Reports of the Academy of Sciences of the USSR
1969. Volume 184, No. 3

UDC 519.21

MATHEMATICS

T. M. MOLCHAN, Yu. I. GOLOSOV

GAUSSIAN STATIONARY PROCESSES WITH ASYMPTOTICALLY POWER-LAW SPECTRUM

(Presented by Academician A. N. Kolmogorov on 21 V 1968)

1. Let \(\xi(\varphi)\), \(\varphi(t)\in K_T\), be a real Gaussian process on the interval \([0,T]\), generalized in the sense of Gelfand—Ito, with zero mean and correlation functional \(\bar B(\varphi,\psi)=M\xi(\varphi)\xi(\psi)\); \(K_T\) is the space of infinitely differentiable functions \(\varphi(t)\) with support in \([0,T]\) and with the usual topology of L. Schwartz.

Denote by \(H_t(\xi)\), \(t\le T\), the Hilbert spaces obtained by closing the linear span of the random variables \(\{\xi(\varphi),\varphi\in K_t\}\) with respect to the scalar product \((\xi(\varphi),\xi(\psi))=B(\varphi,\psi)\). The space \(H_T(\xi)\) is isomorphic to the space of functionals \(H_T(B)\subset K_T'\), called the space with reproducing kernel \(B(\varphi,\psi)\) (see \((^2)\)). The isomorphism \(H_T(\xi)\leftrightarrow H_T(B)\) is effected by the unitary correspondence \(U:\ H_T(\xi)\ni\eta\leftrightarrow (U\eta)(\varphi)=M\eta\xi(\varphi)\in H_T(B)\) and
\[ \langle f_1,f_2\rangle_{H_T(B)}=(U^{-1}f_1,U^{-1}f_2), \]
\(f_1,f_2\in H_T(B)\).

A number of statistical problems, such as extrapolation of the process \(\xi(\varphi)\), description of measures equivalent to the measure induced by the process \(\xi(\varphi)\), etc., are solved by using the spaces \(H_T(\xi)\) and \(H_T(B)\) (see, for example, \((^{2-5})\)). In the present note a method is developed for the analytic description of the spaces \(H_T(\xi)\) and \(H_T(B)\) on the basis of the canonical representation \((^6)\) of the process \(\xi(\varphi)\) and the solution of certain Wiener—Hopf type equations by a method close to that used by M. G. Krein \((^7)\) in inverse problems for an inhomogeneous string. This makes it possible to study \(H_T(\xi)\) and \(H_T(B)\) for processes \(\xi(\varphi)\) that include, as particular cases, the Gaussian process \(\xi(t)\) with stationary increments
\[ M|\xi(t)-\xi(s)|^2=\operatorname{const}\cdot |t-s|^\alpha,\quad \alpha>0, \]
and stationary processes with spectral density
\[ c_1<f(\lambda)(1+|\lambda|)^\mu<c_2,\quad |\mu|<\infty. \]
A somewhat different approach is used in Sec. 5 for the study of homogeneous fields.

1. We note a simple property of the functionals from \(H_T(B)\). Let \(X\supset K_T\) be a linear topological space containing \(K_T\) as an everywhere dense subset, and suppose the topology in \(K_T\) induced from \(X\) is weaker than the topology of \(K_T\); we shall call \(X\) an extension of \(K_T\). If the correlation functional \(B(\varphi,\varphi)\) is continuous in the extension \(X\supset K_T\), then \(H_T(B)\subset X'\), where \(X'\) is the conjugate space to \(X\). This follows from the Cauchy—Bunyakovsky inequality:
   \[ |f(\varphi)|^2=|(U^{-1}f,\xi(\varphi))|^2\le \|U^{-1}f\|^2 B(\varphi,\varphi). \]
   The process \(\xi(\varphi)\) can then, by continuity, be extended to \(X\). It is clear that the maximal extension of \(K_T\) is the closure of \(K_T\) in the metric \(\|\varphi\|^2=B(\varphi,\varphi)\), which we denote by \(L_T^2(B)\). The extension of \(\xi(\varphi)\) to \(L_T^2(B)\) makes it possible to describe the elements of \(H_T(\xi)\) as follows: \(H_T(\xi)\ni\eta=\xi(q)\), \(q\in L_T^2(B)\).

Definition. Let \(E_t\) be a family of projection operators in \(H_T(\xi)\) onto the subspaces \(H_t(\xi)\), \(t\le T\). Following Hida—Cramér, we shall call the process \(\xi(\varphi)\), \(\varphi\in K_T\), a process of multiplicity 1 if there exists an element \(\eta\in H_T(\xi)\) for which the linear span \(\{E_t\eta=\eta_t,\ 0\le t\le T\}\) is dense in \(H_T(\xi)\) (cf. \((^8)\)); the element \(\eta\) will be called cyclic.

Theorem 1. Let \(\xi(\varphi)\), \(\varphi\in K_T\), be a Gaussian process of multiplicity 1; \(\eta\in H_T(\xi)\) a cyclic element, \(\eta_t=E_t\eta=\xi(q_t)\), \(q_t\in L_T^2(B)\cap\)

\(\eta L_t^2(B)\) and \(f_t=U\eta_t\in H_t(B)\), \(f_\varphi(\cdot)=B(\varphi,\cdot)\in H_T(B)\). Then \(\xi(\varphi)\) admits the canonical representation

\[ \xi(\varphi)=\int_0^\tau \frac{df_t(\varphi)}{d\sigma(t)}\,d\eta_t =\int_0^\tau \frac{df_\varphi(q_t)}{d\sigma(t)}\,d\eta_t,\quad \varphi\in K_\tau, \tag{1} \]

where the integral is defined as a stochastic integral with respect to the process \(\eta_t\) with independent increments, \(\sigma(t)=M|\eta_t|^2=B(q_t,q_t)\geq 0\) is a nondecreasing bounded function, and \(df(\cdot)/d\sigma(\cdot)\) is understood as the derivative in the Radon–Nikodym sense.

Theorem 2. Let \(B(\varphi,\psi)\) be the correlation functional of a generalized Gaussian process \(\xi(\varphi)\), \(\varphi\in K_T\), continuous in the extension \(X\supset K_T\). If for some \(f_0\in X'\) and every \(t\), \(0\leq t\leq T\), there exists an element \(q_t\in X_t\), where \(X_t\) is the closure of \(K_t\) in \(X\), such that

\[ B(\varphi,q_t)=f_0(\varphi),\quad \varphi\in K_t, \tag{2} \]

then \(f_0(q_t)=\sigma(t)\geq 0\) defines a nondecreasing bounded function of \(t\), and the functions \(\hat f(t)=f(q_t)\) for \(f\in H_T(B)\) are absolutely continuous with respect to \(\sigma(t)\), moreover \(\hat f(t)\) and \(\sigma(t)\) do not depend on the nonunique choice of \(q_t\). If the linear span of \(\{q_t,0\leq t\leq T\}\) is dense in \(X\), then \(H_T(B)\) is unitarily isomorphic to \(L^2([0,T],d\sigma(t))\), and

\[ \langle f_1,f_2\rangle_{H_T(B)} =\int_0^T \frac{d\hat f_1(t)\,d\hat f_2(t)}{d\sigma(t)}. \tag{3} \]

1. Consider a generalized stationary Gaussian process \(\xi(\varphi)\), \(\varphi\in K_T\), with mean value \(M\xi(\varphi)=0\) and correlation functional
   \(B_\mu(\varphi,\psi)=\int \widetilde\varphi(\lambda)\overline{\widetilde\psi(\lambda)}|\lambda|^{-\mu}d\lambda\), \(\mu<1\), where \(\widetilde\varphi(\lambda)\) is the Fourier transform of the function \(\varphi(t)\).

Theorem 3. 1) The integral equation

\[ \int \widetilde\varphi(\lambda)\,\overline{\widetilde q_t(\lambda)}|\lambda|^{-\mu}d\lambda =\int_0^t \varphi(t)\,dt,\quad \varphi\in K_t, \tag{4} \]

has the solution

\[ q_t(s)=\frac{1}{2\pi}\frac{1}{\Gamma(1-\mu)}(t-s)_+^{-\mu/2}s_+^{-\mu/2}, \quad 0<s\leq T, \tag{5} \]

(\(u_+=u\) for \(u\geq 0\) and \(u_+=0\) for \(u\leq 0\)); the family \(\{q_t(s),0\leq t\leq T\}\) is dense in \(L^2(0,T)\).

2) For \(0\leq\mu<1\), the elements of \(H_T(B_\mu)\) are \(f(\varphi)=\int_0^T f(t)\varphi(t)\,dt\), \(f(t)\in L^{2/(1-\mu)}\), and

\[ \langle f_1,f_1\rangle_{H_T(B_\mu)} =\frac{1}{2\pi}\int_0^T \left|t^{\mu/2}D^{\mu/2}[f(t)t^{-\mu/2}]\right|^2dt, \tag{6} \]

where

\[ D^\alpha[f]=\frac{1}{\Gamma(1-\alpha)}\frac{d}{dt}\int_0^t (t-s)^{-\alpha}f(s)\,ds \]

is the fractional differentiation operator of order \(\alpha\).

3) For \(-1\leq\mu<0\), the elements of \(H_T(B_\mu)\) are singular generalized functions of first order:
\(H_T(B_\mu)\ni f(\varphi)=\int_0^T F(t)\varphi'(t)\,dt\),
\(F(t)\in \operatorname{Lip}((1+\mu)/2)\), if \(-1<\mu<0\), and \(F(t)\in L^p\), \(p>1\), if \(\mu=-1\), \(\left(\frac{d}{dt}F(t)=f(t)\right)\); in this case

\[ \langle f,f\rangle_{H_T(B_\mu)} =\frac{1}{2\pi}\int_0^T \left|t^{\mu/2}\frac{d}{dt}\int_0^t F(s)\,d\left[ \frac{(t-s)^{-\mu/2}s^{-\mu/2}}{\Gamma(1-\mu)} \right]\right|^2dt. \tag{7} \]

Using Theorem 3 and the method developed in (2), it is easy to write out the Radon–Nikodym derivative of the measures \(P_1\) and \(P_2\) corresponding to the processes \(\xi(\varphi)\) and \(\xi(\varphi)+m(\varphi)\), \(m(\varphi)\in H_T(B_\mu)\).

The process \(\xi(\varphi)\) is the derivative of order \(([(1+\mu)/2]+1)\) of the Gaussian process \(x(t)\) with homogeneous increments, invariant with respect to the similarity transformation (8). This observation makes it possible to transfer Theorem 3 also to the process \(x(t)\). Let \(|\mu|<1\), and let \(x(t)\) be a Gaussian process with correlation function
\[ R_\mu(s,t)=k\left(|s|^{1+\mu}+|t|^{1+\mu}-|t-s|^{1+\mu}\right), \qquad k=2\sin\frac{\mu}{2}\pi\Gamma(-1-\mu). \]
Since \(x(t)=\xi(\chi_{0t})\), \(H_T(R_\mu)\) consists of functions \(F(t)=f(\chi_{0t})\), \(f(\varphi)\in H_T(B_\mu)\), with scalar product
\[ \langle F_1(t),F_2(t)\rangle_{H_T(R_\mu)} = \langle f_1,f_2\rangle_{H_T(B_\mu)}. \]
Combining Theorem 1 with Theorem 3, we obtain the canonical representation of the process \(x(t)\) in the form
\[ x(t)=\int_0^t Q(t,s)\,d\eta(s), \]
where
\[ Q(t,s)= \]
\[ = \frac{1}{\pi}\sin\frac{\mu\pi}{2} \begin{cases} \displaystyle \int_s^t(\tau-s)^{\mu/2-1}\tau^{\mu/2}\,d\tau, & \text{for } 0<\mu<1,\\[1.2em] \displaystyle \int_s^t(\tau-s)^{\mu/2-1}\left[\tau^{\mu/2}-s^{\mu/2}\right]\,d\tau +\frac{2}{\mu}(ts-s^2)^{\mu/2}, & \text{for } -1\le \mu<0, \end{cases} \]
and the process with independent increments \(\eta(s)\), for which \(H_t(x)=H_t(\eta)\), \(t\le T\), has the form
\[ \eta(s)= \begin{cases} \displaystyle \int_0^s(s-t)^{-\mu/2}t^{-\mu/2}\,dx(t), & \text{for } 0<\mu<1,\\[1.2em] \displaystyle \int_0^s x(t)\,d\left[(s-t)^{-\mu/2}t^{-\mu/2}\right], & \text{for } -1\le \mu\le 0. \end{cases} \]
Hence, in the usual way, one obtains formulas for prediction of \(x(t)\), first obtained in (9).

1. Let \(\xi_i(\varphi)\), \(\varphi\in K_T\), be Gaussian stationary processes with means \(m_i(\varphi)=0\), correlation functions
   \[ B_i(\varphi,\varphi)=\psi_i(\varphi*\varphi^*),\qquad \psi_i\in K', \]
   \(\varphi^*(x)=\varphi(-x)\), and spectral measures \(F_i(d\lambda)\) (1); \(P_i\) are the measures induced by the processes \(\xi_i(\varphi)\), \(\varphi\in K_T\), in the space \(K_T'\), \(i=1,2\).

Theorem 4. If \(F_i(d\lambda)\) are absolutely continuous and
\[ 0<c_1<\frac{F_1(d\lambda)}{d\lambda}(1+|\lambda|)^{2n+\mu}<c_2 \quad \text{as } \lambda\to\infty, \]
where \(n\) is an integer and \(-1\le\mu<1\), then the measures \(P_i\) are equivalent if and only if
\[ \Delta^{(2n)}(\varphi*\psi^*)\in H_T(B_\mu)\otimes H_T(B_\mu), \]
where
\[ \Delta(\varphi)=\psi_1(\varphi)-\psi_2(\varphi),\qquad \Delta^{(k)}(\varphi)=\Delta(\varphi^{(k)}), \]
in other words:

1) for \(0\le \mu<1\),
\[ \Delta^{(2n)}(\varphi*\psi^*) = \int_0^T\int_0^T \Delta^{(2n)}(s-t)\varphi(s)\psi(t)\,ds\,dt, \]
where
\[ \Delta^{(2n)}(s-t)\in L^{2/(1-\mu)}([0,T]\times[0,T]), \]
and
\[ \int_0^T\int_0^T (st)^\mu \left| D_t^{\mu/2}D_s^{\mu/2} \left[\Delta^{(2n)}(s-t)(st)^{-\mu/2}\right] \right|^2 \,ds\,dt<\infty; \]

2) for \(-1\le\mu<0\), the functional
\[ \Delta^{(2n-2)}(\varphi)=\int_0^T\Delta^{(2n-2)}(t)\varphi(t)\,dt, \]
\[ \Delta^{(2n-2)}(t)\in C[0,T] \]
and
\[ \int_0^T\int_0^T (st)^\mu \left| \frac{\partial^2}{\partial s\,\partial t} \int_0^s\int_0^t \Delta^{(2n-2)}(u-v)\,du\,dv \left[(s-u)(t-v)uv\right]^{-\mu/2} \right|^2 \,ds\,dt<\infty. \]

The proof is based on the results of \((^{2,10})\).

Theorem 5. Let \(f(\lambda)\ge 0\) be a locally summable function and
\(c_1 < f(\lambda)|\lambda|^{(2n+\mu)} < c\) as \(\lambda\to\infty\), \(-1\le \mu<1\), \(n\) an integer. For the existence of a solution of the Wiener–Hopf integral equation

\[ a(t)=\int e^{i\lambda t}\varphi(\lambda)f(\lambda)\,d\lambda,\qquad 0\le t\le T, \]

in the class of functions \(L_T^2(f)\)—the closure of \(\{\varphi(\lambda),\ \widetilde{\varphi}(t)\in K_T\}\), in the metric
\[ \int|\varphi|^2 f\,d\lambda=\|\varphi\|_{L_T^2(f)}^2, \]
it is necessary and sufficient that \(a(t)\) have \(n-1\) derivatives and that
\(a^n(t)\in H_T(B_\mu)\); the \(n\)-th derivative is understood as a generalized one.

Theorems 4 and 5 in the case \(\mu=0\) were obtained by Yu. A. Rozanov \((^{11,12})\).

1. Let \(\xi(\varphi)\), \(\varphi\in K(E^n)\), be a Gaussian homogeneous field with correlation functional
   \[ B(\varphi,\varphi)=\int_{E^n}|\widetilde{\varphi}(\lambda)|^2 f(\lambda)\,d^n\lambda, \]
   and let \(\Omega\) be a bounded domain with smooth boundary in \(E^n\). The notation \(H_\Omega(B)\) has the same meaning as in the one-dimensional case.

Theorem 6. Let \(0<c_1<f(\lambda)(1+|\lambda|^2)^{-\mu}<c_2\), \(\mu>0\); then the space \(H_\Omega(B)\), up to equivalence of norms, coincides with the Sobolev–Slobodetskii space \(W_2^\mu(\Omega)\), whose norm has the form:
\[ \|f\|_{W_2^\mu(\Omega)}^2 = \|f\|_{L^2}^2 + \sum_{|\alpha|=[\mu]}\|f^{(\alpha)}\|_{L^2}^2 + \sum_{|\alpha|=[\mu]} \int_{\Omega\times\Omega} \frac{|f^{(\alpha)}(x)-f^{(\alpha)}(y)|^2}{|x-y|^{n+2\beta}} \,d^n x\,d^n y, \]
where
\[ \beta=\mu-[\mu],\qquad \alpha=(\alpha_1,\ldots,\alpha_n),\qquad |\alpha|=\sum_{i=1}^{n}\alpha_i \]
and
\[ f^\alpha(x)=\frac{\partial^{|\alpha|}}{\partial x_1^{\alpha_1}\cdots\partial x_n^{\alpha_n}}\,f(x) \]
is a generalized derivative in the sense of Sobolev \((^{13})\).

Theorem 6 follows from results of V. M. Babich and L. N. Slobodetskii (see \((^{13})\)).

Remark 1. Theorems 4 and 5, taking Theorem 6 into account, admit obvious reformulations for the case of homogeneous fields, one of which has the spectrum indicated above.

Remark 2. Since
\[ W_2^\mu(\Omega)\otimes W_2^\mu(\Omega)\supset W_2^{2\mu}(\Omega\times\Omega), \]
from Theorem 4 we obtain sufficient conditions for the equivalence of two homogeneous fields, namely,
\[ \Delta(s-t)\in W_2^{2\mu}(\Omega\times\Omega). \]

Remark 3. Comparing Theorems 3, 4, and 6 in the case \(E^n=E^1\) makes it possible to describe the space \(W_2^\mu([0,T])\), \(\mu>0\), in a new way.

The authors express their gratitude to A. M. Yaglom, who drew their attention to the questions considered in this note.

Schmidt Institute of Physics of the Earth,
Academy of Sciences of the USSR

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

Received
21 V 1968

REFERENCES

1. I. M. Gel'fand, N. Ya. Vilenkin, Some Applications of Harmonic Analysis. Generalized Functions, M., 1961.
2. Yu. I. Golosov, A. M. Yaglom, Abstracts of scientific communications, International Congress of Mathematicians, Section II, M., 1967.
3. J. Hajek, Czechosl. Math. J., 12, 404 (1962).
4. E. Parzen, In: Time Series Analysis, Ch. II, N. Y., 1963.
5. G. M. Molchan, Theory of Probability and Its Applications, 12, 747 (1967).
6. H. Cramér, Theory of Probability and Its Applications, 9 (1964).
7. M. G. Krein, DAN, 100, 413 (1955).
8. A. M. Yaglom, Matem. sbornik, 37 (79), 141 (1955).
9. S. V. Grigor'ev, Uch. zap. Kazan. Univ., 125, No. 6, 106 (1966).
10. D. S. Apokolov, Theory of Probability and Its Applications, 12, 698 (1967).
11. Yu. A. Rozanov, Theory of Probability and Its Applications, 11, 170 (1966).
12. Yu. A. Rozanov, Theory of Probability and Its Applications, 8, 241 (1963).
13. L. N. Slobodetskii, Uch. zap. Leningrad Ped. Inst. im. A. I. Herzen, 197, 54 (1958).