Asymptotic Behavior For Markovian Iterated Function Systems

Let (U, d) be a complete separable metric space and (F-n)(n >= 0) a sequence of random functions from U to U. Motivated by studying the stability property for Markovian dynamic models, in this paper, we assume that the random function (F-n)(n >= 0) is driven by a Markov chain X = {X-n, n >= 0}. Under some regularity conditions on the driving Markov chain and the mean contraction assumption, we show that the forward iterations M-n(u) = F-n omicron ... omicron F-1(u), n >= 0, converge weakly to a unique stationary distribution Pi for each u is an element of U, where. denotes composition of two maps. The associated backward iterations (M) over tilde (u)(n) = F-1 omicron ... omicron F-n(u) are almost surely convergent to a random variable (M) over tilde (infinity) which does not depend on u and has distribution Pi. Moreover, under suitable moment conditions, we provide estimates and rate of convergence for d((M) over tilde (infinity), (M) over tilde (u)(n)) and d(M-n(u), M-n(v)), u, v is an element of U. The results are applied to the examples that have been discussed in the literature, including random coefficient autoregression models and recurrent neural network. (C) 2021 Elsevier B.V. All rights reserved.