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Doubly stochastic Hilbertian processes

Part of: Inference from stochastic processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Serge Guillas
Serge Guillas*
Affiliation:
Université Pierre et Marie Curie–Paris VI and École des Mines de Douai
*
∗ Current address: The University of Chicago, Center for Integrating Statistical and Environmental Science, Chicago, IL 60637, USA. Email address: [email protected]
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Abstract

In this paper, we consider a Hilbert-space-valued autoregressive stochastic sequence (Xn) with several regimes. We suppose that the underlying process (In) which drives the evolution of (Xn) is stationary. Under some dependence assumptions on (In), we prove the existence of a unique stationary solution, and with a symmetric compact autocorrelation operator, we can state a law of large numbers with rates and the consistency of the covariance estimator. An overall hypothesis states that the regimes where the autocorrelation operator's norm is greater than 1 should be rarely visited.

Keywords

Hilbertnonlinearautoregressivestability

MSC classification

Primary: 60G10: Stationary processes
Secondary: 62M10: Time series, auto-correlation, regression, etc.
Type
Research Papers
Information
Journal of Applied Probability , Volume 39 , Issue 3 , September 2002 , pp. 566 - 580
Copyright
Copyright © Applied Probability Trust 2002 

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