Asymptotic ( r− 1)-dependent representation for rth order statistic from a stationary sequence

Abstract

Let X 1, X 2,… be a stationary sequence of random variables. Denote by M ( k) n the kth largest value of X 1, X 2, …, X n . We find necessary and sufficient conditions for the existence of an ( r− 1)-dependent stationary sequence X̃ 1, X̃ 2, …(determined by a distribution function G and numbers β 1, β 2,…, β r ⩾ 0, Σ r q=1 β q =1), such that for each k, 1⩽k⩽r, sup χ∈ R 1|P (M (k) n⩽x)minus;P( M ̃ (k) n⩽x)|→0 as n→+∞, where M ̃ (1) n, M ̃ (2) n, …, M ̃ (r) n are order statistics of X ̃ 1, X ̃ 2, …, X ̃ n . If such asymptotic ( G, β 1, β 2, …, β r )-representation exists, then for each k, 1⩽ k⩽ r, there are numbers 0⩽ γ k, j ⩽1, j=1, 2, …, k−1, satisfying sup χ∈ R 1|P(M (k) n⩽x)− P(M (l) n⩽x)( 1 + Σ k−1 j=1 ((-log P(M (1) n⩽ x)) j j! )·γ k,j)|→0 as n→∞. This corresponds to limit theorems for M ( q) n obtained by Hsing (1988). Convergence of all order statistics is also discussed.