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The number of components in a logarithmic combinatorial structure


Arratia, R; Barbour, A D; Tavaré, S (2000). The number of components in a logarithmic combinatorial structure. Annals of Applied Probability, 10(2):331-361.

Abstract

Under very mild conditions, we prove that the number of components in a decomposable logarithmic combinatorial structure has a distribution which is close to Poisson in total variation. The conditions are satisfied for all assemblies, multisets and selections in the logarithmic class.The error in the Poisson approximation is shown under marginally more restrictive conditions to be of exact order $O(1/\log n)$, by exhibiting the penultimate asymptotic approximation; similar results have previously been obtained by Hwang [20], under stronger assumptions.Our method is entirely probabilistic, and the conditions can readily be verified in practice.

Under very mild conditions, we prove that the number of components in a decomposable logarithmic combinatorial structure has a distribution which is close to Poisson in total variation. The conditions are satisfied for all assemblies, multisets and selections in the logarithmic class.The error in the Poisson approximation is shown under marginally more restrictive conditions to be of exact order $O(1/\log n)$, by exhibiting the penultimate asymptotic approximation; similar results have previously been obtained by Hwang [20], under stronger assumptions.Our method is entirely probabilistic, and the conditions can readily be verified in practice.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:Logarithmic combinatorial structures; component counts; total variation approximation; Poisson approximation
Language:English
Date:2000
Deposited On:07 Apr 2010 12:39
Last Modified:05 Apr 2016 13:25
Publisher:Institute of Mathematical Statistics
ISSN:1050-5164
Publisher DOI:https://doi.org/10.1214/aoap/1019487347
Permanent URL: https://doi.org/10.5167/uzh-22060

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