Stable Processes, Mixing, and Distributional Properties. II

In Part I of this paper [Theory Probab. Appl., 52 (2008), pp. 580–593], we considered real-valued stable Lévy processes $ (S_t^{\alpha, \beta,\gamma,\delta})_{t\ge 0}$, where the deterministic numbers $\alpha, \beta, \gamma,\delta$ are, respectively, the stability, skewness, scale, and drift coefficients. Then, allowing $ \beta, \gamma,\delta $ to be random, we introduced the notion of mixed stable processes $ (M_t^{\alpha, \beta, \gamma,\delta})_{t\ge 0}$ and gave a structure of conditionally Lévy processes. In this second part, we provide controls of the (nonmixed) densities $ G_t^{\alpha, \beta, \gamma,\delta}(x)$ when $ x $ goes to the extremities of the support of $ G_t^{\alpha, \beta, \gamma,\delta} $ uniformly in $t,\beta,\gamma,\delta $ and present a Mellin duplication formula on these densities, relative to the stability coefficient $\alpha $. The new representations of the densities give an explicit expression of all the moments of order $0<\rho<\alpha$. We also study the densities $x\mapsto H_s(x)$ of mixed stable variables $M_s^{\alpha,\beta_s,\gamma_s,\delta_s}$ (by families of random variables $(\beta_s,\gamma_s,\delta_s)_{s\in S}$) and give their asymptotic controls in the space variable $x$ uniformly in $s\in S$.