Gibbs Partitions, Riemann-Liouville Fractional Operators, Mittag-Leffler Functions, and Fragmentations Derived From Stable Subordinators

TL;DR

This paper explores the connections between Gibbs partitions derived from stable subordinators, fractional calculus, and special functions like Mittag-Leffler, providing new characterizations and potential applications in random trees, graphs, and urn models.

Contribution

It introduces novel characterizations of Gibbs partitions related to Mittag-Leffler functions and fractional operators, expanding understanding of their structure and applications.

Findings

Characterization of laws associated with nested Poisson-Dirichlet partitions

Interpretation of PD(α,θ) laws within a mixed Poisson framework

Connections to random trees, graphs, and urn models

Abstract

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 $\alpha\in(0,1)$ have infinite Gibbs (product) structure as a characterizing feature. The most notable case are random partitions derived from the two-parameter Poisson-Dirichlet distribution, $\mathrm{PD}(\alpha,\theta)$, which are induced by mixing over variables with generalized Mittag-Leffler distributions, denoted by $\mathrm{ML}(\alpha,\theta).$ 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 $\mathrm{PD}(\alpha,\theta)$ 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 $\mathrm{ML}(\alpha,\theta)$ variables. Notably, this leads to an interpretation of $\mathrm{PD}(\alpha,\theta)$ laws within a mixed Poisson waiting time framework based on $\mathrm{ML}(\alpha,\theta)$ variables, which suggests connections to recent construction of P\'olya urn models with random immigration by Pek\"oz, R\"ollin and Ross(2018). Simplifications in the Brownian case are highlighted.