# Hierarchical structures associated with order functions

Deli Li, Yongcheng Qi

Research output: Contribution to journalArticlepeer-review

## Abstract

Let k and m be two fixed positive integers with k ≥ 3 and 1 < m < k. Let f (x1, x2, ..., xk) = xk : m which is the mth smallest value of k real numbers x1, x2, ..., xk. We call this f the mth order function of k real numbers x1, x2, ..., xk. Let X0 be a nonconstant random variable with cumulative distribution F (x) = P (X0 ≤ x), x ∈ (- ∞, ∞). Let ξ be the unique solution in (0, 1) to the equation B (x) = x where B (x) = ∫0x (k ! / (m - 1) ! (k - m) !) tm - 1 (1 - t)k - m d t, x ∈ [0, 1], which is the (cumulative) Beta distribution function with parameters m and k - m + 1. Write λ1 = sup { x ; F (x) < ξ } and λ2 = inf { x ; F (x) > ξ }. Define the hierarchical sequence of random variables Xn, 1, n ≥ 0 by Xn + 1, j = f (Xn, 1 + (j - 1) k, Xn, 2 + (j - 1) k, ..., Xn, k + (j - 1) k) with { X0, j, j ≥ 1 } being independent random variables identically distributed as X0. In this note it is shown that Xn, 1 over(→, d) G (x) = ξ I[λ1, ∞) (x) + (1 - ξ) I[λ2, ∞) (x) and lim infn → ∞ Xn, 1 = λ1 a.s. and lim supn → ∞ Xn, 1 = λ2 a.s. It follows that limn → ∞ Xn, 1 = λ a.s. if and only if λ1 = λ2 = λ. This result generalizes and improves Propositions 4.3 and 4.4 of Li and Rogers [1999. Asymptotic behavior for iterated functions of random variables. Ann. Appl. Probab. 9, 1175-1201].

Original language English (US) 525-529 5 Statistics and Probability Letters 77 5 https://doi.org/10.1016/j.spl.2006.08.020 Published - Mar 1 2007

### Bibliographical note

Funding Information:
The research of Deli Li was supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada, and the research of Yongcheng Qi was supported in part by NSF Grant DMS-0604176.

## Keywords

• Asymptotic behavior
• Hierarchical structures
• Order functions

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