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Coag-Frag duality for a class of stable Poisson-Kingman mixtures

Exchangeable sequences of random probability measures (partitions of mass) and their corresponding exchangeable bridges play an important role in a variety of areas in probability, statistics and related areas, including Bayesian statistics, physics, finance and machine learning. An area of theoretical as well as practical interest, is the study of coagulation and fragmentation operators on partitions of mass. In this regard, an interesting but formidable question is the identification of operators and distributional families on mass partitions that exhibit interesting duality relations. In this paper we identify duality relations for a large sub-class of mixed Poisson-Kingman models generated by a stable subordinator. Our results are natural generalizations of the duality relations developed in Pitman, Bertoin and Goldschmidt, and Dong, Goldschmidt and Martin for the two-parameter Poisson Dirichlet family. These results are deduced from results for corresponding bridges.