Discrete tempered stable distributions (Q2157412)

A discrete tempered stable (DTS) random variable \(X\) with parameters \(\alpha\in(0,1)\) and \(\eta\geq0\), and with tempering function \(q\) satisfying \(\lim_{x\downarrow0}q(x)=1\) and \(\mathrm{ess}\sup q(x)<\infty\), has probability generating function given by \[ E[s^X]=\exp\left\{-\eta\int_0^\infty\left(1-e^{-(1-s)x}\right)q(x)x^{-1-\alpha}\,dx\right\}\,, \] for \(s\in(0,1]\).The author studies various properties of these DTS distributions, including probabilistic representations, explicit formulas for the mass function and moments, overdispersion, self-decomposability, and unimodality. Simulation of DTS distributions is discussed, making use of the representation of \(X\) as a compound Poisson random variable with compounding distribution a mixture of zero-truncated Poisson distributions. This is then extended to cover simulation of the corresponding bilateral distributions and Lévy processes, and the paper concludes with a simulation study.