Stationary Processes (No. 48)

Abstract

The authors solve the problem for a Gaussian stationary process X(t) under the strong mixing (SM) condition introduced by Rosenblatt [1]. It is proved that in this case the SM function α(τ) is equivalent to the maximal correlation coefficient p(τ) (α(τ) ≤ p(τ) ≤ 2πα(τ)) and the representation (4) for p(τ) is found in terms of the spectral density of the process f(λ), which exists in view of the SM condition. Thus, the problem is solved completely. For example, in the case of integer time the process X t possesses the SM property if and only if where the supremum is taken over all one-sided trigonometric polynomials and ∥ · ∥∞ is the essential supremum norm. This implies that if f(λ) is continuous, f(-π) = f(π), and bounded everywhere, except perhaps at zero, then X t has the SM property. Ibragimov [2] obtained further results on the form of f(λ) when X t possesses the SM property. Helson and Sarason [3] reconsidered this problem as a question in harmonic analysis. Generalizations were also obtained for functions X t with values in a space of matrices [6]. In the case of a random walk on a compact abelian group Rosenblatt [4] showed that a stationary process has the SM property if and only where T is the transition operator of the random walk. A.N. Kolmogorov also posed the problem of finding an effective criterion for complete regularity of a Gaussian stationary process; it was solved by Volkonskii and Rozanov [7]. A student of A.N. Kolmogorov, Leonov [8], studied various SM conditions of random processes using higher semi-invariants.