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Contents of PMS, Vol. 40, Fasc. 1,
pages 119 - 137
DOI: 10.37190/0208-4147.40.1.8
Published online 20.4.2020
 

Random iteration with place dependent probabilities

R. Kapica
Ślęczka

Abstract: We consider Markov chains arising from random iteration of functions \(S_{\theta}:X\to X\), \(\theta \in \Theta\), where \(X\) is a Polish space and \(\Theta\) is an arbitrary set of indices. At \(x\in X\), \(\theta\) is sampled from a distribution \(\vartheta_x\) on \(\Theta\), and the \(\vartheta_x\) are different for different \(x\). Exponential convergence to a unique invariant measure is proved. This result is applied to the case of random affine transformations on \({\mathbb R}^d\), giving the existence of exponentially attractive perpetuities with place dependent probabilities.

2010 AMS Mathematics Subject Classification: Primary 60J05; Secondary 37A25.

Keywords and phrases: random iteration of functions, exponential convergence, invariant measure, perpetuities.

References


[1] G. Alsmeyer, A. Iksanov and U. Rösler, On distributional properties of perpetuities, J. Theoret. Probab. 22 (2009), 666-682.

[2] M. F. Barnsley, S. G. Demko, J. H. Elton and J. S. Geronimo, Invariant measures for Markov processes arising from iterated function systems with place dependent probabilities, Ann. Inst. H. Poincaré 24 (1988), 367-394.

[3] K. Bartkiewicz, A. Jakubowski, T. Mikosch and O. Wintenberger, Stable limits for sums of dependent infinite variance random variables, Probab. Theory Related Fields 150 (2011), 337-372.

[4] S. Brofferio, D. Buraczewski and E. Damek, On the invariant measure of the random difference equation \(X_n=A_nX_{n-1}+B_n\) in the critical case, Ann. Inst. H. Poincaré Probab. Statist. 48 (2012), 377-395.

[5] D. Buraczewski, E. Damek and Y. Guivarc'h, Convergence to stable laws for a class of multidimensional stochastic recursions, Probab. Theory Related Fields 148 (2010), 333-402.

[6] P. Diaconis and D. Freedman, Iterated random functions, SIAM Rev. 41 (1999), 45-76.

[7] P. Embrechts, C. Klüppelberg and T. Mikosch, Modeling Extremal Events for Insurance and Finance, Appl. Math. 33, Springer, New York, 1997.

[8] S. Ethier and T. Kurtz, Markov Processes, Wiley, New York, 1986.

[9] C. M. Goldie and R. A. Maller, Stability of perpetuities, Ann. Probab. 28 (2000), 1195-1218.

[10] A. K. Grincevicjus, On the continuity of the distribution of a sum of dependent variables connected with independent walks on lines, Theory Probab. Appl. 19 (1974), 163-168.

[11] M. Hairer, Exponential mixing properties of stochastic PDEs through asymptotic coupling, Probab. Theory Related Fields 124 (2002), 345-380.

[12] M. Hairer and J. Mattingly, Spectral gaps in Wasserstein distances and the 2D stochastic Navier-Stokes equations, Ann. Probab. 36 (2008), 2050-2091.

[13] M. Hairer, J. Mattingly and M. Scheutzow, Asymptotic coupling and a general form of Harris' theorem with applications to stochastic delay equations, Probab. Theory Related Fields 149 (2011), 223–259.

[14] S. Hille, K. Horbacz, T. Szarek and H. Wojewódka, Limit theorems for some Markov chains, J. Math. Anal. Appl. 443 (2016), 385-408.

[15] K. Horbacz and T. Szarek, Continuous iterated function systems on Polish spaces, Bull. Polish Acad. Sci. Math. 49 (2001), 191-202.

[16] K. Horbacz, The central limit theorem for random dynamical systems, J. Statist. Phys. 164 (2016), 1261–1291.

[17] H. Kesten, Random difference equations and renewal theory for products of random matrices, Acta Math. 131 (1973), 207–248.

[18] T. Komorowski and A. Walczuk, Central limit theorem for Markov processes with spectral gap in the Wasserstein metric, Stoch. Process. Appl. 122 (2012), 2155-2184.

[19] S. Kuksin and A. Shirikyan, A coupling approach to randomly forced nonlinear PDE’s. I, Comm. Math. Phys. 221 (2001), 351-366.

[20] S. Kuksin, A. Piatnitsky and A. Shirikyan, A coupling approach to randomly forced nonlinear PDE’s. II, Comm. Math. Phys. 230 (2002), 81-85.

[21] T. Lindvall, Lectures on the Coupling Method, Wiley, New York, 1992.

[22] S. Łojasiewicz, An Introduction to the Theory of Real Function, Wiley, Chichester, 1998.

[23] S. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, 2nd ed., Cambridge Univ. Press, Cambridge, 2009.

[24] M. Mirek, Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps, Probab. Theory Related Fields 151 (2011), 705-734.

[25] A. Öberg, Approximation of invariant measures for random iterations, Rocky Mountain J. Math. 36 (2006), 273-301.

[26] C. Odasso, Exponential mixing for stochastic PDEs: the non-additive case, Probab. Theory Related Fields 140 (2008), 41-82.

[27] A. Shirikyan, A version of the law of large numbers and applications, in: Probabilistic Methods in Fluids (Swansea, 2002), World Sci., 2003, 263-271.

[28] M. Śęczka, The rate of convergence for iterated function systems, Studia Math. 205 (2011), 201-214.

[29] M. Śęczka, Exponential convergence for Markov systems, Ann. Math. Silesianae 29 (2015), 139-149.

[30] Ö. Stenflo, A note on a theorem of Karlin, Statist. Probab. Lett. 54 (2001), 183-187.

[31] T. Szarek, Invariant measures for nonexpansive Markov operators on Polish spaces, Dissertationes Math. 415 (2003), 62 pp.

[32] W. Vervaat, On a stochastic difference equation and a representation of non-negative infinitely divisible random variables, Adv. Appl. Probab. 11 (1979), 750-783.

[33] I. Werner, Contractive Markov sysems, J. London Math. Soc. 71 (2005), 236-258.

[34] L. Xu, Exponential mixing of 2D SDEs forced by degenerate Levy noises, J. Evolutionary Equations 14 (2014), 249-272.

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