Approximation by the Dickman distribution and quasi-logarithmic combinatorial structures

TL;DR

This paper introduces a refined coupling method to approximate the component spectrum of quasi-logarithmic combinatorial structures using the Dickman distribution, enabling precise asymptotic analysis of their component sizes.

Contribution

It develops a blocking construction to improve coupling rates, facilitating accurate asymptotic approximations for the component spectrum of these structures.

Findings

Derived distributional limit theorems for largest component size

Established asymptotic behavior of small component counts

Enhanced coupling techniques for sums of independent variables

Abstract

Quasi-logarithmic combinatorial structures are a class of decomposable combinatorial structures which extend the logarithmic class considered by Arratia, Barbour and Tavar\'{e} (2003). In order to obtain asymptotic approximations to their component spectrum, it is necessary first to establish an approximation to the sum of an associated sequence of independent random variables in terms of the Dickman distribution. This in turn requires an argument that refines the Mineka coupling by incorporating a blocking construction, leading to exponentially sharper coupling rates for the sums in question. Applications include distributional limit theorems for the size of the largest component and for the vector of counts of the small components in a quasi-logarithmic combinatorial structure.