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

  • Gerold Alsmeyer (a1), Dariusz Buraczewski (a2) and Alexander Iksanov (a3)
Abstract

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

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Corresponding author
* 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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Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
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