Discussion on the paper "On simulation and properties of the stable law" by L. Devroye and L. James

We congratulate the authors for the interesting paper. The reading has been really pleasant and instructive. We discuss briefly only some of the interesting results given in [6] with particular attention to evolution problems. The contribution of the results collected in the paper is useful in a more wide class of applications in many areas of applied mathematics. The paper under discussion is a very well-written and interesting review article by Professors Devroye and James [6] dealing with known and lesser-known properties of stable laws, with methods of simulation for stable random variates, and with related random variables such as the Mittag–Leffler, Linnik, and Lamperti random variables. The main aim of the paper is to review and to collect in a single place simple procedures (one-liners) to generates random deviates from distributions that are in some way related to stable random variables. This is undoubtedly a very important topic at the basis of many techniques in different scientific fields. One can think for example at simulation of stochastic processes, generation of pseudo random numbers, cryptography, Monte Carlo and MCMC techniques, and so forth. A simple example in which generation of random deviates that are functions of stable random variables is needed, and which can make evident the important of the topic, regards the simulation of trajectories of time-fractional point processes such as the fractional Poisson process [2, 10, 11] or the fractional Yule process (fractional pure birth process) [13] (or in general of renewal processes with inter-arrival times distribution related to the stable law). The fractional Poisson process for example can be indeed constructed by exploiting its renewal structure. Let us thus consider a sequence of iid positive-Linnik distributed random variables (T j) n j=1 with parameter μ > 0 (also known as Mittag–Leffler random variables in part of the literature, see for example Gerd and Schreiber [7], Pillai [14], Jose et al. [8]) which we consider as random inter-arrival times between occurrences of point events. We have that for each j = 1, . . . ,n, E e j = μ/(s + μ), P{T j ∈ dt}/dt = μt Eν ,ν(−μt ), ν ∈ (0,1], μ > 0, t > 0. (1) The state probability, that is the probability of attaining level k at time t for the fractional Poisson process N(t), t > 0, easily follows from (1) and reads P{N(t) = k}= (μt) E ν ,νk+1 (−μt), t > 0, k ≥ 0, (2) where E γ ξ,μ (z), for z,ξ,μ,γ ∈ C, R(ξ) > 0, is the three-parameter generalized Mittag–Leffler function [9]. Clearly, the simulation of a trajectory corresponds to the generation of a sequence of independent random variates from positive-Linnik distributions. As suggested by the authors (see Devroye and James [6] but also Devroye [4, 5]) a positive Linnik random variate can be generated as E Sν , (3)