Coagulation Fragmentation Laws Induced By General Coagulations of Two-Parameter Poisson-Dirichlet Processes

Pitman (1999) describes a duality relationship between fragmentation and coagulation operators. An explicit relationship is described for the two-parameter Poisson-Dirichlet laws, say PD(a, b), with parameters (α, θ) and (β, θ/α), wherein PD(α, θ) is coagulated by PD(β, θ/α) for 0 < α < 1, 0 ≤ β < 1 and −β < θ/α. This remarkable explicit agreement was obtained by combinatorial methods via exchangeable partition probability functions (EPPF). It has been noted that such a method is not easy to employ for more general processes. This work discusses an alternative analysis which can feasibly extend the characterizations above to more general models of PD(α, θ) coagulated with some law Q. The analysis exploits distributional relationships between compositions of species sampling random probability measures and coagulation operators. It is shown, based on results of Vershik, Yor and Tsilevich (2004) and James (2002), how the calculation of generalized Cauchy-Stieltjes transforms of random probability measures provides a blueprint to obtain explicit characterizations of PD(α, θ) coagulated with some law Q. We use this to obtain explicit descriptions in the case where Q corresponds to a large class of power tempered Poisson Kingman models described in James (2002). That is, explicit results are obtained for models outside of the PD(β, θ/α) family. We obtain a new proof of Pitman’s result as a by-product. Furthermore, noting an obvious distinction from the class of PD(α, θ) derived from a stable subordinator, we discuss briefly the case of Dirichlet processes coagulated by various Q.