Assemblies are labelled combinatorial objects that can be
decomposed into components. Examples of assemblies include set
partitions, permutations and random mappings. In addition, a
distribution from population genetics called the Ewens sampling
formula may be treated as an assembly. Each assembly has a size
n, and the sum of the sizes of the components sums to
n. When the uniform distribution is put on all assemblies of
size n, the process of component counts is equal in
distribution to a process of independent Poisson variables
Zi conditioned on the event that a
weighted sum of the independent variables is equal to n.
Logarithmic assemblies are assemblies characterized by some
θ > 0 for which
i[ ]Zi → θ.
Permutations and random mappings are logarithmic assemblies; set
partitions are not a logarithmic assembly. Suppose b =
b(n) is a sequence of positive integers for which
b/n → β ε
(0, 1]. For logarithmic assemblies, the total variation
distance db(n) between the laws
of the first b coordinates of the component counting process
and of the first b coordinates of the independent processes
converges to a constant H(β). An explicit
formula for H(β) is given for β
ε (0, 1] in terms of a limit process which depends only
on
the parameter θ. Also, it is shown that
db(n) → 0 if and only if
b/n → 0, generalizing results of Arratia,
Barbour and Tavaré for the Ewens sampling formula. Local limit
theorems for weighted sums of the Zi are
used to prove these results.
Central limit theorem for the prefix exchange distance under Ewens sampling formula
