STABLE PROCESSES, MIXING, AND DISTRIBUTIONAL PROPERTIES. II

摘要

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.