Couplings for irregular combinatorial assemblies

TL;DR

This paper explores coupling methods to demonstrate the smoothness of distributions arising from sums of non-negative integer random variables in irregular combinatorial assemblies, providing insights into their probabilistic structure.

Contribution

It introduces and compares two coupling techniques for proving distribution smoothness in irregular combinatorial structures, extending understanding beyond regular cases.

Findings

Coupling approaches effectively establish distribution smoothness.

Results contrast different coupling methods' effectiveness.

Insights into irregular combinatorial assembly distributions.

Abstract

When approximating the joint distribution of the component counts of a decomposable combinatorial structure that is `almost' in the logarithmic class, but nonetheless has irregular structure, it is useful to be able first to establish that the distribution of a certain sum of non-negative integer valued random variables is smooth. This distribution is not like the normal, and individual summands can contribute a non-trivial amount to the whole, so its smoothness is somewhat surprising. In this paper, we consider two coupling approaches to establishing the smoothness, and contrast the results that are obtained.