On Mittag-Leffler distributions and related stochastic processes

Random variables with Mittag-Leffler distribution can take values either in the set of non-negative integers or in the positive real line. They can be of two different types, one (type-1) heavy-tailed with index α∈(0,1)α∈(0,1), the other (type-2) possessing all its moments. We investigate various stochastic processes where they play a key role, among which: the discrete space/time Neveu branching process, the discrete-space continuous-time Neveu branching process, the continuous space/time Neveu branching process (CSBP) and renewal processes with rare events. Its relation to (discrete or continuous) self-decomposability and branching processes with immigration is emphasized. Special attention will be paid to the Neveu CSBP for its connection with the Bolthausen–Sznitman coalescent. In this context, and following a recent work of Möhle (2015), a type-2 Mittag-Leffler process turns out to be the Siegmund dual to Neveu’s CSBP block-counting process arising in sampling from PD(e−t,0)PD(e−t,0). Further combinatorial developments of this model are investigated.