Geometric infinite divisibility, stability, and self-similarity: an overview
The concepts of geometric infinite divisibility and stability extend the classical properties of infinite divisibility and stability to geometric convolutions. In this setting, a random variable $X$ is geometrically infinitely divisible if it can be expressed as a random sum of $N_p$ components for each $p\in (0,1)$, where $N_p$ is a geometric random variable with mean $1/p$, independent of the components. If the components have the same distribution as that of a rescaled $X$, then $X$ is (strictly) geometric stable. This leads to broad classes of probability distributions closely connected with their classical counterparts. We review fundamental properties of these distributions and discuss further extensions connected with geometric sums, including multivariate and operator geometric stability, discrete analogs, and geometric self-similarity.