On the asymptotic normality of the number of absent s -chains

On the asymptotic normality of the number of absent s -chains -- The asymptotic behaviour of the number of absent s-chains in a polynomial scheme is studied. We consider the case where the length of chains s is constant, s > 2, and the number of trials, the number of outcomes and their probabilities are related by the conditions determining an analogue of the central domain. Some asymptotic expressions for the mean and variance are given and the asymptotic normality is proved. 1. INTRODUCTION In this paper some topics concerning the asymptotic behaviour of the number of absent s-chains in a polynomial scheme are studied. We consider the case where the length of chains is constant, the number of trials and the number of outcomes of the scheme tend to infinity in such a way that 0 < < -- = < 2 < oo, and the probabilities of outcomes pi,...,PN vary in such a way that 0<NPk 2. In this case the statistic is a decomposable statistic on the frequencies of outcomes of the Markov chain generated by s-chains. It seems to us that the time is appropriate to extend the investigations of such statistics since in the case of the proper polynomial scheme, i.e., in the case http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Discrete Mathematics and Applications de Gruyter

On the asymptotic normality of the number of absent s -chains

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Publisher
de Gruyter
Copyright
Copyright © 2009 Walter de Gruyter
ISSN
0924-9265
eISSN
1569-3929
DOI
10.1515/dma.1993.3.1.89
Publisher site
See Article on Publisher Site

Abstract

-- The asymptotic behaviour of the number of absent s-chains in a polynomial scheme is studied. We consider the case where the length of chains s is constant, s > 2, and the number of trials, the number of outcomes and their probabilities are related by the conditions determining an analogue of the central domain. Some asymptotic expressions for the mean and variance are given and the asymptotic normality is proved. 1. INTRODUCTION In this paper some topics concerning the asymptotic behaviour of the number of absent s-chains in a polynomial scheme are studied. We consider the case where the length of chains is constant, the number of trials and the number of outcomes of the scheme tend to infinity in such a way that 0 < < -- = < 2 < oo, and the probabilities of outcomes pi,...,PN vary in such a way that 0<NPk 2. In this case the statistic is a decomposable statistic on the frequencies of outcomes of the Markov chain generated by s-chains. It seems to us that the time is appropriate to extend the investigations of such statistics since in the case of the proper polynomial scheme, i.e., in the case

Journal

Discrete Mathematics and Applicationsde Gruyter

Published: Jan 1, 1993

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