Stable Processes, Mixing, and Distributional Properties. I

In this article, we consider real-valued stable Lйvy processes $(S_t^{\alpha, \beta,\gamma,\delta})_{t\ge 0}$, where $\alpha,\beta,\gamma,\delta$ are, respectively, the stability, skewness, scale, and drift coefficients. We introduce the notion of mixed stable processes $ (M_t^{\alpha, \beta,\gamma,\delta})_{t\ge 0}$ (i.e., we allow the skewness, scale, and drift coefficients to be random). Our mixing procedure gives a structure of conditionally Lйvy processes. This procedure permits us to show that the sum of independent stable processes can be expressed via a mixed stable process.