  A random vector ${\bf X}$ is weakly stable iff for all $a,b\in \mathbb{R}$
there exists a random variable $\Theta$ such that $a{\bf X}+b{\bf
X}'\stackrel{d}{=}{\bf X}\Theta$. This is equivalent (see \cite{MOU}) with the
condition that for all random variables $Q_1,Q_2$ there exists a random
variable $\Theta$ such that $$ X Q_1 + X' Q_2 \stackrel{d}{=} X \Theta, $$
where ${\bf X},{\bf X}',Q_1,Q_2,\Theta$ are independent. In this paper we
define generalized convolution of measures defined by the formula $$ L(Q_1)
\oplus_{\mu} L(Q_2) = L(\Theta), $$ if the equation $(*)$ holds for ${\bf
X},Q_1,Q_2,\Theta$ and $\mu ={\cal L}(\Theta)$. We study here basic properties
of this convolution, basic properties of $\oplus_{\mu}$-infinitely divisible
distributions, $\oplus_{\mu}$-stable distributions and give a series of
examples.
