(s,t)), (s,t\in T), be a (K(\mathcal H))-valued positive definite function; (H) a Hilbert space of (\mathcal H)-valued functions defined on (T); (\langle\cdot,\cdot\rangle) the scalar product in (H).

Definition. The function (R(s,t)), (s,t\in T), is called the reproducing kernel of the space (H) if, for any (t\in T), (x\in\mathcal H), a) the function (R(\cdot,t)x\in H); b) (\langle R(\cdot,t)x,m(\cdot)\rangle=(x,m(t))).

The space (H) is uniquely determined by the function (R(s,t)) as the linear hull of the functions (m_{t,x}(\cdot)=R(\cdot,t)x), closed with respect to the scalar product
[
\langle R(\cdot,s)x,R(\cdot,t)y\rangle=(x,R(s,t)y)
]
(cf. (1)).

Let (R_1(s,t)) and (R_2(s,t)), (s,t\in T), be two (K(\mathcal H))-valued kernels; in the tensor product (\mathcal H\otimes\mathcal H) consider the operators
[
\Phi(s,u;t,v)(x\otimes y)=R_1(s,t)x\otimes R_2(u,v)y,
]
(s,u,t,v\in T,\quad x,y\in\mathcal H). (\Phi(s,u;t,v)), ((s,u),(t,v)\in T\times T), is a (K(\mathcal H\otimes\mathcal H))-valued positive definite function. We denote the kernel (\Phi(s,u;t,v)) by (R_1(s,t)\otimes R_2(u,v)). In accordance with what was said above, the space (H(R_1\otimes R_2)) is spanned by the (\mathcal H\otimes\mathcal H)-valued functions of the form
[
R_1(\cdot,t)x\otimes R_2(\cdot,v)y;
]
if to each such function one assigns the element (R_1(\cdot,t)x\otimes R_2(\cdot,v)y) of (H(R_1)\otimes H(R_2)), then we obtain an isomorphism
[
H(R_1\otimes R_2)\simeq H(R_1)\otimes H(R_2);
]
this circumstance facilitates finding (H(R_1\otimes R_2)) when (H(R_1)) and (H(R_2)) are known. To every element of the form (x\otimes y\in\mathcal H\otimes\mathcal H) there corresponds an operator (A_{x\otimes y}\in\mathcal G(\mathcal H)), defined by the formula
[
A_{x\otimes y}z=(y,z)x,\quad z\in\mathcal H.
]
This correspondence extends to an isomorphism between the spaces (\mathcal H\otimes\mathcal H) and (\mathcal G(\mathcal H)). Thus we may regard functions from (H(R_1\otimes R_2)) as (\mathcal G(\mathcal H))-valued operator functions on (T\times T).

Let (\xi(t)), (t\in T), be an (\mathcal H)-valued Gaussian function on (T), relative to the measure (\mathcal P), with mean value (0) and correlation function (R(s,t)). Denote by (H(\xi(t),\mathcal P)) the linear hull, closed in the sense of mean-square convergence, of the random variables ((x,\xi(t))), (x\in\mathcal H), (t\in T). To any function (\varphi(\cdot)\in H(R)) one can associate a random variable (y\in H(\xi(t),\mathcal P)) such that for any (x\in\mathcal H) we have
[
M(x,\xi(t))y=(x,\varphi(t)),\quad t\in T;
]
this correspondence is an isomorphism between the Hilbert spaces (H(R)) and (H(\xi(t),\mathcal P)); to the function (R(\cdot,t)x) ((x\in\mathcal H,\ t\in T)) there corresponds the random variable ((x,\xi(t))). Denote
[
y=\langle\varphi(t),\xi(t)\rangle.
]
Since the realizations of the random function (\xi(t)) do not belong to (H(R)), (\langle\varphi(t),\xi(t)\rangle) cannot be understood literally as a scalar product; however, in concrete cases this formally written expression can be given meaning if the limiting operations over random—

...random variables entering into it is to be understood in the sense of mean-square convergence.

Denote by (H^2(\xi(t), \mathscr P)) the Hilbert space (H(\zeta(s,t), \mathscr P)), where
(\zeta(s,t)=\xi(s)\otimes \xi(t)-R(s,t)) is a random function with values in (\mathscr H\otimes \mathscr H) and with correlation function (\Phi(s,t;u,v)), where
[
\Phi(s,t;u,v)(x\otimes y)=R(s,u)x\otimes R(t,v)y+R(s,v)y\otimes R(t,u)x.
]
According to what was said above, there is an isomorphism between the spaces (H^2(\xi(t),\mathscr P)) and (H(\Phi)), under which the random variables ((x\otimes y,\zeta(u,v))) correspond to the (\mathscr H\otimes \mathscr H)-valued functions (\Phi(\cdot,\cdot;u,v)(x\otimes y)), (u,v\in T).

1. Let (\xi(t)), (t\in T), be an (\mathscr H)-valued random function on (T), and let (\mathscr B_\xi) be the minimal (\sigma)-algebra with respect to which all random variables (\xi(t)), (t\in T), are measurable; on (\mathscr B_\xi) two measures (\mathscr P_1) and (\mathscr P_2) are given, with respect to which (\xi(t)), (t\in T), is a Gaussian random function with characteristics (m_i(t)) and (R_i(s,t)), (i=1,2). Theorem 1 reduces the question of equivalence of the measures (\mathscr P_1) and (\mathscr P_2) and the computation of the Radon–Nikodym derivative
   [
   \frac{d\mathscr P_2}{d\mathscr P_1}[\xi(t)]
   ]
   to the study of the spaces (H(R_1)) and (H(R_2)).

Theorem 1. The measures (\mathscr P_1) and (\mathscr P_2) are either equivalent or orthogonal. (\mathscr P_1) is equivalent to (\mathscr P_2) if and only if the following conditions below are satisfied: 1)—3), or 1) and 4), or 2) and 5), or 6):

1) (m_2(t)-m_1(t)\in H(R_1));

2) (R_2(\cdot,t)x\in H(R_1)), and if (\langle R_2(\cdot,t)x,\varphi(\cdot)\rangle_1=0)* for all (x\in\mathscr H), then (\varphi(t)\equiv 0);

3) (R_2(s,t)-R_1(s,t)\in H(R_1\otimes R_1));

4) (R_2(s,t)-R_1(s,t)\in H(R_1\otimes R_2));

5) (R_2(s,t)-m_2(s)\otimes m_2(t)-R_1(s,t)+m_1(s)\otimes m_1(t)\in H(R_1\otimes R_1));

6) (R_2(s,t)-m_2(s)\otimes m_2(t)-R_1(s,t)+m_1(s)\otimes m_1(t)\in H(R_1\otimes R_2)).

In the case of equivalence of the measures (\mathscr P_1) and (\mathscr P_2),
[
\frac{d\mathscr P_2}{d\mathscr P_1}[\xi(t)]
=
D\exp\left{\frac12\langle m_1-m_2,m_1-m_2\rangle_1
-\langle \tilde{\xi},m_1-m_2\rangle_1
+\frac12\left\langle A(s,t),\tilde{\xi}(s)\otimes \tilde{\xi}(t)-R_1(s,t)\right\rangle_{11}\right};
]
[
\tilde{\xi}=\xi-m_2;\qquad
A(s,t)=R_1(s,t)-\langle R_1(\cdot,s),R_1(\cdot,t)\rangle_2,
]
i.e.
[
\langle f_1,f_2\rangle_1-\langle f_1,f_2\rangle_2
=
\left\langle A(s,t),f_1(s)\otimes f_2(t)\right\rangle_{11},
\qquad f_i\in H(R_1);
]
[
D=\exp\left{\frac12\operatorname{Sp}\left[\bar A+\ln(I-\bar A)\right]\right},
]
where (\bar A) is the operator in (H(R_1)) defined by the relation
[
\langle A(\cdot,t)x,f(\cdot)\rangle_1=(x,(\bar A f)(t))
]
for any (x\in\mathscr H), (f\in H(R_1)). The quantities entering into the expression
[
\frac{d\mathscr P_2}{d\mathscr P_1}[\xi(t)]
]
are defined under the conditions of equivalence.*

As special cases, this theorem contains the equivalence criteria and expressions for the Radon–Nikodym derivative for measures corresponding to one-dimensional Gaussian functions (see ((2,3,9,12))), and to Gaussian measures in Hilbert spaces ((6,7,11,13)). In paragraphs 3 and 4 some applications of the results set forth are considered.

1. Let (\xi) be an (\mathscr H)-valued Gaussian random variable with zero mathematical expectation and correlation operator (R). Let (R=VV^), where (V\in\mathscr G(\mathscr H)) (one may, for example, put (V=V^=R^{1/2})). Denote by (V^{-1}) the operator equal to (V^{-1}x=y), if (Vy=x), (y\in V\mathscr H), and equal to (V^{-1}x=0), if (Vx=0). An element (m\in H(R)) if and only if (m=Vx), (x\in\mathscr H);
   [
   \langle m_1,m_2\rangle=(V^{-1}m_1,V^{-1}m_2).
   ]
   Formally,
   [
   \langle m,\xi\rangle=(V^{-1}m,V^{-1}\xi).
   ]
   This expression can be given, for example, the following meaning. Let (e_1,e_2,\ldots) be a sequence of eigenvectors of the operator (R), corresponding to the nonzero eigenvalues (\lambda_1,\lambda_2,\ldots). Then
   [
   \langle m_1,m_2\rangle=\sum_i\frac1{\lambda_i}(m_1,e_i)(m_2,e_i),
   \qquad
   \langle m,\xi\rangle=\sum_i\frac1{\lambda_i}(m,e_i)(\xi,e_i)
   ]
   (the series converges in mean and almost everywhere). The space (H(R_1\otimes R_2)), ((R_1,R_2\in K(\mathscr H))), consists of operators (K\in\mathscr G(\mathscr H)) of the form (K=V_1BV_2^*), where (B\in\mathscr G(\mathscr H)),

[
\text{* }\ \langle\cdot,\cdot\rangle_i=\langle\cdot,\cdot\rangle_{H(R_i)},\quad
\langle\cdot,\cdot\rangle_{i,j}=\langle\cdot,\cdot\rangle_{H(R_i\otimes R_j)}.
]

(R_i=V_iV_i^*,\ i=1,2;\ \langle K_1,K_2\rangle_{12}=(B_1,B_2)), and if (R_1=R_2=R), then

[
\langle K_1,K_2\rangle=\sum \frac{1}{\lambda_i^2\lambda_j}(K_1e_i,e_j)(K_2e_i,e_j),
]

[
\langle K,\xi\otimes \xi-R\rangle
=\sum \left(\frac{1}{\lambda_i\lambda_j}-\delta_{ij}\right)
(Ke_i,e_j)(\xi,e_i)(\xi,e_j)
]

(the series converges in mean and almost everywhere).

Theorem 2. Let (\xi) be an (\mathcal H)-valued Gaussian random variable with respect to the measures (\mathcal P_1) and (\mathcal P_2); let (m_i) and (R_i) ((i=1,2)) be its means and correlation operators, and let (R_i=V_iV_i^). The measures (\mathcal P_1) and (\mathcal P_2) are equivalent on (\mathcal B_\xi) if and only if the following conditions are satisfied: 1) (m_2-m_1=V_1x,\ x\in\mathcal H); 2) (\overline{R_1\mathcal H}=\overline{R_2\mathcal H}); 3) (R_2-R_1=V_1BV_1^,\ B\in\mathfrak G(\mathcal H)), or condition 1) and condition 4) (R_2-R_1=V_1CV_2^=V_2C^V_1^,\ C\in\mathfrak G(\mathcal H)), or condition 2) and condition 5) (R_2-m_2\otimes m_2-R_1+m_1\otimes m_1=V_1DV_1^,\ D\in\mathfrak G(\mathcal H)), or condition 6) (R_2-m_2\otimes m_2-R_1+m_1\otimes m_1=V_1EV_2^*,\ E\in\mathfrak G(\mathcal H)). The computation of (\dfrac{d\mathcal P_2}{d\mathcal P_1}[\xi]) is carried out using the formula given in Theorem 1.

The operator

[
A=V_1[I-(V_2^{-1}V_1)^V_2^{-1}V_1]V_1^,\qquad
\bar A=I-V_1(V_2^{-1}V_1)^*V_2^{-1}
]

(cf. ((^6,^7,^{{11}},^{{13}}))).

1. Let (T=[a,b]), (-\infty<a<b<\infty), (\mathcal H=R^n), and let (\Psi(t)) and (\Phi(t)) be operator functions, where (\Psi^{-1}(t)) and (\Phi^{-1}(t)) exist.

Theorem 3. The operator function

[
R(s,t)=\Psi(s)\Phi^(t),\ s\leq t;\qquad R(s,t)=\Phi(s)\Psi^(t),\ s\geq t,
\tag{1}
]

is positive definite if and only if:

1) (R(t,t)=\Psi(t)\Phi^*(t)\in K_+(\mathcal H)) for every (t\in[a,b]);

2) (\Lambda(t)=\Phi^{-1}(t)\Psi(t)\in K_+(\mathcal H),\ t\in[a,b]);

3) (\Lambda(t)-\Lambda(s)\in K_+(\mathcal H)) for (t>s,\ s,t\in[a,b]).

Theorem 4. A nondegenerate mean-continuous (\mathcal H)-valued Gaussian random process (\xi(t)), (t\in[a,b]), is Markovian if and only if its correlation function is representable in the form (1).

From the properties of (\Lambda(t)) it follows that, for almost all (t\in[a,b]), there exists a derivative (\Lambda'(t)\in K_+(\mathcal H)). Below, for convenience, it is assumed that all operators (\Lambda'(t)) are nondegenerate. Introduce the functions (M(t)) and the operator (V):

[
\Lambda'(t)=M(t)M^(t),\qquad \Lambda(a)=VV^
]

(for example, one may take (M(t)=M^(t)=[\Lambda'(t)]^{1/2}), (V=V^=[\Lambda(a)]^{1/2})).

Theorem 5. Let (\xi(t)), (t\in[a,b]), be an (R^n)-valued process with correlation function of the form (1); then

[
w(t)=\int_a^t M^{-1}(t)\,d[\Phi^{-1}(t)\xi(t)]
+V^{-1}\Phi^{-1}(a)\xi(a)
]

is an (R^n)-valued Wiener process with correlation function (I\min(s,t)). Conversely,

[
\xi(t)=\Phi(t)\int_a^t M(t)\,dw(t)+\xi(a).
]

Assume further, for convenience, that (\xi(a)=0). An (R^n)-valued function (m(t)\in H(R)) if and only if the function (\Phi^{-1}(t)m(t)) is absolutely continuous and

[
f(t)=M^{-1}(t)(\Phi^{-1}(t)m(t))'\in L_2.
]

If (m_1(t),m_2(t)\in H(R)), then

[
\langle m_1,m_2\rangle=\int_a^b (f_1(t),f_2(t))\,dt;
\qquad
\langle m,\xi\rangle=\int_a^b
\bigl(M^{-1}(t)(\Phi^{-1}(t)m(t))',
M^{-1}(t)\,d(\Phi^{-1}(t)\xi(t))\bigr).
]

Let (R_1) and (R_2) be two kernels of the form (1). The space (H(R_1\otimes R_2)) consists of operator functions (m(s,t)) such that (\Phi_1^{-1}(s)m(s,t)\Phi_2^{*-1}(t)) is an absolutely continuous function and

[
f(s,t)=M_1^{-1}(s)\frac{\partial^2}{\partial s\,\partial t}
\bigl(\Phi_1^{-1}(s)m(s,t)\Phi_2^{-1}(t)\bigr)M_2^{-1}(t)\in L_2;
]

[
\langle m_1,m_2\rangle_{1,2}
=
\int_a^b\int_a^b
\operatorname{Sp}[f_1(s,t)f_2^*(s,t)]\,ds\,dt.
]

(H^2(\xi(t),\mathcal P)) consists of random variables of the form

[
\langle m(s,t),\xi(s)\times \xi(t)-R(s,t)\rangle=
]

[
= \int_a^b \int_a^b \operatorname{Sp} \left{[\Lambda'(s)]^{-1}
\frac{\partial^2}{\partial s\,\partial t}
\left(\Phi^{-1}(s)m(s,t)\Phi^{-1}(t)\right)[\Lambda'(t)]^{-1}\, d_s d_t\left(\Phi^{-1}(t)\times
\right.\right.
]
[
\left.\left.
{}\times [\xi(s)\otimes \xi(t)]\Phi^{-1}(s)\right)\right},
]

where the double stochastic integral is understood in the sense of K. Itô; it is defined for any function (m(s,t)\in H(R\otimes R)) (cf. ((2^3,5))).

Let (\xi(t)), (t\in [a,b]), be a continuous in the mean nondegenerate (n)-dimensional Gaussian process with characteristics (m_i(t)) and (R_i(s,t)) with respect to the measures (\mathscr P_i), (i=1,2), on (\mathfrak B_\xi). Denote (K_{12}(t)=\Phi_1^{-1}(t)\Phi_2(t)) and (K_{21}(t)=K_{12}^{-1}(t)=\Phi_2^{-1}(t)\Phi_1(t)).

Theorem 6. The measures (\mathscr P_1) and (\mathscr P_2) are equivalent if and only if:

1) the vector function (\Phi_1^{-1}(t)(m_2(t)-m_1(t))) is absolutely continuous and
[
M_1^{-1}(t)[\Phi_1^{-1}(t)(m_2(t)-m_1(t))]'\in L_2;
]

2) (\Lambda_1'(t)K_{21}'(t)=K_{12}(t)\Lambda_2'(t)) almost everywhere on ([a,b]);

3) (K_{12}'(t)) and (K_{21}'(t)) exist almost everywhere;

4) (f(s,t)\in L_2), where (f(s,t)=[M_2^{-1}(t)K_{21}'(t)M_1(s)]^*) for (s\le t), and
[
f(s,t)=M_1^{-1}(s)K_{12}'(t)M_2(t)
]
for (s>t).

The computation of
[
\frac{d\mathscr P_2}{d\mathscr P_1}[\xi(t)]
]
is carried out by the formula of Theorem 1. The operator function
[
A(s,t)=\Phi_1(s)\widetilde A(s,t)\Phi_1^(t),
]
where
[
\widetilde A(s,t)=\int_a^s K_{21}(u)K_{12}'(u)\Lambda_1(\min(t,u))\,du
+\int_a^t \Lambda_1(\min(s,u))K_{12}^{'}(u)\times
]
[
{}\times K_{21}^(u)\,du
-\int_a^b \Lambda_1(\min(s,u))K_{12}^{'}(u)K_{21}^(u)(\Lambda_1'(u))^{-1}K_{21}(u)K_{12}'(u)\times
]
[
{}\times \Lambda_1(\min(t,u))\,du;\qquad
\ln D=\int_a^b\int_t^b
\operatorname{Sp}\,[K_{12}^(u)K_{21}^*(u)(\Lambda_1'(u))^{-1}K_{21}(u)K_{12}'(u)]\,du.
]

Let us consider, in particular, a random process (\xi(t)) which, with respect to the measures (\mathscr P_i), (i=1,2), on (\mathfrak B_\xi), is a stationary Gaussian Markov process with (m_i(t)\equiv0);
[
R_i(s,t)=C_i\exp{|t-s|Q_i}C_i,
]
where (-(Q_i+Q_i^)) and (C_i) are positive definite operators; here
[
\Psi_i(t)=C_i\exp{-tQ_i},\qquad
\Phi_i(t)=C_i\exp{tQ_i^},\qquad
\Lambda_i(t)=\exp{-tQ_i}\exp{-tQ_i^}.
]
It is not hard to verify that (\mathscr P_1) and (\mathscr P_2) are equivalent if and only if
[
C_1(Q_1+Q_1^)C_1=C_2(Q_2+Q_2^)C_2.
]
The measures (\mathscr P_1) and (\mathscr P_2) corresponding to two Gaussian processes with
[
m_i(t)\equiv0,\qquad R_1(s,t)=C\exp{|t-s|Q}C,\qquad R_2(s,t)=B\min(s,t),
]
are equivalent if and only if
[
B=-C(Q+Q^)C.
]

The authors express their gratitude to A. M. Yaglom for his attention and valuable advice.

Institute of Physicotechnical Problems of Power Engineering
Academy of Sciences of the Lithuanian SSR

Institute of Physics and Mathematics
Academy of Sciences of the Lithuanian SSR

Received
14 VI 1968

References

1. H. Aronszajn, Mathematika, 7, 2, 67 (1963).
2. Yu. I. Golosov, DAN, 170, No. 2, 242 (1966).
3. Yu. I. Golosov, On conditions for equivalence and orthogonality of Gaussian measures and on the computation of the Radon–Nikodym derivative. Dissertation, Moscow State University, 1966.
4. Yu. I. Golosov, A. M. Yaglom, Equivalent Gaussian measures related to generalized random processes. Abstracts of the International Congress of Mathematicians, probability theory section, 1967.
5. T. I. Mirskaya, A. S. Pobedinskaya, A. A. Tempelman, Lit. matem. sborn., 7, 3, 459 (1967).
6. Yu. A. Rozanov, DAN, 171, No. 6, 1286 (1966).
7. M. G. Sotsis, Supplement to the book by G. E. Shilov, Fan Dyk Tin’. Integral, Measure and Derivative on Linear Spaces, “Nauka,” 1967, pp. 175–189.
8. U. Beutler, Ann. Math. Stat., 34, 3, 424 (1963).
9. J. Capon, IEEE Trans. on Information Theory, 11, 247 (1965).
10. J. L. Doob, Ann. Math. Stat., 15, 229 (1944).
11. G. Kallianpur, H. Oodaira, The Equivalence and Singularity of Gaussian Measures, Ch. 19, in Time Series Analysis, N. Y., 1963.
12. E. Parzen, Probability Density Functionals and Reproducing Kernel Hilbert Space, Ch. 11, in Time Series Analysis, N. Y., 1963.
13. T. S. Pitcher, J. Soc. Ind. and Appl. Math., 14, 3, 228 (1966).