# Nested subclasses of the class of $α$-selfdecomposable distributions - Mathematics > Probability

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Abstract: A probability distribution $\mu$ on $\mathbb R ^d$ is selfdecomposable if itscharacteristic function $\widehat\muz, z\in\mathbb R ^d$, satisfies that forany $b>1$, there exists an infinitely divisible distribution $ ho b$satisfying $\widehat\muz = \widehat\mu b^{-1}z\widehat ho bz$. Thisconcept has been generalized to the concept of $\alpha$-selfdecomposability bymany authors in the following way. Let $\alpha\in\mathbb R$. An infinitelydivisible distribution $\mu$ on $\mathbb R ^d$ is $\alpha$-selfdecomposable, iffor any $b>1$, there exists an infinitely divisible distribution $ ho b$satisfying $\widehat\muz = \widehat \mub^{-1}z^{b^{\alpha}}\widehat ho bz$. By denoting the class of all$\alpha$-selfdecomposable distributions on $\mathbb R ^d$ by$L^{\leftangle\alpha ightangle}\mathbb R ^d$, we define in this paper asequence of nested subclasses of $L^{\leftangle\alpha ightangle}\mathbb R^d$, and investigate several properties of them by two ways. One is by usinglimit theorems and the other is by using mappings of infinitely divisibledistributions.

Author: ** Makoto Maejima, Yohei Ueda**

Source: https://arxiv.org/