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Ergodic theorems for Markov chains represented by iterated function systems

Stenflo O.

Bulletin of the Polish Academy of Sciences. Mathematics

2001

Vol. 49, no 1

27-43

We consider Markov chains represented in the form Xn+1 = f (Xn, In), where {In} is a sequence of independent, identically distributed (i.i.d.) random variables, and where f is a measurable function. Any Markov chain {Xn} on a Polish state space may be represented in this form i.e. can be considered as arising from an iterated function system (IFS). A distributional ergodic theorem, including rates of convergence in the Kantorovich distance is proved for Markov chains under the condition that an IFS representation is "stochastically contractive" and "stochastically bounded". We apply this result to prove our main theorem giving upper bounds for distances between invariant probability measures for iterated function systems. We also give some examples indicating how ergodic theorems for Markov chains may be proved by finding contractive IFS representations. These ideas are applied to some Markov chains arising from iterated function systems with place dependent probabilities.

Iterated function systems generated by strict contractions and place-dependent probabilities

Zaharopol R.

Bulletin of the Polish Academy of Sciences. Mathematics

2000

Vol. 48, no 4

429-438

Let (w[sub l], w[sub 2],...,w[sub k];p[sub 1],p[sub 2],...p[sub k]) be an iterated function system (IFS for short) with continuous place-dependent probabilities, defined on a metric space (X, d). Assume that every closed ball in X is compact. Our main result is that the IFS has an attractive probability measure whenever the following three conditions are satisfied: (1) w[sub i] : X --> X is a strict contraction for every i = 1,...,k. (2) sum[...]p[sub i](x)p[sub i](y) > 0 for every x, y [belongs to] X. (3) There exists p > 0 in R such that [...] for every x, y [belongs to] X and j = l, 2,...,k. Note that we do not require the p[sub i]'s to be even uniformly continuous. This research was motivated by a question of Barnsley, Demko, Elton, and Geronimo, [1, p. 373], concerning IFS which satisfy only condition (1). We construct a family C of IFS which we use to answer the question. Our main result allows us to distinguish IFS in C which possess attractive probabilities.