# Large deviations for proportions of observations which fall in random sets determined by order statistics

## Details

Serval ID

serval:BIB_974A2C159C44

Type

**Article**: article from journal or magazin.

Collection

Publications

Institution

Title

Large deviations for proportions of observations which fall in random sets determined by order statistics

Journal

Methodology and Computing in Applied Probability

ISSN

1387-5841 (Print)

1573-7713 (Electronic)

1573-7713 (Electronic)

Publication state

Published

Issued date

2013

Peer-reviewed

Oui

Volume

15

Number

4

Pages

875-896

Language

english

Abstract

Let {X (n) :n a parts per thousand yenaEuro parts per thousand 1} be independent random variables with common distribution function F and consider , where h aaEuro parts per thousand{1,...,n}, X (1:k) a parts per thousand currency signaEuro parts per thousand a <-aEuro parts per thousand a parts per thousand currency signaEuro parts per thousand X (k:k) are the order statistics of the sample X (1),...,X (k) and D is some suitable Borel set of the real line. In this paper we prove that, if F is continuous and strictly increasing in the essential support of the distribution and if for some lambda aaEuro parts per thousand[0,1], then satisfies the large deviation principle. As a by product we derive the large deviation principle for order statistics . We also present results for the special case of Bernoulli distributed random variables with mean p aaEuro parts per thousand(0,1), and we see that the large deviation principle holds only for p a parts per thousand yenaEuro parts per thousand 1/2. We discuss further almost sure convergence of and some related quantities.

Keywords

Almost sure convergence, Bernoulli law, Near maximum, Point process, Relative entropy

Web of science

Create date

03/05/2012 10:28

Last modification date

20/08/2019 14:59