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On the strong stability of ergodic iterations

Part of: Stochastic processes Special processes Markov processes

Published online by Cambridge University Press:  05 November 2024

László Györfi ,
Attila Lovas Open the ORCID record for Attila Lovas [Opens in a new window]  and
Miklós Rásonyi Open the ORCID record for Miklós Rásonyi [Opens in a new window]
László Györfi*
Affiliation:
Budapest University of Technology and Economics
Attila Lovas*
Affiliation:
HUN-REN Alfréd Rényi Institute of Mathematics and Budapest University of Technology and Economics
Miklós Rásonyi*
Affiliation:
HUN-REN Alfréd Rényi Institute of Mathematics
*
*Postal address: Department of Computer Science and Information Theory, Faculty of Electrical Engineering and Informatics, Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary. Email: [email protected]
**Postal address: Department of Analysis and Operations Research, Faculty of Natural Sciences, Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary. Email: [email protected]
***Postal address: HUN-REN Alfréd Rényi Institute of Mathematics, Reáltanoda street 13-15, H-1053, Budapest. Email: [email protected]
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Abstract

We revisit processes generated by iterated random functions driven by a stationary and ergodic sequence. Such a process is called strongly stable if a random initialization exists for which the process is stationary and ergodic, and for any other initialization the difference of the two processes converges to zero almost surely. Under some mild conditions on the corresponding recursive map, without any condition on the driving sequence we show the strong stability of iterations. Several applications are surveyed such as generalized autoregression and queuing. Furthermore, new results are deduced for Langevin-type iterations with dependent noise and for multitype branching processes.

Keywords

Iterated random functionsstrong stabilitystationary and ergodic sequencesrecursive mapsgeneralized autoregression

MSC classification

Primary: 60G10: Stationary processes
Secondary: 60K37: Processes in random environments 60K25: Queueing theory 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Original Article
Information
Journal of Applied Probability , Volume 62 , Issue 1 , March 2025 , pp. 284 - 297
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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