  A random vector ${\bf X}$ is weakly stable iff for all $a,b \in \mathbb{R}$
there exists a random variable $\Theta$ such that $a{\bf X} + b {\bf X}'
\stackrel{d}{=} {\bf X} \Theta$, where $X'$ is an independent copy of $X$ and
$\Theta$ is independent of $X$. This is equivalent (see [12]) with the
condition that for all random variables $Q_1, Q_2$ there exists a random
variable $\Theta$ such that $$ {\bf X} Q_1 + {\bf X}' Q_2 \stackrel{d}{=} {\bf
X} \Theta, \quad \quad \quad \quad \quad (\ast) $$ where ${\bf X}, {\bf X}',
Q_1, Q_2, \Theta$ are independent. In this paper we define weak generalized
convolution of measures defined by the formula $$ {\mathcal L}(Q_1)
\otimes_{\mu} {\mathcal L}(Q_2) = {\mathcal L}(\Theta), $$ if the equation
$(\ast)$ holds for ${\bf X}, Q_1, Q_2, \Theta$ and $\mu = {\mathcal L}(X)$. We
study here basic properties of this convolution and basic properties of
distributions which are infinitely divisible in the sense of this convolution.
The main result of this paper is the analog of the L\'evy-Khintchine
representation theorem for $\otimes_{\mu}$-infinitely divisible distributions.

Probability