The Annals of Applied Probability

Asymptotic behavior for iterated functions of random variables

Deli Li and T. D. Rogers

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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.

Article information

Ann. Appl. Probab., Volume 9, Number 4 (1999), 1175-1201.

First available in Project Euclid: 21 August 2002

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Zentralblatt MATH identifier

Primary: 60F15: Strong theorems 60F99: None of the above, but in this section
Secondary: 58F11 62G30: Order statistics; empirical distribution functions

Asymptotic behavior hierarchical models law of large numbers order statistics weighted sums


Li, Deli; Rogers, T. D. Asymptotic behavior for iterated functions of random variables. Ann. Appl. Probab. 9 (1999), no. 4, 1175--1201. doi:10.1214/aoap/1029962869.

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