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This note studies spectral densities of stationary Gaussian processes under two dependence weakening conditions: strong mixing and a stronger conditional probability condition. It shows that the latter condition is equivalent to eventual independence of the past and future sigma algebras, yielding a spectral characterization in terms of squared moduli of trigonometric polynomials in discrete time and entire functions of exponential type in continuous time. For strongly mixing Gaussian processes, the paper derives restrictions on the spectral density, including absence of jump discontinuities, limits on possible singular growth near a point, and constraints on the orders of zeros, which must bracket an even integer. An additional discrete-time consequence relates strong mixing to the existence of a polynomial satisfying an integrability condition involving the reciprocal spectral density.
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MATHEMATICS
I. A. IBRAGIMOV
ON THE SPECTRAL FUNCTIONS OF SOME CLASSES OF STATIONARY GAUSSIAN PROCESSES
(Presented by Academician A. N. Kolmogorov on 25 XI 1960)
- Let \(x(t)\) be a stationary Gaussian process. Denote by \(\mathfrak M_a^b\) the \(\sigma\)-algebra of events generated by the random variables \(x(t)\), \(a \le t \le b\). It is said that the process \(x(t)\) has the property of strong mixing \((^1)\) if, as \(\tau \to \infty\),
\[ \sup_{\substack{A\in \mathfrak M_{-\infty}^{0}\\ B\in \mathfrak M_{\tau}^{\infty}}} \left|{\bf P}(AB)-{\bf P}(A){\bf P}(B)\right| =\alpha(\tau)\downarrow 0 . \tag{1} \]
The fulfillment of (1) means that, for large \(\tau\), events determined by the beginning and the end of the process \(x(t)\) become weakly dependent.
In the author’s papers \((^2,^3)\) the following condition of weakening of dependence between \(\mathfrak M_{-\infty}^{0}\), \(\mathfrak M_{\tau}^{\infty}\) was also used: for every \(B\in \mathfrak M_{\tau}^{\infty}\), with probability 1,
\[ \left|{\bf P}(B\mid \mathfrak M_{-\infty}^{0})-{\bf P}(B)\right| \le \varphi(\tau)\downarrow 0 \quad\text{as }\tau\to\infty . \tag{2} \]
In the present note a number of theorems are formulated concerning the properties of the spectral function \(F(\lambda)\) of a Gaussian process \(x(t)\) satisfying requirements (1) or (2). Since the spectral function \(F(\lambda)\) of a Gaussian process possessing property (1) or (2) is absolutely continuous, below we shall everywhere speak not of it, but of the spectral density (s.d.) \(f(\lambda)=F'(\lambda)\) of the process.
- Theorem 1. In order that a stationary Gaussian process \(x(t)\) satisfy condition (2), it is necessary and sufficient that, for sufficiently large \(\tau\), \(\tau>\tau_0\), the \(\sigma\)-algebras \(\mathfrak M_{-\infty}^{0}\), \(\mathfrak M_{\tau}^{\infty}\) be independent.
Restating the independence condition for the \(\sigma\)-algebras \(\mathfrak M_{-\infty}^{0}\), \(\mathfrak M_{\tau}^{\infty}\) in spectral language, we arrive at the following variant of Theorem 1:
Theorem 1′. In order that the function \(f(\lambda)\) be the s.d. of a stationary Gaussian process satisfying condition (2), it is necessary and sufficient that it be the square of the modulus of some trigonometric polynomial—in the discrete case, or that it be an entire transcendental function of exponential type with exponents \(\le \sigma<\infty\), nonnegative and summable on \((-\infty,\infty)\), in the continuous-time case.
- Theorem 2. The spectral density \(f(\lambda)\) of a stationary Gaussian process \(x(t)\) possessing the property of strong mixing has no discontinuities of the first kind.
In the proof of this and the following theorems, an inequality obtained in \((^4)\) is essentially used,
\[ \alpha(\tau)\le \rho(\tau)\le 2\pi\alpha(\tau), \]
\[ \rho(\tau)= \sup_{\substack{\xi\in \mathfrak M_{-\infty}^{0},\, \eta\in \mathfrak M_{\tau}^{\infty}}} \frac{{\bf E}(\xi-{\bf E}\xi)(\eta-{\bf E}\eta)} {\sqrt{{\bf E}(\xi-{\bf E}\xi)^2\,{\bf E}(\eta-{\bf E}\eta)^2}}, \tag{3} \]
where by \(\mathfrak{M}_a^b\) is denoted the closed linear hull (in the mean-square sense) of the quantities \(x(t)\), \(a \leqslant t \leqslant b\).
Theorem 3. If \(f(\lambda)\) is the spectral density of a stationary Gaussian process \(x(t)\) having the property of strong mixing, then for all \(\delta > 0\)
\[ \lim_{\lambda \to \lambda_0} f(\lambda)|\lambda-\lambda_0|^\delta = 0, \]
in other words, it is impossible that
\[ \lim_{\lambda \to \lambda_0} f(\lambda)|\lambda-\lambda_0|^\delta = \infty . \]
Moreover, if on some set \(\Lambda\)
\[ \lim_{\lambda \to \lambda_0,\ \lambda \in \Lambda} f(\lambda)|\lambda-\lambda_0|^\delta = \infty, \]
then
\[ \lim_{\varepsilon \to 0} \frac{\operatorname{mes}\bigl(\Lambda \cap (\lambda_0-\varepsilon,\lambda_0+\varepsilon)\bigr)} {2\varepsilon} =0 . \]
- Let us call, respectively, the order, the upper order, and the lower order of the zero \(\lambda_0\) of the function \(f(\lambda)\):
\[ k(\lambda_0)=\lim_{\lambda \to \lambda_0}\frac{\log f(\lambda)} {\log|\lambda-\lambda_0|}, \]
\[ \overline{k}(\lambda_0)=\overline{\lim_{\lambda \to \lambda_0}} \frac{\log f(\lambda)} {\log|\lambda-\lambda_0|}, \]
\[ \underline{k}(\lambda_0)=\underline{\lim_{\lambda \to \lambda_0}} \frac{\log f(\lambda)} {\log|\lambda-\lambda_0|}. \tag{*} \]
Theorem 4. If \(f(\lambda)\) is the spectral density of a stationary Gaussian process possessing the property of strong mixing, then between the upper and lower order of the zero \(\lambda_0\) of the function \(f(\lambda)\) there necessarily lies an even integer
\[ \underline{k}(\lambda_0) \leqslant 2n \leqslant \overline{k}(\lambda_0),\qquad n=0,1,\ldots,\infty, \]
i.e. the true order of the zero \(\lambda_0\), \(k(\lambda_0)\), can only be an integer and even**.
Remark. If \(x(t)\) is a stationary Gaussian process possessing the property of strong mixing, then with the aid of (3) it is easy to show that there exists an interval \((-T,T)\) such that
\[ \inf_{\xi} E|x(0)-\xi|^2=\sigma_T^2>0, \]
where the infimum is taken over all \(\xi\) belonging to the closed (in the mean-square sense) linear hull of the quantities \(x(t)\), \(t \geqslant T\).
Therefore, from the results of A. M. Yaglom \((^5)\) for processes with discrete time, the following additional characteristic of the zeros of their spectral density is obtained:
If \(f(\lambda)\) is the spectral density of a stationary Gaussian process \(x(t)\) with discrete time, \(t=\ldots,-1,0,1,\ldots\), possessing the property of strong mixing, then there exists a polynomial
\[ P(e^{i\lambda})=\sum_{0\leqslant k<T}\alpha_k e^{i\lambda k}, \]
* This definition was pointed out to me by A. N. Kolmogorov.
** The formulation of Theorem 4 given here belongs to A. N. Kolmogorov.
that
\[ \int_{-\pi}^{\pi} \frac{|P(e^{i\lambda})|^2}{f(\lambda)}\, d\lambda < \infty . \]
I express my deep gratitude to A. N. Kolmogorov for a number of valuable comments.
Leningrad State University
named after A. A. Zhdanov
Received
19 XI 1960
References
- M. Rosenblatt, Proc. Nat. Acad. Sci. USA, 42, 43 (1956).
- I. A. Ibragimov, DAN, 125, No. 4 (1959).
- I. A. Ibragimov, Vestn. LGU, No. 1 (1960).
- A. N. Kolmogorov, Yu. A. Rozanov, Theory of Probability and Its Applications, 5, issue 3 (1960).
- A. M. Yaglom, UMN, 4, issue 4 (1949).