On Limiting Distributions of Order Statistics with Variable Ranks from Stationary Sequences

Abstract

Let $\{\xi_n\}$ be a stationary sequence and $\xi^{(n)}_1 \leq \cdots \leq \xi^{(n)}_n$ be the order statistics of $\xi_1,\cdots, \xi_n$. In this paper the limiting distribution of $\{\xi^{(n)}_{k_n}\}$, where $\{k_n\}$ satisfies $\min(k_n, n - k_n) \rightarrow \infty$, is determined under appropriate conditions. Further results for some special $\{k_n\}$ that satisfy $k_n/n \rightarrow \lambda \in \lbrack 0, 1\rbrack$ are also obtained. These results are applied to discussing the limiting distributions of corresponding order statistics from $m$-dependent stationary sequences and stationary normal sequences.