Nested subclasses of the class of \(\alpha\)-selfdecomposable distributions

Abstract

A probability distribution \(\mu\) on \(\mathbb R ^d\) is selfdecomposable if its characteristic function \(\widehat\mu(z), z\in\mathbb R ^d\), 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 \(\alpha\in\mathbb R\). 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 \(\widehat\mu(z) = \widehat \mu (b^{-1}z)^{b^{\alpha}}\widehat\rho_b(z)\). By denoting the class of all \(\alpha\)-selfdecomposable distributions on \(\mathbb R ^d\) by \(L^{\leftangle\alpha\rightangle}(\mathbb R ^d)\), we define in this paper a sequence of nested subclasses of \(L^{\leftangle\alpha\rightangle}(\mathbb R ^d)\), 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.