$\newcommand{\R}{\mathbb{R}}$

The fundamental paper in this area is the one by [Sharpe][1]. 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 nn $\R^d$ is stable. 


[1]: https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=2ahUKEwiO5KTD2NjcAhVIQ6wKHfyQD3kQFjAAegQIABAB&url=http%3A%2F%2Fwww.ams.org%2Ftran%2F1969-136-00%2FS0002-9947-1969-0238365-3%2FS0002-9947-1969-0238365-3.pdf&usg=AOvVaw10G_cIEAPk6JZJ3Yu2wS4l