Properties of stationary distributions of a sequence of generalized Ornstein–Uhlenbeck processes

✍ Scribed by Alexander Lindner; Ken-iti Sato

• Publisher: John Wiley and Sons
• Year: 2011
• Tongue: English
• Weight: 277 KB
• Volume: 284
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200910223

✦ Synopsis

Abstract

The infinite (in both directions) sequence of the distributions μ^(k)^ of the stochastic integrals \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\int _0^{\infty -} c^{-N_{t-}^{(k)}} dL_t^{(k)}$\end{document} for integers k is investigated. Here c > 1 and (N^(k)^~t~, L~t~^(k)^), 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~t~^(0)^) is marginally Poisson and μ^(0)^ has been studied by Lindner and Sato (Ann. Probab. 37 (2009), 250–274). The distributions μ^(k)^ are the stationary distributions of a sequence of generalized Ornstein–Uhlenbeck processes structurally related in a particular way. Continuity properties of μ^(k)^ are shown to be the same as those of μ^(0)^. The dependence on k of infinite divisibility of μ^(k)^ is clarified. The problem to find necessary and sufficient conditions in terms of c, p, q, and r for μ^(k)^ 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^(k)^ are also studied. The distributions μ^(k)^ and their symmetrizations are c^−1^‐decomposable, and it is shown that, for each k ≠ 0, μ^(k)^ 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).