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

TL;DR

This paper explores the relationships between coagulation and fragmentation laws in two-parameter Poisson-Dirichlet processes, extending existing duality results to more general models using advanced probabilistic methods.

Contribution

It introduces an alternative analysis method that generalizes coagulation laws beyond the classical PD families, utilizing distributional relationships and recent transform techniques.

Findings

Explicit descriptions for coagulation laws with power tempered Poisson Kingman models.

Extension of duality relationships to broader classes of models.

Application of Cauchy-Stieltjes transforms in coagulation-fragmentation analysis.

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.