Sums of independent Poisson subordinators and their connections with strictly $\alpha$-stable processes of the Ornstein–Uhlenbeck type

We give a construction by which the discrete time of a sequence of independent identically distributed random variables changes with the Poisson time. The Poisson time is independent of this sequence. We name similarly defined process with continuous time as the random index process. We establish a number of properties of the random index processes. We study the asymptotic of sums of independent identically distributed random index processes for the case when elements of the initial sequence have the strictly $\alpha$-stable distribution. By calculated characteristic functions we establish the relationships of these sums with the strictly $\alpha$-stable processes of the Ornstein–Uhlenbeck Type. Bibl. – 4 titles.