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Doubly stochastic Hilbertian processes

Part of: Stochastic processes Inference from 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: guillas@galton.uchicago.edu
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Abstract

In this paper, we consider a Hilbert-space-valued autoregressive stochastic sequence (X n ) with several regimes. We suppose that the underlying process (I n ) which drives the evolution of (X n ) is stationary. Under some dependence assumptions on (I n ), 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.

Information

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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