ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

12. Probability and Statistics

1991 MS Classification: 60, 43 Makagon, Andrzej, Hampton University, USA Periodically Correlated Processes and Mackey’s Imprimitivity Theorem. A second orded stochastic process X : R → H is called Periodically Correlated (PC) if X is continuous and for every s the correlation K(s + t, t) = (X(t + s), X(t)) is periodic in t with the same period T (Gladyshev, 1963). Gladyshev proved that a continuous function K(s, t) is the correlation function of a PC process iff the matrix function with entries K^{k,j}(t) = exp (−2πikt/T)a_{k−j}(t), where a_{j}(t) = (1/T) ∫_{0}^{T} exp (−2πiju/T)K(t + u, u)du, is positive definite, i. e. iff it is the correlation function of some infinite dimensional stationary process Y^{k}(t). In this note a method for an explicit reconstruction of the process X from Y^{k}(t) (and vice versa ) is presented. The construction leads to complete characterization of correlation functions of not necessary continuous PC processes. It turns out that this characterization is equivalent to a special case of Mackey’s Imprimitivity Theorem.