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. There can be of two different types, one (type-1) heavy-tailed with index 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 [49], 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 ) . Further combinatorial developments of this model are investigated.