Asymptotic Behavior for Iterated Functions of Random Variables

Let$\mathscr D \subseteq (-\infty, \infty)$be a closed domain and set$\xi = \inf\{x;x \in \mathscr D\}$. Let the sequence X(n)= {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 knnon-negative constants αn,i, 1 ≤ i ≤ knsuch that Σkn i=1αn,i= 1 and fn(x1,..., xkn ) ≤ Σkn i=1αn,ixi, ∀(x1,..., xkn ) ∈ Dkn . If Πn j=1max1≤ i≤ kj αj,i→ 0 as n → ∞ and $E(X^{(0)}_1 \vee 0) < \infty$, then, for some$\lambda \in \mathscr D \cup \{\xi\}, E(X^{(n)}_1) \downarrow \lambda$as n → ∞ and X(n) 1→Pλ. We conclude with various examples, comments and open questions and discuss further how our results can be applied to models arising in mathematical physics.