Asymptotic behavior for iterated functions of random variables

Let D.(..,.) be closed domain and set .=infx;x.D. Let the sequence Xn=X(n)j;j.1,n.1 be associated with the sequence of measurable iterated functions fn(x1,x2,.,xkn):Dkn.D(kn.2),n.1 and some initial sequence X(0)=X(0)j;j.1 of stationary and m-dependent random variables such that P(X(0)1.D)=1 and X(n)j=fn(X(n.1)(j.1)kn+1,.,X(n.1)jkn),j.1,n.1. This paper studies the asymptotic behavior for the hierarchical sequence X(n)1;n.0. We establish general asymptotic results for such sequences under some surprisingly relaxed conditions. Suppose that, for each n.1, there exist kn non-negative constants .n,i,1.i.kn such that .kni=1.n,i=1 and fn(x1,.,xkn)..kni=1.n,ixi,.(x1,.,xkn).Dkn. If \Pi_{j=1}^n \max_{1\leqi\leqk_j \alpha_{j, i} \rightarrow 0 as n.. and E(X(n)1).. as n.. and X(n)1.P.. We conclude with various examples, comments and open questions and discuss further how our results can be applied to models arising in mathematical physics.