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
Author(s)
Hashorva E., Macci C., Pacchiarotti B.
ISSN
1387-5841 (Print)
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
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Create date
03/05/2012 10:28
Last modification date
20/08/2019 14:59
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