Total Variation Asymptotics for Refined Poisson Process Approximations of Random Logarithmic Assemblies

Abstract

Assemblies are decomposable combinatorial objects characterized by a sequence m_i that counts the number of possible components of size i. Permutations on n elements, mappings from a set containing n elements into itself, 2-regular graphs on n vertices, and set partitions on a set of size n are all assemblies with natural decompositions. Logarithmic assemblies are characterized by constants θ > 0 and κ₀ > 0 such that m_iκⁱ₀/ (i−1)! → θ. Random mappings, permutations and 2-regular graphs are all logarithmic assemblies, but set partitions are not.

Given a logarithmic assembly, all representatives having total size n are chosen uniformly and a component counting process C(n) = (C₁(n), C₂(n), …, C_n(n)) is defined, where C_i(n) is the number of components of size i. Our results also apply to C(n) distributed as the Ewens sampling formula with parameter θ. Denote the component counting process up to size at most b by C_b(n) = (C₁(n), C₂(n), …, C_b(n)). It is natural to approximate C_b by Z_b = (Z₁, Z₂, …, Z_b), the b-dimensional process of independent Poisson variables Z_i for which the ith variable has expectation [ ]Z_i = m_iκⁱ₀ exp((1−θ)i/n)/i!. We find asymptotics for the total variation distance between C_b(n) and Z_b.