Abstract Generated abstract
The paper studies spectral conditions associated with strong mixing for stationary Gaussian processes in discrete and continuous time, using analytic functions built from the logarithm of the spectral density and Hardy space approximation methods. For discrete time, it relates mixing to properties of the conjugate function of the logarithmic spectral density, gives sufficient continuity and approximation criteria, and shows that prescribed rates of decay of the mixing coefficient can be realized. For continuous time, it establishes analogous necessary and sufficient type conditions involving factorization of the spectral density, smoothness or modulus conditions on auxiliary functions, and approximation by entire functions of finite degree, including examples of smooth spectral densities that fail to be strongly mixing. The final section extends these results to generalized stationary Gaussian processes by comparing mixing coefficients under polynomial changes of the spectral density.
Full Text
Reports of the Academy of Sciences of the USSR
1965. Volume 161, No. 1
MATHEMATICS
I. A. IBRAGIMOV
ON A STRONG MIXING CONDITION FOR STATIONARY GAUSSIAN PROCESSES
(Presented by Academician Yu. V. Linnik on 10 X 1964)
In the present note we study the properties of the spectral density (s.d.) \(f(\lambda)\) of a stationary Gaussian process \(x(t)\) satisfying the strong mixing condition \((^{1-4})\). In contrast to the purely real methods of note \((^3)\), the methods of the present paper are connected with the study of the function \(\Gamma(z)\) associated with \(f(\lambda)\), analytic in the disk (upper half-plane) (see below).
1. Discrete time. In this section we consider processes \(x(t)\) with discrete time \(t=0,\pm1,\ldots\). In view of \((^2)\), the mixing coefficient is
\[ \alpha(\tau)=\sup\left|\int_{-\pi}^{\pi} e^{i\tau\lambda}\varphi(\lambda)f(\lambda)\,d\lambda\right|, \qquad \tau=0,1,\ldots, \tag{1} \]
where the supremum is taken over all continuous functions \(\varphi(\lambda)\), analytically continuable inside the disk, for which
\[ \int_{-\pi}^{\pi}|\varphi(\lambda)|f(\lambda)\,d\lambda=1. \]
A process \(x(t)\) satisfying the strong mixing condition is, of course, regular and, by a well-known theorem of A. N. Kolmogorov,
\[ \int_{-\pi}^{\pi}\ln f(\lambda)\,d\lambda \]
converges.
Put
\[ \Gamma(z)=\exp\left\{\frac{1}{2\pi}\int_{-\pi}^{\pi}\ln f(\lambda)\, \frac{e^{i\lambda}+z}{e^{i\lambda}-z}\,d\lambda\right\}, \qquad |z|<1. \]
This function is analytic in the disk \(|z|<1\). Its radial boundary values exist almost everywhere, and almost everywhere \(\left|\Gamma(e^{i\lambda})\right|=f(\lambda)\). From Beurling’s results \((^5)\) (\(\Gamma(z)\) is an outer function) it follows that the set of functions \(\{\varphi(\lambda)\Gamma(e^{i\lambda})\}\), where \(\varphi(\lambda)\) runs through all polynomials in nonnegative powers of \(e^{i\lambda}\), is dense in the unit sphere of the Hardy space \(H_1\). Therefore, putting
\[ f(\lambda)=\Gamma(e^{i\lambda})\exp\{-i\arg\Gamma(e^{i\lambda})\}, \]
we find from (1) that
\[ \alpha(\tau)=\sup\left|\int_{-\pi}^{\pi}e^{i\tau\lambda}\varphi(\lambda) \exp\{-i\arg\Gamma(e^{i\lambda})\}\,d\lambda\right|, \tag{2} \]
where the supremum is taken over all elements \(\varphi\) of the unit sphere of \(H_1\).
Put \(\ln f(\lambda)=q(\lambda)\). Obviously,
\[ \arg\Gamma(e^{i\lambda})=\tilde q(\lambda), \]
where, in general, \(\tilde q\) denotes the function conjugate (trigonometrically) to \(q\).
Studying the behavior of the integral in (2) for functions \(\varphi(\lambda)\) of the form
\[ \frac{1}{2\pi N}\left(\frac{e^{iN(\lambda-\mu)}-1}{e^{i(\lambda-\mu)}-1}\right)^2, \]
one can prove the following theorem.
Theorem 1. If \(\alpha(\tau)\to0\), the function \(\tilde q(\lambda)=\ln f(\lambda)\) has no discontinuities of the first kind, except for jumps whose magnitude is a multiple of \(2\pi\).
This theorem makes it possible to derive in another way the consequence of remark \({}^{(3)}\) on the order of the zeros of the spectral density \(f(\lambda)\), relating in addition the behavior of the zeros of \(f(\lambda)\) to the smoothness of \(\tilde q(\lambda)\).
Theorem 2. If the function \(e^{-i\tilde q(\lambda)}\) is continuous, the process \(x(t)\) satisfies the strong mixing condition. Moreover,
\[ \alpha(\tau)=O\left(E_{\tau-1}\left(e^{-i\tilde q}\right)\right), \]
where \(E_\tau(\psi)\) denotes the value of the best approximation of the function \(\psi(\lambda)\) by trigonometric polynomials of degree \(\leqslant \tau\).
From Theorem 2 follows Corollary 1, which makes it possible to construct processes \(x(t)\) with a discontinuous spectral density \(f(\lambda)\) that satisfy the strong mixing condition.
Corollary 1. If at least one of the functions \(q(\lambda)\), \(\tilde q(\lambda)\) is continuous, the process \(x(t)\) satisfies the strong mixing condition, and
\[ \alpha(\tau)=O\left(\min\left(E_{\tau-1}(q),E_{\tau-1}(\tilde q)\right)\right). \]
For example, if
\[ f(\lambda)=e^{q(\lambda)}=\exp\left\{\sum_1^\infty \frac{\cos k\lambda}{k\ln k}\right\}, \]
then the process \(x(t)\) satisfies the strong mixing condition, since
\[ \tilde q(\lambda)=\sum_1^k \frac{\sin k\lambda}{k\ln k} \]
is continuous. Here \(f(\lambda)\) has a logarithmic singularity, and
\[ \alpha(\tau)=O\left(E_\tau(\tilde q)\right)=O\left((\ln \tau)^{-1}\right). \]
Theorem 3. Let \(\gamma(\tau)\) be an arbitrary sequence monotonically decreasing to zero. There exists a stationary Gaussian process \(x(t)\) for which
\[ \frac{\alpha(\tau)}{\gamma(\tau)}\sim 1. \]
For the proof it is enough to consider the process with spectral density
\[ f(\lambda)=A+\sum_1^\infty[\gamma(k)-\gamma(k+1)]\cos k\lambda, \]
where \(A\) is chosen so that \(f(\lambda)>m>0\).
2. Continuous time. In this section we consider processes with continuous time, \(-\infty<t<\infty\). Now
\[ \alpha(\tau)=\sup\left|\int_{-\infty}^{\infty} e^{i\tau\lambda}\varphi(\lambda)f(\lambda)\,d\lambda\right|,\qquad \tau\geqslant 0, \tag{3} \]
where the supremum is taken over functions \(\varphi(\lambda)\) analytically continuable into the upper half-plane and satisfying the condition
\[ \int_{-\infty}^{\infty}|\varphi(\lambda)|f(\lambda)\,d\lambda=1. \]
We first give analogues of Theorems 1 and 2 from \({}^{(3)}\).
Theorem 4. Whatever the number \(a\), \(0<a<\infty\), the spectral density \(f(\lambda)\) of a process \(x(t)\) satisfying the strong mixing condition is representable in the form
\[ f(\lambda)=|P_a(\lambda)|^2 g_a(\lambda), \]
where \(P_a(\lambda)\) is a polynomial with real zeros, and the primitive \(G_a(\lambda)\) of the function \(g_a(\lambda)\) satisfies the following condition: as \(h\to 0\), uniformly in \(\lambda\in[-a,a]\),
\[ |G_a(\lambda+h)+G_a(\lambda-h)-2G_a(\lambda)|=o\left(G_a(\lambda+h)-G_a(\lambda)\right). \tag{4} \]
From this theorem it follows that Corollaries 1–3 of remark \({}^{(3)}\) remain valid also for processes \(x(t)\) with continuous time.
Theorem 5. Let the spectral density of a process \(x(t)\) satisfying the strong mixing condition be representable in the form
\[ f(\lambda)=|B(\lambda)|^2 g(\lambda), \tag{5} \]
where \(B(\lambda)\) is a square-integrable entire function of finite degree, having no nonreal zeros in some strip containing the real axis, and the function \(g(\lambda)\) is bounded above and below, \(0<m<g(\lambda)<M\). Then, for the primitive \(G(\lambda)\) of the function \(g(\lambda)\), condition (4) is satisfied uniformly for all \(\lambda\in(-\infty,\infty)\).
Theorem 6. Let the s.d. \(f(\lambda)\) be representable in the form (5), where \(B(\lambda)\), \(g(\lambda)\) are the same as in Theorem 5. If
\[ \sum_{n=1}^{\infty}\omega^2(2^{-n})<\infty, \]
\[ \omega(h)=\sup_{\lambda,\,s\le h} \frac{|G(\lambda+s)+G(\lambda-s)-2G(\lambda)|}{G(\lambda+s)-G(\lambda)}, \]
then the process \(x(t)\) satisfies the strong mixing condition.
Theorem 5 makes it possible to give examples of processes \(x(t)\) whose s.d. is arbitrarily smooth, has regular behavior at infinity \((\asymp (1+\lambda^2)^p\), and which, nevertheless, do not satisfy the strong mixing condition. Such, for example, are processes with s.d.
\[ f(\lambda)=\frac{1+\sin^2\lambda^2}{(1+\lambda^2)^p},\qquad p>\frac12. \]
In order to transfer the results of Sec. 1 to the case of continuous time, put
\[ \Gamma(z)=\exp\left\{\frac{1}{\pi i}\int_{-\infty}^{\infty} \frac{\ln f(\lambda)}{1+\lambda^2}\frac{1+z\lambda}{\lambda-z}\,d\lambda\right\}, \qquad \operatorname{Im} z>0, \]
\[ q(\lambda)=\frac{\ln f(\lambda)}{1+\lambda^2},\qquad Q(\lambda)=(1+\lambda^2)\tilde q(\lambda) =\frac{1}{\pi}(1+\lambda^2)\int_{-\infty}^{\infty} \frac{\ln f(x)}{1+x^2}\frac{dx}{x-\lambda}. \]
Arguments similar to those used in Sec. 1 lead to the following theorems.
Theorem 7. If the function \(\exp\{iQ(\lambda)\}\) is uniformly continuous, then \(\alpha(\tau)\to0\), and moreover \(\alpha(\tau)\le A_\tau(e^{iQ})\), where \(A_\tau(\psi)\) is the value of the best approximation of the function \(\psi(\lambda)\) by entire functions of finite degree \(\le \tau\).
Theorem 8. In order that \(\alpha(\tau)=O(\tau^{-r-\beta})\), \(r=0,1,\ldots;\ 0<\beta<1\), it is sufficient that \(f(\lambda)\) be representable in the form
\[ f(\lambda)=|B(\lambda)|^2g(\lambda), \]
where \(B(\lambda)\) is a square-integrable entire function of finite degree, and \(g(\lambda)\) has the following properties:
\[ \text{1) }\quad \int_{-\infty}^{\infty}\frac{|\ln g(\lambda)|}{1+\lambda^2}\,d\lambda<\infty; \]
2) the function \(\ln g(\lambda)\) has an \(r\)-th derivative satisfying a Hölder condition of order \(\beta\).
Theorem 9. Let the s.d. \(f(\lambda)\) be representable in the form
\[ f(\lambda)=|B(\lambda)|^2g(\lambda), \]
where:
1) \(B(\lambda)\) is a bounded entire function of finite degree;
2) for some real \(p\), \(-\infty<p<\infty\), \(|\lambda|^p g(\lambda)\asymp1\), \(\lambda\to\infty\);
3) the function \(\ln g(\lambda)\) is uniformly continuous, or, equivalently,
\[ \sup_{\lambda,\, t \le h}\frac{|g(\lambda+t)-g(\lambda)|}{g(\lambda)} = \omega(h)\downarrow 0,\quad h\to 0. \]
Then the process \(x(t)\) satisfies the strong mixing condition, and moreover
\(a(\tau)=O(\omega(1/\tau))\).
3. Generalized processes. Let now \(x(\varphi)\) be a generalized stationary Gaussian process in the sense of K. Itô— I. M. Gelfand (see \({}^{6}\)). Naturally, we shall say that it satisfies the strong mixing condition if
\[ |Ex(\varphi)x(\psi)| \le \alpha(\tau)\bigl(E|x(\varphi)|^2E|x(\psi)|^2\bigr)^{1/2}, \quad \alpha(\tau)\downarrow 0, \]
where the supports of the functions \(\varphi\) and \(\psi\) are located respectively in \((-\infty,0]\), \([\tau,\infty)\), \(\tau\ge 0\). The mixing coefficient \(\alpha(\tau)\) in this case too is defined by equality (3), where now \(f(\lambda)\) is the s.d. of the process \(x(\varphi)\).
Lemma 1. The mixing coefficients of generalized stationary processes with s.d. \(f(\lambda)\) and \(f(\lambda)(1+\lambda^2)^p\), \(-\infty<p<\infty\), differ by a quantity of order \(O(e^{-c\tau})\), \(c=c(p)>0\).
It follows from this lemma that all the theorems of Sec. 2 remain valid also for generalized processes.
Leningrad State University
named after A. A. Zhdanov
Received
1 X 1964
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