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January 2009 Continuity properties and infinite divisibility of stationary distributions of some generalized Ornstein–Uhlenbeck processes
Alexander Lindner, Ken-iti Sato
Ann. Probab. 37(1): 250-274 (January 2009). DOI: 10.1214/08-AOP402


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).


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Alexander Lindner. Ken-iti Sato. "Continuity properties and infinite divisibility of stationary distributions of some generalized Ornstein–Uhlenbeck processes." Ann. Probab. 37 (1) 250 - 274, January 2009.


Published: January 2009
First available in Project Euclid: 17 February 2009

zbMATH: 1173.60007
MathSciNet: MR2489165
Digital Object Identifier: 10.1214/08-AOP402

Primary: 60E07 , 60G10 , 60G30 , 60G51

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

Rights: Copyright © 2009 Institute of Mathematical Statistics


Vol.37 • No. 1 • January 2009
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