Pitman(2003)(and subsequently Gnedin and Pitman (2006) showed that a large class of random partitions of the integers derived from a stable subordinator of index $α\in(0,1)$ have infinite Gibbs (product) structure as a characterizing feature. The most notable case are random partitions derived from the two-parameter Poisson-Dirichlet distribution, $\mathrm{PD}(α,θ)$, which are induced by mixing over variables with generalized Mittag-Leffler distributions, denoted by $\mathrm{ML}(α,θ).$ Our aim in this work is to provide indications on the utility of the wider class of Gibbs partitions as it relates to a study of Riemann-Liouville fractional integrals and size-biased sampling, decompositions of special functions, and its potential use in the understanding of various constructions of more exotic processes. We provide novel characterizations of general laws associated with two nested families of $\mathrm{PD}(α,θ)$ mass partitions that are constructed from notable fragmentation operations described in Dong, Goldschmidt and Martin(2006) and Pitman(1999), respectively. These operations are known to be related in distribution to various constructions of discrete random trees/graphs in $[n],$ and their scaling limits, such as stable trees. A centerpiece of our work are results related to Mittag-Leffler functions, which play a key role in fractional calculus and are otherwise Laplace transforms of the $\mathrm{ML}(α,θ)$ variables. Notably, this leads to an interpretation of $\mathrm{PD}(α,θ)$ laws within a mixed Poisson waiting time framework based on $\mathrm{ML}(α,θ)$ variables, which suggests connections to recent construction of Pólya urn models with random immigration by Peköz, Röllin and Ross(2018). Simplifications in the Brownian case are highlighted.
Convergence of random sums and statistics constructed from samples with random sizes to the Linnik and Mittag-Leffler distributions and their generalizations
Scuola Superiore Meridionale（南方高等学院）；Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Università degli Studi di Napoli Federico II（那不勒斯费德里科二世大学数学与应用雷纳托·卡丘波利系）
