On Limiting Distributions of Intermediate Order Statistics from Stationary Sequences

Abstract

Let $X_1, X_2, \cdots$, be a sequence of random variables and write $X^{(n)}_k$ for the $k$th largest among $X_1, X_2, \cdots, X_n$. If $\{k_n\}$ is a sequence of integers such that $k_n \rightarrow \infty, k_n/n \rightarrow 0$, the sequence $\{X^{(n)}_{k_n}\}$ is referred to as the sequence of intermediate order statistics corresponding to the intermediate rank sequence $\{k_n\}$. The possible limiting distributions for $X^{(n)}_{k_n}$ have been characterized (under mild restrictions) by various authors when the random variables $X_1, X_2, \cdots$ are independent and identically distributed. In this paper we consider the case when the $\{X_n\}$ form a stationary sequence and obtain a natural dependence restriction under which the "classical" limits still apply. It is shown in particular that the general dependence restriction applies to normal sequences when the covariance sequence $\{r_n\}$ converges to zero as fast as an appropriate power $n^{-\rho}$ as $n \rightarrow \infty$.