Continuity properties and infinite divisibility of stationary distributions of some generalized Ornstein-Uhlenbeck processes. (English) Zbl 1173.60007

The authors study the distribution \(\mu \) of \[ \int_{0}^{\infty -}e^{-(\log c)N_{s-}dY_{s}}, \] where \(c>1\) is a constant, and \(N_t\) and \(Y_t\) are Poisson processes, and such that \((N_{t},Y_{t})\) is a bivariate Lévy process. Conditions are obtained under which \(\mu\) is infinitely divisible and/or absolutely continuous. The conditions are in terms of the parameters \(p\), \(q\) and \(r\), defined as point masses of the Lévy measure of \((N_{t},Y_{t})\); if \(r=0\), then \(N_{t}\) and \(Y_{t}\) are independent. The infinite divisibility is closely connected to the infinite divisibility of certain mixtures of geometric distributions. The results on continuity properties hinge on very special results, that can not be explained here, and on c being or not being a PV number or a PS number, where PV stands for Pisot–Vijayaraghavan and PS for Peres–Solomyak.


60E07 Infinitely divisible distributions; stable distributions
60G10 Stationary stochastic processes
60G30 Continuity and singularity of induced measures
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