## The Annals of Probability

- Ann. Probab.
- Volume 10, Number 3 (1982), 653-662.

### On Limiting Distributions of Intermediate Order Statistics from Stationary Sequences

Vernon Watts, Holger Rootzen, and M. R. Leadbetter

#### 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 $n \rightarrow \infty$.

#### Article information

**Source**

Ann. Probab., Volume 10, Number 3 (1982), 653-662.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176993774

**Digital Object Identifier**

doi:10.1214/aop/1176993774

**Mathematical Reviews number (MathSciNet)**

MR659535

**Zentralblatt MATH identifier**

0487.62015

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

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

**Keywords**

Order statistics stationary processes ranks intermediate ranks

#### Citation

Watts, Vernon; Rootzen, Holger; Leadbetter, M. R. On Limiting Distributions of Intermediate Order Statistics from Stationary Sequences. Ann. Probab. 10 (1982), no. 3, 653--662. doi:10.1214/aop/1176993774. https://projecteuclid.org/euclid.aop/1176993774