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 Levy processes (S(i)(alpha,beta,gamma,delta))(t >= 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 >= 0) and gave a structure of conditionally Levy 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 bar right arrow 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)is an element of S) and give their asymptotic controls in the space variable x uniformly in s is an element of S.