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

Abstract

The infinite (in both directions) sequence of the distributions $\mu^{(k)}$ of the stochastic integrals $\int_0^{\infty-}c^{-N_{t-}^{(k)}} dL_t^{(k)}$ for integers $k$ is investigated. Here $c>1$ and $(N_t^{(k)},L_t^{(k)})$, $t\geq0$, is a bivariate compound Poisson process with L\'evy measure concentrated on three points $(1,0)$, $(0,1)$, $(1,c^{-k})$. The amounts of the normalized L\'evy measure at these points are denoted by $p$, $q$, $r$. For $k=0$ the process $(N_t^{(0)},L_t^{(0)})$ is marginally Poisson and $\mu^{(0)}$ has been studied by Lindner and Sato (Ann. Probab. 37 (2009), 250-274). The distributions $\mu^{(k)}$ are the stationary distributions of a sequence of generalized Ornstein-Uhlenbeck processes structurally related in some way. Continuity properties of $\mu^{(k)}$ are shown to be the same as those of $\mu^{(0)}$. The problem to find necessary and sufficient conditions in terms of $c$, $p$, $q$, and $r$ for $\mu^{(k)}$ 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 $\mu^{(k)}$ are also studied. The distributions $\mu^{(k)}$ and their symmetrizations are $c^{-1}$-decomposable, and it is shown that, for each $k\neq 0$, $\mu^{(k)}$ 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 $[0,\infty)$ can have its quasi-L\'evy measure concentrated on $(-\infty,0)$.