A characterization of oscillatory processes and their prediction

Abstract

The oscillatory stochastic processes recently studied by Priestley are characterized as deformed stationary curves in a Hilbert space. This characterization leads to the simple time domain proof of prediction and moving average representation for these stochastic processes in terms of the associated stationary curve. In [6], A. N. Kolmogorov studied second order stationary processes as curves in Hilbert space. The idea again occurred in Cramer [2] for nonstationary processes. In this note we characterize the oscillatory processes introduced by Priestley [8] as deformed stationary curves in a Hilbert space and give a simple geometric solution for the prediction problem for such curves. As a consequence of this one can easily derive the analytic results on prediction in [1], thus providing a simple and more general solution for the prediction problem of oscillatory processes. Also our characterization makes the definition of the time dependent spectral distribution an obvious generalization of the stationary case. For the sake of being specific we consider throughout the continuous parameter case. 1. DEFINITION. Let H be a Hilbert-space and R be the space of real numbers with the usual topology. We say that (a) x is a continuous curve in H, if x is a continuous map of R into H. (b) y is a stationary continuous curve in H if (i) y is a continuous curve in H, (ii) (y(t), y(s)) is a function of t-s. With every curve x in H we associate the following subspaces of H: Hx(t) = S{x(r), 7 C t},2 Hx(+ oo) = {x(-), c RI, H$(-oo) = n H$(t). Received by the editors April 26, 1971. AMS 1970 subject classqiications. Primary 60G25; Secondary 47A50.