Selfdecomposability and selfsimilarity: a concise primer. (English) Zbl 1395.60018
Summary: We summarize the relations among three classes of laws: infinitely divisible, selfdecomposable and stable. First we look at them as the solutions of the Central Limit Problem; then their role is scrutinized in relation to the Lévy and the additive processes with an emphasis on stationarity and selfsimilarity. Finally we analyze the Ornstein-Uhlenbeck processes driven by Lévy noises and their selfdecomposable stationary distributions, and we end with a few particular examples.
MSC:
| 60E07 | Infinitely divisible distributions; stable distributions |
| 60G18 | Self-similar stochastic processes |
| 82C31 | Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics |
| 60-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory |
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