Wold decomposition, prediction and parameterization of stationary processes with infinite variance (Q1094748)

A discrete time stochastic process \(\{X_ t\}\) is said to be a p- stationary process \((1<p\leq 2)\) if \[ E| \sum ^{n}_{k=1}b_ kX_{t_ k+h}| ^ p=E| \sum ^{n}_{k=1}b_ kX_{t_ k}| ^ p, \] for all integers \(n\geq 1\), \(t_ 1,...,t_ n\), 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 \(\alpha\) \((1<\alpha \leq 2)\), and \(p^{th}\) order 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 scalars which are used as parameters of the process \(\{X_ t\}\). 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.