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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 (F )
  n n>0  be a random sequence of i.i.d. global Lipschitz functions on a complete separable metric space (X,d) with Lipschitz constants L ,L ,...
 1  2 For n > 0 , denote by M x = F o ...o F (x)
  n    n       1 and M^x = F o ...o F (x)
  n    1       n the associated sequences of forward and backward iterations, respectively. If E log+ L  < 0
       1 (mean contraction) and E log+ d(F (x ),x )
         1  0  0 is finite for some x   (-  X
 0  , then it is known (see [9]) that, for each x  (-  X  , the Markov chain M x
  n  converges weakly to its unique stationary distribution p , while M^x
  n  is a.s. convergent to a random variable ^M
  oo  which does not depend on x and has distribution p . In [2], renewal theoretic methods have been successfully employed to provide convergence rate results for M^x
  n  , which then also lead to corresponding assertions for   x
M n  via   x d  x
M n= M^n  for all n and x , where d
= 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          -1
limn --> oo  n  log l(Fn o ...o F1) is assumed (here l(F) denotes the Lipschitz constant of F ). 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 g ladder epochs, Lyapunov exponent.

References

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[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.

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