  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.
