CONTINUITY PROPERTIES AND INFINITE DIVISIBILITY OF STATIONARY DISTRIBUTIONS OF SOME GENERALIZED ORNSTEIN-UHLENBECK PROCESSES

Properties of the law mu of the integral f(0)(infinity) C(-N1)-dY(1), are studied, where c > 1 and {(N(1), Y(1)). t >= 0} is a bivariate Levy 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 Levy 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 <= 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 p is absolutely continuous whenever q >= C(1)p >= 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) <= C(3)p. Here, if {N(t)} and {Y(t)} are independent, then r = 0 and q = b/(a + b).