UNIVERSITY OF WROC£AW

Main Page
Contents of previous volumes
Forthcoming papers
General Information
Instructions for authors

VOLUMES
39.2 39.1 38.2 38.1 37.2 37.1 36.2
36.1 35.2 35.1 34.2 34.1 33.2 33.1
32.2 32.1 31.2 31.1 30.2 30.1 29.2
29.1 28.2 28.1 27.2 27.1 26.2 26.1
25.2 25.1 24.2 24.1 23.2 23.1 22.2
22.1 21.2 21.1 20.2 20.1 19.2 19.1
18.2 18.1 17.2 17.1 16.2 16.1 15
14.2 14.1 13.2 13.1 12.2 12.1 11.2
11.1 10.2 10.1 9.2 9.1 8 7.2
7.1 6.2 6.1 5.2 5.1 4.2 4.1
3.2 3.1 2.2 2.1 1.2 1.1 .imap
 WROC£AW UNIVERSITY OF SCIENCE AND TECHNOLOGY

 Contents of PMS, Vol. 29, Fasc. 2, pages 321 - 336 ASYMPTOTIC BEHAVIOR OF ULTIMATELY CONTRACTIVE ITERATED RANDOM LIPSCHITZ FUNCTIONS Gerold Alsmeyer Gerd Hölker Abstract: Let be a random sequence of i.i.d. global Lipschitz functions on a complete separable metric space with Lipschitz constants For , denote by and the associated sequences of forward and backward iterations, respectively. If (mean contraction) and is finite for some , then it is known (see [9]) that, for each , the Markov chain converges weakly to its unique stationary distribution , while is a.s. convergent to a random variable which does not depend on and has distribution . In [2], renewal theoretic methods have been successfully employed to provide convergence rate results for , which then also lead to corresponding assertions for via for all and , where means equality in law. Here our purpose is to demonstrate how these methods are extended to the more general situation where only ultimate contraction, i.e. an a.s. negative Lyapunov exponent is assumed (here denotes the Lipschitz constant of ). This not only leads to an extension of the results from [2] but in fact also to improvements of the obtained convergence rate. 2000 AMS Mathematics Subject Classification: Primary: 60J05; Secondary: 60K05, 60G17. Keywords and phrases: Random Lipschitz function, ultimately contractive, forward iterations, backward iterations, stationary distribution, Prokhorov metric, level ladder epochs, Lyapunov exponent. References [1]   G. Alsmeyer, On the Harris recurrence of iterated random Lipschitz functions and related convergence rate results, J. Theoret. Probab. 16 (2003), pp. 217-247. [2]   G. Alsmeyer and C. D. Fuh, Limit theorems for iterated random functions by regenerative methods, Stochastic Process. Appl. 96 (2001), pp. 123-142. Corrigendum: 97 (2002), pp. 341-345. [3]   L. Arnold and H. Crauel, Iterated function systems and multiplicative ergodic theory, in: Diffusion Processes and Related Problems in Analysis 2, M. Pinsky and V. Wihstutz (Eds.), Birkhäuser, Boston 1992, pp. 283-305. [4]   M. Babillot, P. Bougerol and L. Elie, The random difference equation in the critical case, Ann. Probab. 25 (1997), pp. 478-493. [5]   M. F. Barnsley and J. H. Elton, A new class of Markov processes for image encoding, Adv. in Appl. Probab. 20 (1988), pp. 14-32. [6]   M. Benda, A central limit theorem for contractive dynamical systems, J. Appl. Probab. 35 (1998), pp. 200-205. [7]   Y. S. Chow and T. L. Lai, Some one-sided theorems on the tail distribution of sample sums with applications to the last time and largest excess of boundary crossings, Trans. Amer. Math. Soc. 208 (1975), pp. 51-72. [8]   P. Diaconis and D. Freedman, Iterated random functions, SIAM Rev. 41 (1999), pp. 45-76. [9]   J. H. Elton, A multiplicative ergodic theorem for Lipschitz maps, Stochatic  Process. Appl. 34 (1990), pp. 39-47. [10]   A. Gut, Stopped Random Walks: Limit Theorems and Applications, Springer, New York 1988. [11]   J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), pp. 713-747. [12]   G. Letac, A contraction principle for certain Markov chains and its applications, Contemp. Math. 50 (1986), pp. 263-273. [13]   D. Silvestrov and Ö. Stenflo, Ergodic theorems for iterated function systems controlled by regenerative sequences, J. Theoret. Probab. 11 (1998), pp. 589-608. [14]   D. Steinsaltz, Locally contractive iterated function systems, Ann. Probab. 27 (1999), pp. 1952-1979.