$\newcommand{\R}{\mathbb{R}}$
The fundamental paper in this area is the one by Sharpe. By Theorem 1 in that paper, a full probability measure $\mu$ on $\R^d$ is (operator) stable if and only if for each natural $n$ there are a $d\times d$ real matrix $B_n$ and a vector $b_n\in\R^d$ such that $X_1+\cdots+X_n$ equals $B_nX+b_n$ in distribution, where $X,X_1,X_2,\ldots$ are independent random vectors in $\R^d$ each with distribution $\mu$. (A measure $\mu$ on $\R^d$ is called full if the linear span of its support is $\R^d$.)
If a random vector $X$ in $\R^d$ has a stable distribution, we may say that $X$ itself is stable. Examples of stable random vectors include random vectors with independent stable coordinates. Also, if a a random vector $X$ in $\R^d$ is stable, then any nonsingular affine transformation $BX+b$ of $X$ is also obviously stable (here $B$ is any nonsingular $d\times d$ real matrix and $b\in\R^n$). In particular, any Gaussian distribution on $\R^d$ is stable.