Wold decomposition, prediction and parameterization of stationary processes with infinite variance

Abstract

A discrete time stochastic process {Χt} is said to be a p-stationary process (1<p≦2)if $$E\left| {\sum\limits_{k = 1}^n {b_k X_{tk + h} } } \right|^p = E\left| {\sum\limits_{k = 1}^n {b_k X_{tk} } } \right|^p $$ , for all integers n≧1, t 1,...t n,h and scalars b 1,...b n.The class of p-stationary processes includes the class of second-order weakly stationary stochastic processes, harmonizable stable processes of order α (1<α≦2), and p thorder strictly stationary processes. For any nondeterministic process in this class a finite Wold decomposition (moving average representation) and a finite predictive decomposition (autoregressive representation) are given without alluding to any notion of “covariance” or “spectrum”. These decompositions produce two unique (interrelated) sequences of scalar which are used as parameters of the process {Χt}. It is shown that the finite Wold and predictive decomposition are all that one needs in developing a Kolmogorov-Wiener type prediction theory for such processes.