Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-21T14:17:50.607Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Markov processes Limit theorems

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
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Babillot, M., Bougerol, P. and Elie, L. (1997). The random difference equation X n = A n X n-1 + B n in the critical case. Ann. Prob. 25, 478–493. Google Scholar
[2] Bauernschubert, E. (2013). Perturbing transient random walk in a random environment with cookies of maximal strength. Ann. Inst. H. Poincaré Prob. Statist. 49, 638–653. Google Scholar
[3] Benda, M. (1998). Schwach kontraktive dynamische systeme. Doctoral thesis. Ludwig-Maximilians-Universität München. Google Scholar
[4] Brofferio, S. (2003). How a centred random walk on the affine group goes to infinity. Ann. Inst. H. Poincaré Prob. Statist. 39, 371–384. Google Scholar
[5] Brofferio, S. and Buraczewski, D. (2015). On unbounded invariant measures of stochastic dynamical systems. Ann. Prob. 43, 1456–1492. Google Scholar
[6] Buraczewski, D. (2007). On invariant measures of stochastic recursions in a critical case. Ann. Appl. Prob. 17, 1245–1272. Google Scholar
[7] Buraczewski, D. and Iksanov, A. (2015). Functional limit theorems for divergent perpetuities in the contractive case. Electron. Commun. Prob. 20, 10. CrossRefGoogle Scholar
[8] Buraczewski, D., Damek, E. and Mikosch, T. (2016). Stochastic Models with Power-Law Tails. The Equation X = AX + B . Springer, Cham. Google Scholar
[9] Burton, R. M. and Rösler, U. (1995). An L 2 convergence theorem for random affine mappings. J. Appl. Prob. 32, 183–192. Google Scholar
[10] Denisov, D., Korshunov, D. and Wachtel, V. (2016). At the edge of criticality: Markov chains with asymptotically zero drift. Preprint. Available at https://arxiv.org/abs/1612.01592. Google Scholar
[11] Erickson, K. B. (1973). The strong law of large numbers when the mean is undefined. Trans. Amer. Math. Soc. 185, 371–381. Google Scholar
[12] Goldie, C. M. and Maller, R. A. (2000). Stability of perpetuities. Ann. Prob. 28, 1195–1218. Google Scholar
[13] Grincevičius, A. (1976). Limit theorems for products of random linear transformations on the line. Lith. Math. J. 15, 568–579. CrossRefGoogle Scholar
[14] Hitczenko, P. and Wesołowski, J. (2011). Renorming divergent perpetuities. Bernoulli 17, 880–894. Google Scholar
[15] Iksanov, A. (2016). Renewal Theory for Perturbed Random Walks and Similar Processes. Birkhäuser, Cham. CrossRefGoogle Scholar
[16] Iksanov, A., Pilipenko, A. and Samoilenko, I. (2017). Functional limit theorems for the maxima of perturbed random walk and divergent perpetuities in the M 1-topology. Extremes 20, 567–583. Google Scholar
[17] Kellerer, H. G. (1992). Ergodic behaviour of affine recursions I: Criteria for recurrence and transience. Tech. Rep., Universität München, Germany. Available at http://www.mathematik.uni-muenchen.de/~kellerer/. Google Scholar
[18] Kesten, H. and Maller, R. A. (1996). Two renewal theorems for general random walks tending to infinity. Prob. Theory Relat. Fields 106, 1–38. Google Scholar
[19] Pakes, A. G. (1983). Some properties of a random linear difference equation. Austral. J. Statist. 25, 345–357. CrossRefGoogle Scholar
[20] Peigné, M. and Woess, W. (2011). Stochastic dynamical systems with weak contractivity properties I. Strong and local contractivity. Colloq. Math. 125, 31–54. CrossRefGoogle Scholar
[21] Rachev, S. T. and Samorodnitsky, G. (1995). Limit laws for a stochastic process and random recursion arising in probabilistic modelling. Adv. Appl. Prob. 27, 185–202. Google Scholar
[22] Stromberg, K. R. (1981). Introduction to Classical Real Analysis. Wadsworth, Belmont, CA. Google Scholar
[23] Tong, J. C. (1994). Kummer's test gives characterizations for convergence or divergence of all positive series. Amer. Math. Monthly 101, 450–452. Google Scholar
[24] Zeevi, A. and Glynn, P. W. (2004). Recurrence properties of autoregressive processes with super-heavy-tailed innovations. J. Appl. Prob. 41, 639–653. Google Scholar
[25] Zerner, M. P. W. (2016). Recurrence and transience of contractive autoregressive processes and related Markov chains. Preprint. Available at https://arxiv.org/abs/1608.01394v2. Google Scholar