The Number of Components in a Logarithmic Combinatorial Structure

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. The Annals of Applied Probability 2000, Vol. 10, No. 2, 331–361 THE NUMBER OF COMPONENTS IN A LOGARITHMIC COMBINATORIAL STRUCTURE By Richard Arratia,1 A. D. Barbour2 and Simon Tavaré1 University of Southern California, Universität Zürich and University of Southern California 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.