# 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.

## 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.

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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics Physical Sciences > Statistics and Probability Social Sciences & Humanities > Statistics, Probability and Uncertainty Logarithmic combinatorial structures, component counts, total variation approximation, Poisson approximation English 2000 07 Apr 2010 12:39 26 Jun 2022 22:36 Institute of Mathematical Statistics 1050-5164 Hybrid https://doi.org/10.1214/aoap/1019487347