## The Annals of Probability

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

Shihong Cheng

#### 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.

#### Article information

Source
Ann. Probab., Volume 13, Number 4 (1985), 1326-1340.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176992816

Digital Object Identifier
doi:10.1214/aop/1176992816

Mathematical Reviews number (MathSciNet)
MR806229

Zentralblatt MATH identifier
0584.60032

JSTOR