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

Part of: Stochastic processes Nonparametric inference

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]
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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 

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