2005/07/31 by Rui Dong, Christina Goldschmidt, Jean Martin +1 · 1 citation
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #Random Matrices and Applications #Stochastic processes and statistical mechanics #math.CO #math.PR #msc:05C05 #msc:60C05 #msc:60G51 #msc:60J05
paper · pdf · doi:10.1214/105051606000000655
published as Annals of Applied Probability 2006, Vol. 16, No. 4, 1733-1750 · Published at http://dx.doi.org/10.1214/105051606000000655 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2006/11/01 · arxiv created 2007/02/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31
In this paper we give a new example of duality between fragmentation and coagulation operators. Consider the space of partitions of mass (i.e., decreasing sequences of nonnegative real numbers whose sum is 1) and the two-parameter family of Poisson–Dirichlet distributions PD (α,θ) that take values in this space. We introduce families of random fragmentation and coagulation operators Fragα and Coagα,θ, respectively, with the following property: if the input to Fragα has PD (α,θ) distribution, then the output has PD (α,θ+1) distribution, while the reverse is true for Coagα,θ. This result may be proved using a subordinator representation and it provides a companion set of relations to those of Pitman between PD (α,θ) and PD (αβ,θ). Repeated application of the Fragα operators gives rise to a family of fragmentation chains. We show that these Markov chains can be encoded naturally by certain random recursive trees, and use this representation to give an alternative and more concrete proof of the coagulation–fragmentation duality.