Tokyo Journal of Mathematics

Nested Subclasses of the Class of $\alpha$-selfdecomposable Distributions

Makoto MAEJIMA and Yohei UEDA

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Abstract

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.

Article information

Source
Tokyo J. Math., Volume 34, Number 2 (2011), 383-406.

Dates
First available in Project Euclid: 30 January 2012

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1327931393

Digital Object Identifier
doi:10.3836/tjm/1327931393

Mathematical Reviews number (MathSciNet)
MR2918913

Zentralblatt MATH identifier
1236.60018

Subjects
Primary: 60E07: Infinitely divisible distributions; stable distributions
Secondary: 60G51: Processes with independent increments; Lévy processes 60F05: Central limit and other weak theorems

Citation

MAEJIMA, Makoto; UEDA, Yohei. Nested Subclasses of the Class of $\alpha$-selfdecomposable Distributions. Tokyo J. Math. 34 (2011), no. 2, 383--406. doi:10.3836/tjm/1327931393. https://projecteuclid.org/euclid.tjm/1327931393


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