Annals of Probability

Continuity properties and infinite divisibility of stationary distributions of some generalized Ornstein–Uhlenbeck processes

Alexander Lindner and Ken-iti Sato

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Properties of the law μ of the integral 0cNtdYt are studied, where c>1 and {(Nt, Yt), t≥0} is a bivariate Lévy process such that {Nt} and {Yt} are Poisson processes with parameters a and b, respectively. This is the stationary distribution of some generalized Ornstein–Uhlenbeck process. The law μ is parametrized by c, q and r, where p=1−qr, q, and r are the normalized Lévy measure of {(Nt, Yt)} at the points (1, 0), (0, 1) and (1, 1), respectively. It is shown that, under the condition that p>0 and q>0, μc, q, r is infinitely divisible if and only if rpq. The infinite divisibility of the symmetrization of μ is also characterized. The law μ is either continuous-singular or absolutely continuous, unless r=1. It is shown that if c is in the set of Pisot–Vijayaraghavan numbers, which includes all integers bigger than 1, then μ is continuous-singular under the condition q>0. On the other hand, for Lebesgue almost every c>1, there are positive constants C1 and C2 such that μ is absolutely continuous whenever qC1pC2r. For any c>1 there is a positive constant C3 such that μ is continuous-singular whenever q>0 and max {q, r}≤C3p. Here, if {Nt} and {Yt} are independent, then r=0 and q=b/(a+b).

Article information

Ann. Probab., Volume 37, Number 1 (2009), 250-274.

First available in Project Euclid: 17 February 2009

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Zentralblatt MATH identifier

Primary: 60E07: Infinitely divisible distributions; stable distributions 60G10: Stationary processes 60G30: Continuity and singularity of induced measures 60G51: Processes with independent increments; Lévy processes

Decomposable distribution generalized Ornstein–Uhlenbeck process infinite divisibility Lévy process Peres–Solomyak (P.S.) number Pisot–Vijayaraghavan (P.V.) number symmetrization of distribution


Lindner, Alexander; Sato, Ken-iti. Continuity properties and infinite divisibility of stationary distributions of some generalized Ornstein–Uhlenbeck processes. Ann. Probab. 37 (2009), no. 1, 250--274. doi:10.1214/08-AOP402.

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