On Mittag-Leffler distributions and related stochastic processes. (English) Zbl 1355.60115
Summary: 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 \(\alpha \in(0, 1)\), 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 M. Möhle [ALEA, Lat. Am. J. Probab. Math. Stat. 12, No. 1, 35–53 (2015; Zbl 1329.60271)], a type-2 Mittag-Leffler process turns out to be the Siegmund dual to Neveu’s CSBP block-counting process arising in sampling from \(\mathrm{PD}(e^{- t}, 0)\). Further combinatorial developments of this model are investigated.
MSC:
| 60J80 | Branching processes (Galton-Watson, birth-and-death, etc.) |
| 60E05 | Probability distributions: general theory |
| 60E07 | Infinitely divisible distributions; stable distributions |
| 60G18 | Self-similar stochastic processes |
| 60K05 | Renewal theory |
Keywords:
Mittag-Leffler random variables and processes; stochastic growth models; Neveu branching process with infinite mean; immigration and self-decomposability; renewal process; Bolthausen-Sznitman coalescentCitations:
Zbl 1329.60271References:
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