Limit distributions for doubly stochastically rarefied renewal processes and their properties

A limit theorem is proved for doubly stochastically rarefied renewal processes. It is shown that under rather general conditions, as limit laws in limit theorems for mixed geometric random sums, there appear mixed exponential and mixed Laplace distributions. Some known and new properties of these distributions are reviewed. Also, some nonobvious properties of special representatives of these classes (the Weibull, Mittag-Leffler, Linnik, and other distributions) are described.