Let $\mathscr{D} \subseteq (-\infty, \infty)$ be closed domain and
set $\xi = \inf{x;x \epsilon \mathscr{D}}$. Let the sequence $\mathscr{X}^n =
{X_j^{(n)}; j \geq 1}, n \geq 1$ be associated with the sequence of measurable
iterated functions $f_n(x_1, x_2,\dots, x_{k_n}): \mathscr{D}^{k_n} \rightarrow
\mathscr{D} (k_n \geq 2), n \geq 1$ and some initial sequence
$\mathscr{X}^{(0)} = {X_j^{(0)}; j \geq 1}$ of stationary and
m-dependent random variables such that $P(X_1^{(0)} \epsilon
\mathscr{D}) = 1$ and $X_j^{(n)} = f_n(X_{(j-1)k_n+1}^{(n-1)},\dots,
X_{jk_n}^{(n-1)}), j \geq 1, n \geq 1$. This paper studies the asymptotic
behavior for the hierarchical sequence ${X_1^{(n)}; n \geq 0}$. We establish
general asymptotic results for such sequences under some surprisingly relaxed
conditions. Suppose that, for each $n \geq 1$, there exist $k_n$ non-negative
constants $\alpha_{n, i}, 1 \leq i \leq k_n$ such that $\Sigma_{i=1}^{k_n}
\alpha_{n, i} = 1$ and $f_n(x_1,\dots, x_{k_n}) \leq \Sigma_{i=1}^{k_n}
\alpha_{n, i}x_i, \forall(x_1,\dots, x_{k_n}) \epsilon \mathscr{D}^{k_n}$. If
$\Pi_{j=1}^n \max_{1\leqi\leqk_j \alpha_{j, i} \rightarrow 0$ as $n \rightarrow
\infty$ and $E(X_1^{(n)}) \downarrow \lambda$ as $n \rightarrow \infty$ and
$X_1^{(n)} \rightarrow_P \lambda$. We conclude with various examples, comments
and open questions and discuss further how our results can be applied to models
arising in mathematical physics.