Asymptotic behavior for Markovian iterated function systems

Let (U,d) be a complete separable metric space and (Fn)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 (Fn)n≥0 is driven by a Markov chain X={Xn,n≥0}. Under some regularity conditions on the driving Markov chain and the mean contraction assumption, we show that the forward iterations Mnu=Fn∘⋯∘F1(u), n≥0, converge weakly to a unique stationary distribution Π for each u∈U, where ∘ denotes composition of two maps. The associated backward iterations M̃nu=F1∘⋯∘Fn(u) are almost surely convergent to a random variable M̃∞ which does not depend on u and has distribution Π. Moreover, under suitable moment conditions, we provide estimates and rate of convergence for d(M̃∞,M̃nu) and d(Mnu,Mnv), u,v∈U. The results are applied to the examples that have been discussed in the literature, including random coefficient autoregression models and recurrent neural network.