On the sum of independent generalized Mittag–Leffler random variables and the related fractional processes

Abstract

We obtain the distribution of the sum of independent and non-identically distributed generalized Mittag–Leffler random variables. We then apply this result to study some related fractional point processes. We present their explicit probability mass functions as well as their connections with the fractional integral/differential equations. In the case of a point process with Mittag–Leffler distributed waiting times which alternate two indexes and two rates we also study the conditional arrival times and we show an application to the telegraph process.