Abstract: Let $\mathbb{X} = \{X_t\}$ be an infinitely divisible stationary process. A good measure of the asymptotic dependence stucture of $\mathbb{X}$ is provided by the limit of $\rho_{\mathbb{X}}(t)$ as $t \rightarrow \infty$, where $\rho_{\mathbb{X}}(t)$ is equal to the joint characteristic function of $(X_t , X_0 )$ minus the product of the characteristic functions of $X_t$ and $X_0$. An interesting case is when $\rho_{\mathbb{X}}(t)\rightarrow 0$; which roughly says that, as time becomes large, the future of the random phenomenon (represented by $\mathbb{X}$) is becoming independent of it's past. In this talk, we shall discuss the rate at which this occurs by providing the rate of decay of $\rho_{\mathbb{X}}$ when $\mathbb{X}$ is an Ornstein-Uhlenbeck $(r, \alpha)$ -semi-stable process. It will be pointed out that the results obtained generalize and complement the corresponding results for Ornstein-Uhlenbeck $\alpha$-stable and Ornstein-Uhlenbeck Gaussian processes. Other works related to those noted above and to the mixing stationary infinitely divisible processes will also be discussed.