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Convergence Properties in Certain Occupancy Problems Including the Karlin-Rouault Law

Part of: Distribution theory - Probability Limit theorems Distribution theory Sampling theory, sample surveys

Published online by Cambridge University Press:  14 July 2016

Estáte V. Khmaladze
Estáte V. Khmaladze*
Affiliation:
Victoria University of Wellington
*
∗ Postal address: School of Mathematics, Statistics and Operations Research, Victoria University of Wellington, PO Box 600, Wellington, 2052, New Zealand. Email address: [email protected]
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Abstract

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Let x denote a vector of length q consisting of 0s and 1s. It can be interpreted as an ‘opinion’ comprised of a particular set of responses to a questionnaire consisting of q questions, each having {0, 1}-valued answers. Suppose that the questionnaire is answered by n individuals, thus providing n ‘opinions’. Probabilities of the answer 1 to each question can be, basically, arbitrary and different for different questions. Out of the 2q different opinions, what number, μn , would one expect to see in the sample? How many of these opinions, μn (k), will occur exactly k times? In this paper we give an asymptotic expression for μn / 2q and the limit for the ratios μn (k)/μn , when the number of questions q increases along with the sample size n so that n = λ2q , where λ is a constant. Let p(x ) denote the probability of opinion x . The key step in proving the asymptotic results as indicated is the asymptotic analysis of the joint behaviour of the intensities np(x ). For example, one of our results states that, under certain natural conditions, for any z > 0, ∑1 {np( x) > z} = d n z −u ,d n = o(2q ).

Keywords

Number of unique outcomessparse tablesKarlin-Rouault lawZipf's lawGood-Turing indexlarge deviationscontiguity

MSC classification

Secondary: 62D05: Sampling theory, sample surveys 62E20: Asymptotic distribution theory 60E05: Distributions 60F10: Large deviations
Type
Research Papers
Information
Journal of Applied Probability , Volume 48 , Issue 4 , December 2011 , pp. 1095 - 1113
Copyright
Copyright © Applied Probability Trust 2011 

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