Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-24T11:27:15.088Z Has data issue: false hasContentIssue false
  • English
  • Français

A unique representation theorem for the conditional expectation of stationary processes and application to dynamic estimation problems

Part of: Nonparametric inference Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Marco Campi
Marco Campi*
Affiliation:
Università di Brescia
*
∗Postal address: Dipartimento di Elettronica per l'Automazione, Università di Brescia, via Branze, 38, 25123 Brescia, Italy. e-mail address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, multivariate strict sense stationary stochastic processes are considered. It is shown that there exists a universal function by means of which the conditional expectation of any stationary process with respect to its past can be represented. This requires no ergodicity assumptions. The important implications of this result in the evaluation of the achievable performance in certain dynamic estimation problems with incomplete statistical information are also discussed.

Keywords

STATIONARITYCONDITIONAL EXPECTATIONUNIQUENESSDYNAMIC ESTIMATION PROBLEMS

MSC classification

Primary: 60G10: Stationary processes
Secondary: 62G05: Estimation
Type
Research Article
Information
Journal of Applied Probability , Volume 34 , Issue 2 , June 1997 , pp. 372 - 380
Copyright
Copyright © Applied Probability Trust 1997 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Benveniste, A. and Goursat, G. (1984) Blind equalizers. IEEE Trans. Commun. 32, 871–883.Google Scholar
Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
Giannakis, G. B. and Mendel, J. M. (1989) Identification of non-minimum phase systems using higher order statistics. IEEE Trans. Acoustic. Speech Signal Proc. 37, 360–377.CrossRefGoogle Scholar
Halmos, P. R. (1956) Lectures on Ergodic Theory. Mathematical Society of Japan, Tokyo.Google Scholar
Haykin, S. (1994) Blind Deconvolution. Prentice Hall, New Jersey.Google Scholar
Macchi, O. M. and Bershad, N. J. (1991) Adaptive recovery of chirped sinusoid in noise, part 1: performance of the RLS algorithm. IEEE Trans. Signal Proc. 39, 583–594.CrossRefGoogle Scholar
Widrow, B. and Stearns, S. D. (1985) Adaptive Signal Processing. Prentice Hall, New Jersey.Google Scholar