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

Abstract

Properties of the law $\mu$ of the integral ${\int }_0^{\infty }{c}^{-N_{t-}}d{Y}_t$ are studied, where $c>1$ and $\{(N_t,Y_t),t\ge 0\}$ is a bivariate Lévy process such that $\{N_t\}$ and $\{Y_t\}$ are Poisson processes with parameters $a$ and $b$, respectively. This is the stationary distribution of some generalized Ornstein--Uhlenbeck process. The law $\mu$ is parametrized by $c$, $q$ and $r$, where $p=1-q-r$, $q$, and $r$ are the normalized Lévy measure of $\{(N_t,Y_t)\}$ at the points $(1,0)$, $(0,1)$ and $(1,1)$, respectively. It is shown that, under the condition that $p>0$ and $q>0$, ${\mu }_{c,q,r}$ is infinitely divisible if and only if $r\le pq$. The infinite divisibility of the symmetrization of $\mu$ is also characterized. The law $\mu$ 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 $\mu$ is continuous-singular under the condition $q>0$. On the other hand, for Lebesgue almost every $c>1$, there are positive constants $C_1$ and $C_2$ such that $\mu$ is absolutely continuous whenever $q\ge C_{1}p\ge C_{2}r$. For any $c>1$ there is a positive constant $C_3$ such that $\mu$ is continuous-singular whenever $q>0$ and $\max \{q,r\}\le C_{3}p$. Here, if $\{N_t\}$ and $\{Y_t\}$ are independent, then $r=0$ and $q=b/(a+b)$.

Published in at http://dx.doi.org/10.1214/08-AOP402 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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