The asymptotic dependence behavior of Ornstein-Uhlenbeck semi-stable processes

Let X = {Xt} be an infinitely divisible stationary process. A good measure of the asymptotic dependence structure of X is provided by the limit of ρX(t) as t → ∞, where ρX(t) is equal to the joint characteristic function of (Xt, X0) minus the product of the characteristic functions of Xt and X0. An interesting case is when ρX(t) → 0; which roughly says that, as time becomes large, the future of the random phenomenon (represented by X) is becoming independent of its past. In this paper, we study the rate of decay of ρX(t) (as t →∞) when X is an Ornstein-Uhlenbeck (r, α)-semi-stable process. The results obtained here generalize and complement the corresponding results for Ornstein-Uhlenbeck α-stable and Ornstein-Uhlenbeck (Gaussian) processes.