Geodesic
Sums of independent Poisson subordinators and their connections with strictly $\alpha$-stable processes of the Ornstein–Uhlenbeck type
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 13, Tome 361 (2008), pp. 123-137 
O. V. Rusakov. Sums of independent Poisson subordinators and their connections with strictly $\alpha$-stable processes of the Ornstein–Uhlenbeck type. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 13, Tome 361 (2008), pp. 123-137. http://geodesic.mathdoc.fr/item/ZNSL_2008_361_a8/
@article{ZNSL_2008_361_a8,
     author = {O. V. Rusakov},
     title = {Sums of independent {Poisson} subordinators and their connections with strictly $\alpha$-stable processes of the {Ornstein{\textendash}Uhlenbeck} type},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {123--137},
     year = {2008},
     volume = {361},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2008_361_a8/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

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.

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[2] O. E. Barndorff-Nielsen, “Superposition of Ornstein-Uhlenbeck type processes”, Teoriya veroyatn. i ee primen., 45:2 (2000), 289–311 | MR | Zbl

[3] O. E. Barndorff-Nielsen, N. Shephard, “Non-Gaussian Ornstein–Uhlenbeck – based models and some of their uses in financial economics”, J. Royal Statist. Soc., 63:2 (2001), 167–241 | DOI | MR | Zbl

[4] Z. J. Jurec, J. D. Mason, Operator-Limit Distributions in Probability Theory, Wiley, New York, 1993 | MR