Probability Theory and Related Fields

, Volume 79, Issue 1, pp 145–164

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

Authors

  • A. G. Miamee
    • Department of Mathematical SciencesNorthern Illinois University
  • M. Pourahmadi
    • Department of Mathematical SciencesNorthern Illinois University
Article

DOI: 10.1007/BF00319110

Cite this article as:
Miamee, A.G. & Pourahmadi, M. Probab. Th. Rel. Fields (1988) 79: 145. doi:10.1007/BF00319110

Summary

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, t1,...tn,h and scalars b1,...bn.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 pthorder 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.

Copyright information

© Springer-Verlag 1988