15 research outputs found
Regenerative partition structures
We consider Kingman's partition structures which are regenerative with
respect to a general operation of random deletion of some part. Prototypes of
this class are the Ewens partition structures which Kingman characterised by
regeneration after deletion of a part chosen by size-biased sampling. We
associate each regenerative partition structure with a corresponding
regenerative composition structure, which (as we showed in a previous paper)
can be associated in turn with a regenerative random subset of the positive
halfline, that is the closed range of a subordinator. A general regenerative
partition structure is thus represented in terms of the Laplace exponent of an
associated subordinator. We also analyse deletion properties characteristic of
the two-parameter family of partition structures
Moments of convex distribution functions and completely alternating sequences
We solve the moment problem for convex distribution functions on in
terms of completely alternating sequences. This complements a recent solution
of this problem by Diaconis and Freedman, and relates this work to the
L\'{e}vy-Khintchine formula for the Laplace transform of a subordinator, and to
regenerative composition structures.Comment: Published in at http://dx.doi.org/10.1214/193940307000000374 the IMS
Collections (http://www.imstat.org/publications/imscollections.htm) by the
Institute of Mathematical Statistics (http://www.imstat.org
Constrained exchangeable partitions
For a class of random partitions of an infinite set a de Finetti-type
representation is derived, and in one special case a central limit theorem for
the number of blocks is shown
A reversible allelic partition process and Pitman sampling formula
We introduce a continuous-time Markov chain describing dynamic allelic
partitions which extends the branching process construction of the Pitman
sampling formula in Pitman (2006) and the birth-and-death process with
immigration studied in Karlin and McGregor (1967), in turn related to the
celebrated Ewens sampling formula. A biological basis for the scheme is
provided in terms of a population of individuals grouped into families, that
evolves according to a sequence of births, deaths and immigrations. We
investigate the asymptotic behaviour of the chain and show that, as opposed to
the birth-and-death process with immigration, this construction maintains in
the temporal limit the mutual dependence among the multiplicities. When the
death rate exceeds the birth rate, the system is shown to have reversible
distribution identified as a mixture of Pitman sampling formulae, with negative
binomial mixing distribution on the population size. The population therefore
converges to a stationary random configuration, characterised by a finite
number of families and individuals.Comment: 17 pages, to appear in ALEA , Latin American Journal of Probability
and Mathematical Statistic
Characterizations of exchangeable partitions and random discrete distributions by deletion properties
We prove a long-standing conjecture which characterises the Ewens-Pitman
two-parameter family of exchangeable random partitions, plus a short list of
limit and exceptional cases, by the following property: for each , if one of individuals is chosen uniformly at random, independently
of the random partition of these individuals into various types, and
all individuals of the same type as the chosen individual are deleted, then for
each , given that individuals remain, these individuals are
partitioned according to for some sequence of random partitions
that does not depend on . An analogous result characterizes the
associated Poisson-Dirichlet family of random discrete distributions by an
independence property related to random deletion of a frequency chosen by a
size-biased pick. We also survey the regenerative properties of members of the
two-parameter family, and settle a question regarding the explicit arrangement
of intervals with lengths given by the terms of the Poisson-Dirichlet random
sequence into the interval partition induced by the range of a neutral-to-the
right process.Comment: 29 page
Asymptotic laws for compositions derived from transformed subordinators
A random composition of appears when the points of a random closed set
are used to separate into blocks
points sampled from the uniform distribution. We study the number of parts
of this composition and other related functionals under the assumption
that , where is a
subordinator and is a diffeomorphism. We derive the
asymptotics of when the L\'{e}vy measure of the subordinator is regularly
varying at 0 with positive index. Specializing to the case of exponential
function , we establish a connection between the asymptotics
of and the exponential functional of the subordinator.Comment: Published at http://dx.doi.org/10.1214/009117905000000639 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Asymptotics of the allele frequency spectrum associated with the Bolthausen-Sznitman coalescent
We work in the context of the infinitely many alleles model. The allelic
partition associated with a coalescent process started from n individuals is
obtained by placing mutations along the skeleton of the coalescent tree; for
each individual, we trace back to the most recent mutation affecting it and
group together individuals whose most recent mutations are the same. The number
of blocks of each of the different possible sizes in this partition is the
allele frequency spectrum. The celebrated Ewens sampling formula gives precise
probabilities for the allele frequency spectrum associated with Kingman's
coalescent. This (and the degenerate star-shaped coalescent) are the only
Lambda coalescents for which explicit probabilities are known, although they
are known to satisfy a recursion due to Moehle. Recently, Berestycki,
Berestycki and Schweinsberg have proved asymptotic results for the allele
frequency spectra of the Beta(2-alpha,alpha) coalescents with alpha in (1,2).
In this paper, we prove full asymptotics for the case of the
Bolthausen-Sznitman coalescent.Comment: 26 pages, 1 figur