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

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

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: estate.khmaladze@msor.vuw.ac.nz
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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 2 q different opinions, what number, μ n , would one expect to see in the sample? How many of these opinions, μ n (k), will occur exactly k times? In this paper we give an asymptotic expression for μ n / 2 q and the limit for the ratios μ n (k)/μ n , when the number of questions q increases along with the sample size n so that n = λ2 q , 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(2 q ).

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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