Translator Disclaimer
November 1999 Asymptotic behavior for iterated functions of random variables
Deli Li, T. D. Rogers
Ann. Appl. Probab. 9(4): 1175-1201 (November 1999). DOI: 10.1214/aoap/1029962869


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.


Download Citation

Deli Li. T. D. Rogers. "Asymptotic behavior for iterated functions of random variables." Ann. Appl. Probab. 9 (4) 1175 - 1201, November 1999.


Published: November 1999
First available in Project Euclid: 21 August 2002

zbMATH: 0963.60029
MathSciNet: MR1728559
Digital Object Identifier: 10.1214/aoap/1029962869

Primary: 60F15, 60F99
Secondary: 58F11, 62G30

Rights: Copyright © 1999 Institute of Mathematical Statistics


Vol.9 • No. 4 • November 1999
Back to Top