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Null recurrence and transience of random difference equations in the contractive case

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  30 November 2017

Gerold Alsmeyer ,
Dariusz Buraczewski  and
Alexander Iksanov
Gerold Alsmeyer*
Affiliation:
University of Münster
Dariusz Buraczewski*
Affiliation:
University of Wrocław
Alexander Iksanov*
Affiliation:
Taras Shevchenko National University of Kyiv and University of Wrocław
*
* Postal address: Institute of Mathematical Stochastics, Department of Mathematics and Computer Science, University of Münster, Einsteinstrasse 62, D-48149 Münster, Germany. Email address: gerolda@math.uni-muenster.de
** Postal address: Institute of Mathematics, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland. Email address: dbura@math.uni.wroc.pl
*** Postal address: Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, 01601 Kyiv, Ukraine. Email address: iksan@univ.kiev.ua
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Abstract

Given a sequence (Mk, Qk)k ≥ 1 of independent and identically distributed random vectors with nonnegative components, we consider the recursive Markov chain (Xn)n ≥ 0, defined by the random difference equation Xn = MnXn - 1 + Qn for n ≥ 1, where X0 is independent of (Mk, Qk)k ≥ 1. Criteria for the null recurrence/transience are provided in the situation where (Xn)n ≥ 0 is contractive in the sense that M1 ⋯ Mn → 0 almost surely, yet occasional large values of the Qn overcompensate the contractive behavior so that positive recurrence fails to hold. We also investigate the attractor set of (Xn)n ≥ 0 under the sole assumption that this chain is locally contractive and recurrent.

Keywords

Attractor setnull recurrenceperpetuityrandom difference equationtransience

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60F15: Strong theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 4 , December 2017 , pp. 1089 - 1110
Copyright
Copyright © Applied Probability Trust 2017 

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