This article is cited in 4 papers

Asymptotic distributions of multivariate intermediate order statistics

S. Cheng^a, L. de Haan^b, J. Yang<sup>a

a</sup> Peking University, China
^b Erasmus University Rotterdam, The Netherlands

Abstract: Let $\{X_n=(X_n^{(1)},\ldots,X_n^{(d)}),n\ge 1\}$ be independent identically distributed random vectors with a common nondegenerate distribution function and for each $n\ge 1$ and each $k=1,\ldots,d$, denote $X_{1;n}^{(k)}\le\cdots\le X_{n;n}^{(k)}$ as the order statistics of $X_1^{(k)},\ldots,X_n^{(k)}$. Suppose that ranks $r_n=(r_n^{(1)},\ldots,r_n^{(d)})$ satisfy $r_n^{(k)} \to\infty$ nondecreasingly, $r_n^{(k)}/n\to 0$ and $r_n^{(k)}/\sum_{l=1}^d r_n^{(l)}\to m^{(k)}>0$ for each $k=1,\ldots,d$ and let $X_{r_n;n}= (X_{r_n^{(1)};n}^{(1)},\ldots,X_{r_n^{(d)};n}^{(d)})$. This paper is to find out the class of limiting distributions of $\{X_{r_n;n}\}$ after suitable normalizing and centering, and give necessary and sufficient conditions of weak convergence.

Keywords: multivariate intermediate order statistics, asymptotic distributions.

Received: 29.12.1994

Language: English

DOI: 10.4213/tvp3205

 English version: Theory of Probability and its Applications, 1997, 41:4, 646–656

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