On Mittag-Leffler distributions and related stochastic processes

Abstract

International audienceRandom 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 P D e −t , 0. Further combinatorial developments of this model are investigated. 1. Sibuya random variables (rvs) and related branching processes We first investigate a class of integral-valued rvs that will show important for our general purpose. 1.1. Sibuya rvs and related ones. We start with their definition and main properties. • One parameter Sibuya(α) rv. Let X α ≥ 1 be an integer-valued random variable with support N = {1, 2, ....} defined as follows: X α = inf (l ≥ 1 : B α (l) = 1) , where (B α (l)) l≥1 is a sequence of independent Bernoulli rvs obeying P (B α (l) = 1) = α/l where α ∈ (0, 1). It is thus the first epoch of a success in a Bernoulli trial when the probability of success is inversely proportional to the number of the trial. X α is called a Sibuya(α) rv. Then P (X α = k) = (−1

The Tempered Discrete Linnik distribution

“…X DS is achieved as δ → ∞. In addition, it should be remarked that the Discrete Linnik law is defined for some negative δ in such a way that λ ≤ |δ|(1 − γ) and, in this case, the distribution is also named Generalized Sibuya (Huillet, 2016). However, the results of this Section are mainly given for positive δ.…”

“…where γ ∈]0, 1] (for a recent survey of this law, see Huillet, 2016). Indeed, the Sibuya distribution is a special case of the (shifted) Negative Binomial Beta distribution introduced by Sibuya (1979) with parameters given by 1, γ and (1 − γ).…”

COVID-19 X-ray images classification based on enhanced fractional-order cuckoo search optimizer using heavy-tailed distributions

“…Using the heavy-tailed distributions, it is possible to escape from non-prominent regions of the search space [41] . The distributions used in to enhance the FO-CS are the Mittag-Leffler distribution (ML ) [43] , the Pareto distribution (P ) [44] , the Cauchy distribution (C ) [45] , and the Weibull distribution (W ) [46] . The FO-CS based on heavy-tailed distributions is proposed as an alternative method for solving features selection problems.…”

On the Multi-Index Mittag-Leffler Functions and Their Mellin Transforms

“…Recently the interest to M-L type functions and their generalizations has grown up in view of their important role in fractional calculus and related integral and differential equations of fractional order (as their solutions) and applications [35], for example in modelling some evolution problems [7], fractional diffusion processes [24], nonlinear waves, etc. The M-L function enjoys also applications in the stochastic processes, statistical distributions and conditional expectations as the so-called M-L (probability) density, see for example Mathai, Haubold et al [27], [8], [9], etc.…”

“…The emphasize is given to the Mellin transform images of the considered M-L type functions, via their Mellin-Barnes type integral representations. It is well known that the Mellin transform plays important role in the theory of special functions, see for example: [23] (for the contributions of S. Pincherle to Mellin-Barnes integrals); the book by Marichev [25] (devoted to methods of evaluation of integrals of special functions via the Mellin transform); also in fractional calculus -see works by Butzer-Kilbas-Trujillo [1], Luchko-Kiryakova [22], in probability and statistics [27], [8], [9], etc.…”