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Stable Processes, Mixing, and Distributional Properties. II
W. Jedidi Université Pierre & Marie Curie, Paris VI
Abstract:
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
$s\in S$.
Keywords:
stable processes, conditionally PIIS, Mellin convolution, density, derivatives, uniform controls. Received: 23.06.2005
Language: English
DOI:
10.4213/tvp2485
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