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Stationary processes of Ornstein-Uhlenbeck type. (English) Zbl 0532.60065
Probability theory and mathematical statistics, Proc. 4th USSR-Jap. Symp., Tbilisi/USSR 1982, Lect. Notes Math. 1021, 541-551 (1983).
[For the entire collection see Zbl 0509.00020.]
In the first part, the authors deal with the class of operator self- decomposable measures. The main result says that the class \(OL({\mathbb{R}}^ d,Q)\) coincides with limit distributions of Ornstein-Uhlenbeck type processes. Also, infinitesimal generators associated with these processes are characterized. These theorems are proved in the authors’ paper, Operator self-decomposable distributions as limit distributions of processes of Ornstein-Uhlenbeck type. Stochastic Processes Appl. 17, 73- 100 (1984). The second part of the paper is devoted to Urbanik classes \(OL_ m({\mathbb{R}}^ d,Q)\), \(m=1,2,3,...\). These classes of limit distributions for the case of Banach spaces are investigated by the reviewer in Bull. Pol. Acad. Sci., Ser. Sci. Math. 31, 51-62 (1983).
Reviewer: Z.Jurek

60J35 Transition functions, generators and resolvents
60H05 Stochastic integrals
60E07 Infinitely divisible distributions; stable distributions