UNIVERSITY
OF WROCŁAW
 
Main Page
Contents
Online First
General Information
Instructions for authors


VOLUMES
43.1 42.2 42.1 41.2 41.1 40.2 40.1
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
 
 
WROCŁAW 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 (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.

Download:    Abstract    Full text   Abstract + References