Classes of measures closed under mixing and convolution. Weak stability
Volume 167 / 2005
Abstract
For a random vector $X$ with a fixed distribution $\mu$ we construct a class of distributions ${\cal M}(\mu)= \{ \mu\circ\lambda: \lambda\in{\cal P}\}$, which is the class of all distributions of random vectors $X {\mit\Theta}$, where $ {\mit\Theta}$ is independent of $X$ and has distribution $\lambda$. The problem is to characterize the distributions $\mu$ for which ${\cal M}(\mu)$ is closed under convolution. This is equivalent to the characterization of the random vectors $X$ such that for all random variables ${\mit\Theta}_1, {\mit\Theta}_2$ independent of $X, X^{\prime}$ there exists a random variable ${\mit\Theta}$ independent of $X$ such that \[ X {\mit\Theta}_1 + X^{\prime}{\mit\Theta}_2 \stackrel{d}{=} X {\mit\Theta}. \] We show that for every $X$ this property is equivalent to the following condition: \[ \forall a,b \in {\mathbb R} \exists {\mit\Theta} \hbox{ independent of } X, \quad aX + b X^{\prime}\stackrel{d}{=} X {\mit\Theta}. \] This condition reminds the characterizing condition for symmetric stable random vectors, except that ${\mit\Theta}$ is here a random variable, instead of a constant. The above problem has a direct connection with the concept of generalized convolutions and with the characterization of the extreme points for the set of pseudo-isotropic distributions.
