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The stochastic equation Yt+1 = AtYt + Bt with non-stationary coefficients

Part of: Stochastic systems and control Limit theorems Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Ulrich Horst
Ulrich Horst*
Affiliation:
Humboldt-Universität zu Berlin
*
∗ Postal address: Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany. Email address: horst@mathematik.hu-berlin.de
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Abstract

In this paper, we consider the stochastic sequence {Yt}t∊ℕ defined recursively by the linear relation Yt+1 = AtYt + Bt in a random environment which is described by the non-stationary process {(At, Bt)}t∊ℕ. We formulate sufficient conditions on the environment which ensure that the finite-dimensional distributions of {Yt}t∊ℕ converge weakly to the finite-dimensional distributions of a unique stationary process. If the driving sequence {(At, Bt)}t∊ℕ becomes stationary in the long run, then we can establish a global convergence result. This extends results of Brandt (1986) and Borovkov (1998) from the stationary to the non-stationary case.

Keywords

stochastic difference equationstochastic stabilityergodicity

MSC classification

Primary: 60G35: Signal detection and filtering 93E15: Stochastic stability 60F99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 1 , March 2001 , pp. 80 - 94
Copyright
Copyright © by the Applied Probability Trust 2001 

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References

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