UDC 519.21

MATHEMATICS

I. A. IBRAGIMOV, V. N. SOLEV

ON A CONDITION FOR REGULARITY OF A GAUSSIAN STATIONARY PROCESS

(Presented by Academician Yu. V. Linnik on 9 VII 1968)

1. Let \(\xi(t)\) be a stationary, in the narrow sense, process with discrete or continuous time. Denote by \(\mathfrak M_a^b\) the \(\sigma\)-algebra of events generated by the course of the process on the interval \([a,b]\), \(-\infty \le a < b \le \infty\). A. N. Kolmogorov proposed the following condition of weak dependence of events generated by the past and the future of the process (see \((^1)\)):

\[ \beta(\tau)=E\left\{\sup_{A\in \mathfrak M_{-\infty}^{0}} \left|P\{A\mid \mathfrak M_\tau^\infty\}-P\{A\}\right|\right\} \xrightarrow[\tau\to\infty]{}0. \tag{1} \]

Denote by \(P_{0,\tau}\{\cdot\}\) the measure induced by the process \(\xi(t)\) on the \(\sigma\)-algebra \(\mathfrak M_{-\infty}^{0}\cup \mathfrak M_\tau^\infty\), and by \(P_{1,\tau}\{\cdot\}\) the measure on the same \(\sigma\)-algebra defined, for \(A\in\mathfrak M_{-\infty}^{0}\), \(B\in\mathfrak M_\tau^\infty\), by the equality

\[ P_{1,\tau}\{AB\}=P_{0,\tau}\{A\}P_{0,\tau}\{B\}. \]

V. A. Volkonskii and Yu. A. Rozanov \((^2)\) found that

\[ \beta(\tau)=\tfrac12 \operatorname{Var}[P_{0,\tau}-P_{1,\tau}]. \tag{2} \]

Starting from (2), they obtained conditions, expressed in terms of the spectral density (s.d.), sufficient for a stationary Gaussian process with discrete time to satisfy condition (1). In the present note we give a scheme of proof of Theorem 1 formulated below, which makes it possible to obtain a complete description of stationary Gaussian processes with discrete time satisfying condition (1).

Theorem 1. In order that a stationary Gaussian process with discrete time satisfy condition (1), it is necessary and sufficient that it have an s.d. \(f(\lambda)\) representable in the form

\[ f(\lambda)=|P(e^{i\lambda})|^2 f_1(\lambda), \tag{3} \]

where:

1) \(P(z)\) is a polynomial with roots on the circle \(|z|=1\);

2)
\[ \ln f_1(\lambda)\sim \sum_{-\infty}^{\infty} f_k e^{ik\lambda}, \]
where

\[ \sum_{-\infty}^{\infty} |k|\, |f_k|^2<\infty. \tag{4} \]

The proof of this theorem is contained in §§ 2, 3. There also some results are given for processes with continuous time. In § 4 results are given on the rate of decrease of \(\beta(\tau)\).

2. We note that if a stationary, in the narrow sense, process satisfies condition (1), then it is regular in the sense of A. N. Kolmogorov (see \((^1)\)) and, consequently, has an s.d. \(f(\lambda)\) satisfying, in the case of discrete time,

\[ \int_{-\pi}^{\pi}\ln f(\lambda)\,d\lambda>- \infty . \tag{5} \]

It follows from condition (5) (see (3)) that there exists an outer function \(g(z)\in H^2\) inside the unit disk such that \(\lvert g(e^{i\lambda})\rvert^2=f(\lambda)\). Put

\[ G(\lambda)=\frac{\overline{g(e^{i\lambda})}}{g(e^{i\lambda})}\sim \sum_{-\infty}^{\infty} g_k e^{ik\lambda}. \tag{6} \]

Theorem 2. In order that a stationary Gaussian process with discrete time satisfy condition (1), it is necessary and sufficient that it have a spectral density \(f(\lambda)\) satisfying condition (5), and that the function \(G(\lambda)\), defined in (6), be such that

\[ \sum_{-\infty}^{0} |k|\, |g_k|^2<\infty . \tag{7} \]

In the case of a process with continuous time, regularity implies the existence for the process of a spectral density \(f(\lambda)\) satisfying the condition

\[ \int_{-\infty}^{\infty}\frac{\ln f(\lambda)}{1+\lambda^2}\,d\lambda<\infty . \tag{5'} \]

It follows from condition (5′) (see (3)) that there exists an outer function \(g(z)\in H^2\) in the right half-plane such that \(\lvert g(i\lambda)\rvert^2=f(\lambda)\). Put

\[ G(\lambda)=\frac{\overline{g(i\lambda)}}{g(i\lambda)}\,\frac{1}{1-i\lambda}\sim \int_{-\infty}^{\infty}\hat G(s)e^{is\lambda}\,ds . \tag{6'} \]

Theorem 2′. In order that a Gaussian stationary process \((t)\) with continuous time satisfy condition (1), it is necessary and sufficient that it have a spectral density \(f(\lambda)\) satisfying condition (5′), and that the function \(G(\lambda)\), defined in (6′), be such that

\[ \int_{-\infty}^{\infty}|\hat G(s)|^2 |s|\,ds<\infty . \tag{7} \]

We give the scheme of the proof of Theorem 2.

Necessity. From condition (1) it follows that the process \(\xi(t)\) has a spectral density \(f(\lambda)\) satisfying condition (5). Denote by \(H_a^b(i)\), \(i=0,1,\ -a,b\ge 0\), the Hilbert space of random variables \(\eta(\omega)\) of the form

\[ \eta=\sum_{t\in[a,b]} c(t)\xi(t) \]

with scalar product

\[ (\eta_1,\eta_2)=\int_{\Omega}\eta_1(\omega)\overline{\eta_2(\omega)}\,P_{i,\theta}\{d\omega\}. \]

Consider the linear operator \(B_n\) from \(H_0^n(0)\) into \(H_0^n(1)\) such that
\((x,y)_0=(B_nx,y)_1,\ x,y\in H_0^n(i)\). We calculate \(\|B_n-E\|\)—the Hilbert–Schmidt norm of the operator \(B_n-E\). If \(T\) is a linear isometry of the space \(H_0^n(0)\) into \(L_2(-\pi,\pi)\) such that \(T\xi(t)=e^{it\lambda}g(\lambda)\), then, by Beurling’s theorem ((3), p. 101), we have \(TH_{-\infty}^n(i)=e^{in\lambda}H^2\). It is easy to see that \(TH_0^\infty=\frac{g}{\bar g}H^2\). Hence

\[ \|B_n-E\|^2=\sum_k |g_{k+n}|^2 k . \tag{8} \]

From condition (1) follows the mutual absolute continuity of the Gaussian measures \(P_{0,n}\), \(P_{1,n}\) for sufficiently large \(n\). Hence, using one

Feldman’s result (see \((^4)\)), we infer that \(\|B_n-E\|<\infty\). From the last inequality and relation (8), (7) follows.

Sufficiency. Relation (8) implies the mutual absolute continuity of the measures mentioned above, if \(H_0^\infty(0)\cap H_{-\infty}^n=\{0\}\). Therefore, from relations (7), (8), and Theorem 4.1 of \((^2)\), relation (1) follows at once.

1. We finish the proof of Theorem 1.

Sufficiency. It is verified directly that (7) follows from (3).

Necessity. From condition (1) it follows that

\[ \alpha(\tau)=\sup_{A\in\mathfrak M_{-\infty}^{0},\ B\in\mathfrak M_{\tau}^{\infty}} \left|P\{AB\}-P\{A\}P\{B\}\right|\xrightarrow[\tau\to\infty]{}0. \tag{9} \]

Denote by \(L_2(f)\) the Hilbert space of functions square-integrable with weight \(f\), with scalar product

\[ (\varphi,\psi)_f=\int_{-\pi}^{\pi}\varphi(\lambda)\overline{\psi(\lambda)}f(\lambda)\,d\lambda. \]

By a theorem of A. N. Kolmogorov and Yu. A. Rozanov \((^5)\), condition (9) is equivalent to the condition

\[ \rho(\tau;f)=\sup_{\varphi,\psi}\left|(e^{i\tau\lambda}\varphi,\overline{\psi})_f\right|\xrightarrow[\tau\to\infty]{}0, \]

where the supremum is taken over all polynomials \(\varphi,\psi\in H^2\) with norms \(\|\varphi\|_f=\|\psi\|_f=1\). Hence, on the basis of the results of papers \((^6,^7)\), it follows that the function \(f(\lambda)\) can be written in the form \(|P(e^{i\lambda})|^2 f_1(\lambda)\), where \(P(z)\) is a polynomial with roots on the circle \(|z|=1\), \(\rho(1;1/f_1)=\rho<1\). It is not difficult to see that \(\overline{P}/P=ae^{-ik\lambda}\), \(|a|=1\), \(k\) is the degree of \(P\), so that together with the function \(g(z)\) satisfying condition (7), the outer function \(g_1(z)\), where \(|g_1(e^{i\lambda})|^2=f_1(\lambda)\), satisfies condition (7). The latter condition can be written in the form

\[ \sum \inf_A \left\|\frac{\overline{g_1}}{g_1}-e^{-in\lambda}A\right\|_2^2<\infty, \tag{10} \]

where \(\|\cdot\|_2\) is the norm in \(L_2(-\pi,\pi)\), and the infimum is taken over all \(A\in H^2\). Choose a sequence of polynomials \(A_n\in H^2\) for which the series (10) converges. Put \(e^{-in\lambda}g_1A_n=Q_n(e^{-in\lambda})+B_n\), where \(Q_n\) is a polynomial of degree \(n\), and all \(B_n\in H^2\). Then

\[ \left\|\overline{g_1}/g_1-e^{-in\lambda}A_n\right\|^2 =\left\|(g_1-Q_n)-B_n\right\|_{1/f_1}^2 \ge (1-\rho)\left\|g-\overline{Q}_n\right\|_{1/f_1}^2. \]

Consequently, the series

\[ \sum \inf_{P_n}\left\|g_1-\overline{P}_n\right\|_{1/f_1}^2, \tag{11} \]

converges, where the infimum is taken over all polynomials \(P_n(e^{i\lambda})\) of degree not exceeding \(n\).

Introduce the polynomials \(\varphi_\nu(z)\), orthonormal with weight \(1/f_1\). From the properties of such polynomials \((^8)\) one can infer that convergence of the series (11) is equivalent to the inequality

\[ \sum_{1}^{\infty}\nu\,|\varphi_\nu(0)|^2<\infty. \]

Let \(\varphi_\nu^*(z)=z^\nu\overline{\varphi_\nu(1/z)}\). The polynomials \(\varphi_{\nu,n}(z)\), orthonormal with weight \(|\varphi_n^*(e^{i\lambda})|^{-2}\), have the form \(\varphi_{\nu,n}(z)=\varphi_\nu(z)\), \(\nu\le n\); \(\varphi_{\nu,n}(z)=z^{\nu-n}\varphi_n(z)\), \(\nu\ge n\). Consequently,

\[ \sum_{\nu=1}^{\infty}\nu\,|\varphi_{\nu,n}(0)|^2 \le \sum_{\nu=1}^{\infty}\nu\,|\varphi_\nu(0)|^2. \tag{12} \]

Denote by \(D_{\nu,n}\) the \(\nu\)-th Toeplitz determinant constructed with respect to the weight \(|\varphi_n^*|^{-2}\). Let

\[ G(h)=\exp\left\{\frac{1}{2\pi}\int_{-\pi}^{\pi}\ln h(\lambda)\,d\lambda\right\}. \]

By a theorem of G. Szegő \(\left({}^{8}\right.\), p. 101),

\[ \lim_{\nu\to\infty}\frac{D_{\nu,n}}{\left(G\left(|\varphi_n^*|^{-2}\right)\right)^{\nu+1}} = \exp\left\{\frac{1}{\pi}\iint_{|z|\leq 1} \left|\frac{\varphi_n^{*\,\prime}(z)}{\varphi_n^*(z)}\right|^2\,d\sigma\right\}. \]

From this and from (12) it follows that all the integrals on the right-hand side of (13) are uniformly bounded. As \(n\to\infty\), \(\varphi_n^*(z)\to g_1(z)\) uniformly in every disk \(|z|\leq r<1\). Therefore

\[ \sum_{-\infty}^{\infty}|k|\,|f_k|^2 = \frac{2}{\pi}\iint_{|z|\leq 1} \left|\frac{g_1'(z)}{g_1(z)}\right|^2\,d\sigma = \lim_{n\to\infty}\frac{2}{\pi}\iint_{|z|\leq 1} \left|\frac{\varphi_n^{*\,\prime}(z)}{\varphi_n^*(z)}\right|^2\,d\sigma <\infty. \]

1. In the case of continuous time, one can give conditions sufficient for condition (1) to hold.

Theorem 3. Let the spectral density \(f(\lambda)\) of a stationary Gaussian process with continuous time be represented in the form \(|B(i\lambda)|^2 f_1(\lambda)\), where \(B(z)\) is an entire function of finite degree, and the function \(f_1(\lambda)\) is such that: 1) \(0<m\leq f_1(\lambda)\leq M\); 2) the function \(\hat f(\lambda)\)—the Fourier transform of \(f_1(\lambda)/(1+\lambda^2)\)—satisfies the inequality

\[ \int_{-\infty}^{\infty}|\lambda|\,|\hat f(\lambda)|^2\,d\lambda<\infty. \]

Then the process \(\xi(t)\) satisfies condition (1).

In the case of a process with discrete time, one can, just as was done in paper \(\left({}^{9}\right)\) (see Theorems 3, 4, 5), completely describe the class of those spectral densities for which \(\beta(n)\) decreases sufficiently rapidly. We give one such theorem.

Theorem 4. In order that \(\beta(n)=O\left(n^{1/2-r-\alpha}\right)\), where \(r\) is an integer, \(r+\alpha>1/2\), \(0<\alpha<1\), it is necessary and sufficient that the spectral density \(f(\lambda)\) be representable in the form \(|P(e^{i\lambda})|^2 f_1(\lambda)\), where \(P(e^{i\lambda})\) is a trigonometric polynomial, and the function \(f_1(\lambda)\) is positive, \(f_1(\lambda)\geq m>0\), and has an \(r\)-th derivative satisfying a Hölder condition of order \(\alpha\).

Leningrad State University
named after A. A. Zhdanov

Leningrad Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
30 XI 1967

REFERENCES

\({}^{1}\) V. A. Volkonskii, Yu. A. Rozanov, Theory of Probability and Its Applications, 4, no. 2 (1959).
\({}^{2}\) V. A. Volkonskii, Yu. A. Rozanov, ibid., 6, no. 2 (1961).
\({}^{3}\) K. Hoffman, Banach Spaces of Analytic Functions, Moscow, 1963.
\({}^{4}\) Yu. A. Rozanov, Theory of Probability and Its Applications, 8, no. 3 (1963).
\({}^{5}\) A. N. Kolmogorov, Yu. A. Rozanov, ibid., 5, no. 2 (1960).
\({}^{6}\) I. A. Ibragimov, ibid., 10, no. 1 (1965).
\({}^{7}\) H. Helson, G. Szegő, Ann. Mat. Pura ed Appl., 51, 107 (1960).
\({}^{8}\) U. Grenander, G. Szegő, Toeplitz Forms and Their Applications, Moscow, 1963.
\({}^{9}\) I. A. Ibragimov, Dokl. Akad. Nauk SSSR, 147, no. 6 (1962).