Abstract
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.
Citation
Deli Li. T. D. Rogers. "Asymptotic behavior for iterated functions of random variables." Ann. Appl. Probab. 9 (4) 1175 - 1201, November 1999. https://doi.org/10.1214/aoap/1029962869
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