Hierarchical structures associated with order functions

Let k and m be two fixed positive integers with k ≥ 3 and 1 < m < k. Let f (x₁, x₂, ..., x_k) = x_{k : m} which is the mth smallest value of k real numbers x₁, x₂, ..., x_k. We call this f the mth order function of k real numbers x₁, x₂, ..., x_k. 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) = ∫₀^x (k ! / (m - 1) ! (k - m) !) t^{m - 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 X_{n, 1}, n ≥ 0 by X_{n + 1, j} = f (X_{n, 1 + (j - 1) k}, X_{n, 2 + (j - 1) k}, ..., X_{n, k + (j - 1) k}) with { X_{0, j}, j ≥ 1 } being independent random variables identically distributed as X₀. In this note it is shown that X_{n, 1} over(→, d) G (x) = ξ I_{[λ1, ∞)} (x) + (1 - ξ) I_{[λ2, ∞)} (x) and lim inf_{n → ∞} X_{n, 1} = λ₁ a.s. and lim sup_{n → ∞} X_{n, 1} = λ₂ a.s. It follows that lim_{n → ∞} X_{n, 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].