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December 2011 Nested Subclasses of the Class of $\alpha$-selfdecomposable Distributions
Makoto MAEJIMA, Yohei UEDA
Tokyo J. Math. 34(2): 383-406 (December 2011). DOI: 10.3836/tjm/1327931393


A probability distribution $\mu$ on $\mathbf{R}^d$ is selfdecomposable if its characteristic function $\widehat\mu(z), z\in\mathbf{R}^d$, satisfies that for any $b>1$, there exists an infinitely divisible distribution $\rho_b$ satisfying $\widehat\mu(z) = \widehat\mu(b^{-1}z)\widehat\rho_b(z)$. This concept has been generalized to the concept of $\alpha$-selfdecomposability by many authors in the following way. Let $\alpha\in\mathbf{R}$. An infinitely divisible distribution $\mu$ on $\mathbf{R}^d$ is $\alpha$-selfdecomposable, if for any $b>1$, there exists an infinitely divisible distribution $\rho_b$ satisfying $\widehat\mu(z) = \widehat\mu(b^{-1}z)^{b^{\alpha}}\widehat\rho_b(z)$. By denoting the class of all $\alpha$-selfdecomposable distributions on $\mathbf{R}^d$ by $L^{(\alpha)}(\mathbf{R}^d)$, we define in this paper a sequence of nested subclasses of $L^{(\alpha)}(\mathbf{R}^d)$, and investigate several properties of them by two ways. One is by using limit theorems and the other is by using mappings of infinitely divisible distributions.


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Makoto MAEJIMA. Yohei UEDA. "Nested Subclasses of the Class of $\alpha$-selfdecomposable Distributions." Tokyo J. Math. 34 (2) 383 - 406, December 2011.


Published: December 2011
First available in Project Euclid: 30 January 2012

zbMATH: 1236.60018
MathSciNet: MR2918913
Digital Object Identifier: 10.3836/tjm/1327931393

Primary: 60E07
Secondary: 60F05, 60G51

Rights: Copyright © 2011 Publication Committee for the Tokyo Journal of Mathematics


Vol.34 • No. 2 • December 2011
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