Nested subclasses of the class of $α$-selfdecomposable distributions

Abstract

A probability distribution $μ$ on $\mathbb R ^d$ is selfdecomposable if its characteristic function $\widehatμ(z), z\in\mathbb R ^d$, satisfies that for any $b>1$, there exists an infinitely divisible distribution $ρ_b$ satisfying $\widehatμ(z) = \widehatμ(b^{-1}z)\widehatρ_b(z)$. This concept has been generalized to the concept of $α$-selfdecomposability by many authors in the following way. Let $α\in\mathbb R$. An infinitely divisible distribution $μ$ on $\mathbb R ^d$ is $α$-selfdecomposable, if for any $b>1$, there exists an infinitely divisible distribution $ρ_b$ satisfying $\widehatμ(z) = \widehat μ(b^{-1}z)^{b^α}\widehatρ_b(z)$. By denoting the class of all $α$-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.