Nested subclasses of the class of $\alpha$-selfdecomposable distributions

Abstract

A probability distribution $\mu$ on ${\mathbb{R}}^d$ is selfdecomposable if its characteristic function $\hat{\mu }(z),z\in {\mathbb{R}}^d$, satisfies that for any $b>1$, there exists an infinitely divisible distribution ${\rho }_b$ satisfying $\hat{\mu }(z)=\hat{\mu }(b^{-1}z){\hat{\rho }}_b(z)$. This concept has been generalized to the concept of $\alpha$-selfdecomposability by many authors in the following way. Let $\alpha \in \mathbb{R}$. An infinitely divisible distribution $\mu$ on ${\mathbb{R}}^d$ is $\alpha$-selfdecomposable, if for any $b>1$, there exists an infinitely divisible distribution ${\rho }_b$ satisfying $\hat{\mu }(z)=\hat{\mu }(b^{-1}z{)}^{b^{\alpha }}{\hat{\rho }}_b(z)$. By denoting the class of all $\alpha$-selfdecomposable distributions on ${\mathbb{R}}^d$ by $L^{\leftangleα\rightangle}(\mathbb R ^d)$, we define in this paper a sequence of nested subclasses of $L^{\leftangleα\rightangle}(\mathbb R ^d)$, 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.