## The Annals of Probability

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

### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

Secondary: 60G10: Stationary processes 60G15: Gaussian processes

**Keywords**

Order statistics stationary sequences limiting distributions variable rank sequences

#### Citation

Cheng, Shihong. On Limiting Distributions of Order Statistics with Variable Ranks from Stationary Sequences. Ann. Probab. 13 (1985), no. 4, 1326--1340. doi:10.1214/aop/1176992816. https://projecteuclid.org/euclid.aop/1176992816