Powiadomienia systemowe
- Sesja wygasła!
Tytuł artykułu
Autorzy
Wybrane pełne teksty z tego czasopisma
Identyfikatory
Warianty tytułu
Języki publikacji
Abstrakty
We consider Markov chains arising from random iteration of functions Sθ : X → X, θ ϵ Θ, where X is a Polish space and Θ is an arbitrary set of indices. At x ϵ X, θ is sampled from a distribution ϑx on Θ, and the ϑx 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 Rd, giving the existence of exponentially attractive perpetuities with place dependent probabilities.
Czasopismo
Rocznik
Tom
Strony
119--137
Opis fizyczny
Bibliogr. 34 poz.
Twórcy
autor
- Faculty of Applied Mathematics, AGH University of Science and Technology, al. A. Mickiewicza 30, 30-059 Kraków, Poland
autor
- Institute of Mathematics, University of Silesia in Katowice, Bankowa 14, 40-007 Katowice, Poland
Bibliografia
- [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 Xn = AnXn−1 + Bn 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. Grincevičjus, 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. Ślęczka, The rate of convergence for iterated function systems, Studia Math. 205 (2011), 201-214.
- [29] M. Ślęczka, 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 systems, 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.
Uwagi
Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-e5f9970f-c67e-41fe-bdee-e315cb94a2f0