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.