期刊名称：Latin American Journal of Probability and Mathematical Statistics
电子版ISSN：1980-0436
出版年度：2020
卷号：17
期号：1
页码：375
DOI：10.30757/ALEA.v17-15
出版社：Instituto Nacional De Matemática Pura E Aplicada
摘要：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 classical birth-and-death process with immigration of 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 a 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.
