Abstract
Assemblies are the decomposable combinatorial constructions characterized by the exponential formula for generating functions: $\Sigma p(n)s^n/n! = \exp(\Sigma m_is^i/i!)$. Here $p(n)$ is the total number of constructions that can be formed from a set of size $n$, and $m_n$ is the number of these structures consisting of a single component. Examples of assemblies include permutations, graphs, 2-regular graphs, forests of rooted or unrooted trees, set partitions and mappings of a set into itself. If an assembly is chosen uniformly from all possibilities on a set of size $n$, the counts $C_i(n)$ of components of size $i$ are jointly distributed like independent nonidentically distributed Poisson variables $Z_i$ conditioned on the event $Z_1 + 2Z_2 + \cdots + nZ_n = n$. We consider assemblies for which the process of component-size counts has a nontrivial limit distribution, without renormalizing. These include permutations, mappings, forests of labelled trees and 2-regular graphs, but not graphs and not set partitions. For some of these assemblies, the distribution of the component sizes may be viewed as a perturbation of the Ewens sampling formula with parameter $\theta$. We consider $d_b(n)$, the total variation distance between $(Z_1, \ldots, Z_b)$ and $(C_1(n),\ldots,C_b(n))$, counting components of size at most $b$. If the generating function of an assembly satisfies a mild analytic condition, we can determine the decay rate of $d_b(n)$. In particular, for $b = b(n) = o(n/\log n)$ and $n \rightarrow \infty, d_b(n) = o(b/n)$ if $\theta = 1$ and $d_b(n) \sim c(b)b/n$ if $\theta \neq 1$. The constant $c(b)$ is given explicitly in terms of the $m_i: c(b) = |1 - \theta|\mathbb{E}|T_{0b} - \mathbb{E}T_{0b}|/(2b)$, where $T_{0b} = Z_1 + 2Z_2 + \cdots + bZ_b$. Finally, we show that for $\theta \neq 1$ there is a constant $c_\theta$ such that $c(b) \sim c_\theta b$ as $b \rightarrow \infty$. Our results are proved using coupling, large deviation bounds and singularity analysis of generating functions.
Citation
Richard Arratia. Dudley Stark. Simon Tavare. "Total Variation Asymptotics for Poisson Process Approximations of Logarithmic Combinatorial Assemblies." Ann. Probab. 23 (3) 1347 - 1388, July, 1995. https://doi.org/10.1214/aop/1176988188
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