Asymptotic behavior for iterated functions of random variables

Let D subset of or equal to (-infinity, infinity) be a dosed domain and set xi = inf{x; x is an element of D}. Let the sequence X-(n) = {X-j((n)); j greater than or equal to 1}, n greater than or equal to 1 be associated with the sequence of measurable iterated functions f (n)(x(1), x(2), ..., x(kn)): D-kn -> D (k(n) greater than or equal to 2), n greater than or equal to 1 and some initial sequence X-(0) = {X-j((0)); j greater than or equal to 1} of stationary and m-dependent random variables such that P(X-1((0)) is an element of D) = 1 and X-j((n)) = f(n)(X-(j-1)kn+1((n-1)),..., X-jkn((n-1))), j greater than or equal to 1, n greater than or equal to 1 . This paper studies the asymptotic behavior for the hierarchical sequence {X-1((n)); n greater than or equal to 0}. We establish general asymptotic results for such sequences under some surprisingly relaxed conditions. Suppose that, for each n greater than or equal to 1, there exist k(n) non-negative constants alpha(n, i), 1 less than or equal to i less than or equal to k(n) such that Sigma(i=1)(kn) alpha(n, i) = 1 and f(n)(x(1),..., x(kn)) less than or equal to Sigma(i = 1)(kn) alpha(n, i)x(i), For All(x(1),..., x(kn)) is an element of D-kn. If IIj = 1n max(1 less than or equal to i less than or equal to k j) alpha(j, i) -> 0 as n -> infinity and E(X-1((0)) boolean OR 0) < infinity, then, for some lambda is an element of D boolean OR {xi}, E(X-1((n))) down arrow lambda qw n -> infinity and X-1((n)) -> (P) lambda. We conclude with various examples, comments and open questions and discuss further how our results can be applied to models arising in mathematical physics.