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Coagulation–fragmentation duality, Poisson–Dirichlet distributions and random recursive trees

2006
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The Annals of Applied Probability
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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 $\operatorname {PD}(\alpha,\theta)$ that take values in this space. We introduce families of random fragmentation and coagulation operators $\mathrm {Frag}_{\alpha}$ and $\mathrm {Coag}_{\alpha,\theta}$, respectively, with

doi:10.1214/105051606000000655
fatcat:gg6ds5xibzgk7ecsdjdk4zssf4