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May 2000 The number of components in a logarithmic combinatorial structure
Richard Arratia, A. D. Barbour, Simon Tavaré
Ann. Appl. Probab. 10(2): 331-361 (May 2000). DOI: 10.1214/aoap/1019487347


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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Richard Arratia. A. D. Barbour. Simon Tavaré. "The number of components in a logarithmic combinatorial structure." Ann. Appl. Probab. 10 (2) 331 - 361, May 2000.


Published: May 2000
First available in Project Euclid: 22 April 2002

MathSciNet: MR1768242
Digital Object Identifier: 10.1214/aoap/1019487347

Primary: 05A16 , 60C05 , 60F05

Keywords: component counts , Logarithmic combinatorial structures , Poisson approximation , total variation approximation

Rights: Copyright © 2000 Institute of Mathematical Statistics


Vol.10 • No. 2 • May 2000
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