## Tokyo Journal of Mathematics

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

#### 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

https://projecteuclid.org/euclid.tjm/1327931393

Digital Object Identifier
doi:10.3836/tjm/1327931393

Mathematical Reviews number (MathSciNet)
MR2918913

Zentralblatt MATH identifier
1236.60018

#### 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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