Properties of Stationary Distributions of a Sequence of Generalized Ornstein–uhlenbeck Processes

The infinite (in both directions) sequence of the distributions μ of the stochastic integrals ∫∞− 0 c−N (k) t− dL (k) t for integers k is investigated. Here c > 1 and (N (k) t , L (k) t ), t ≥ 0, is a bivariate compound Poisson process with Lévy measure concentrated on three points (1, 0), (0, 1), (1, c−k). The amounts of the normalized Lévy measure at these points are denoted by p, q, r. For k = 0 the process (N (0) t , L (0) t ) is marginally Poisson and μ has been studied by Lindner and Sato (Ann. Probab. 37 (2009), 250–274). The distributions μ are the stationary distributions of a sequence of generalized Ornstein–Uhlenbeck processes structurally related in some way. Continuity properties of μ are shown to be the same as those of μ. The dependence on k of infinite divisibility of μ is clarified. The problem to find necessary and sufficient conditions in terms of c, p, q, and r for μ to be infinitely divisible is somewhat involved, but completely solved for every integer k. The conditions depend on arithmetical properties of c. The symmetrizations of μ are also studied. The distributions μ and their symmetrizations are c−1-decomposable, and it is shown that, for each k 6= 0, μ and its symmetrization may be infinitely divisible without the corresponding factor in the c−1-decomposability relation being infinitely divisible. This phenomenon was first observed by Niedbalska-Rajba (Colloq. Math. 44 (1981), 347–358) in an artificial example. The notion of quasi-infinite divisibility is introduced and utilized, and it is shown that a quasi-infinitely divisible distribution on [0,∞) can have its quasi-Lévy measure concentrated on (−∞, 0).