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

Abstract

Pitman~(1999) describes a duality relationship between fragmentation and coagulation operators. An explicit relationship is described for the two-parameter Poisson-Dirichlet laws, with parameters {\footnotesize $(\alpha,\theta)$} and $(\beta,\theta/\alpha)$, wherein $PD(\alpha, \theta)$ is coagulated by $PD(\beta,\theta/\alpha)$ for $0<\alpha<1$, $0 \leq\beta<1$ and $-\beta<\theta/\alpha$. This remarkable explicit agreement was obtained by combinatorial methods via exchangeable partition probability functions~(EPPF). This work discusses an alternative analysis which can feasibly extend the characterizations above to more general models of $PD(\alpha,\theta)$ coagulated with some law $Q$. The analysis exploits distributional relationships between compositions of species sampling random probability measures and coagulation operators and recent work on Cauchy-Stieltjes transforms of random probability measures by Vershik, Yor and Tsilevich (2004) and James (2002). We use this to obtain explicit descriptions in the case where {\footnotesize $Q$} corresponds to a large class of power tempered Poisson Kingman models analyzed in James~(2002). That is, explicit results are obtained for models outside of the $PD(\beta,\theta/\alpha)$ family.