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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

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.

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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

Information

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

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

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

Rights: Copyright © 1999 Institute of Mathematical Statistics

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Vol.9 • No. 4 • November 1999
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