The Annals of Probability

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

Shihong Cheng

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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


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