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

Part of: Stochastic processes Limit theorems Stochastic systems and control

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

Borovkov, A. A. (1998). Ergodicity and Stability of Stochastic Processes. John Wiley, New York.Google Scholar
Brandt, A. (1986). The stochastic equation Y_n+1=A_nY_n + B_n with stationary coefficients. Adv. Appl. Prob. 18, 211–220.Google Scholar
Brandt, A., Franken, L., and Lisek, B. (1990). Stationary Stochastic Models. John Wiley, New York.Google Scholar
Breiman, L. (1968). Probability. Addison-Wesley, Reading.Google Scholar
Ethier, S. N., and Kurtz, T. G. (1986). Markov Processes Characterization and Convergence. John Wiley, New York.CrossRefGoogle Scholar
Föllmer, H. (1979). Tail structure of Markov chains on infinite product spaces. Z. Wahrscheinlichkeitsth. 50, 273–285.CrossRefGoogle Scholar
Horst, U. (2000). Asymptotics of locally and globally interacting Markov processes arising in microstructure models of financial markets. Doctoral Thesis, Humboldt-Universität zu Berlin.Google Scholar
Iosefescu, M., and Theodorescu, M. (1968). Random Processes and Learning. Springer, Berlin.Google Scholar
Vervaat, W. (1979). On a stochastic difference equation and a representation of non-negative infinitely divisible random variables. Adv. Appl. Prob. 11, 750–783.CrossRefGoogle Scholar