I know that one can generalise the classical CLT in terms of heavy tail distributions, namely, for any i.i.d. random variables $X_i$, $$\frac{X_1+\cdots+X_n}{n^{1/\alpha}}\rightarrow S(\alpha,\beta,\gamma,\delta)$$ in distribution sense, whenever $X_i$ belongs to the domain of attraction of its corresponding limit. When $\alpha$ takes $2$, this law becomes the classical CLT.

To push this to multivariate versions, I also see https://en.wikipedia.org/wiki/Multivariate_stable_distribution, in which is multivariate stable distribution is introduced.

Where can I find a reference for such multivariate generalised central limit theorem? A textbook reference would be great.

P.S. This is also an existing question in MSE: https://math.stackexchange.com/questions/3786947/reference-for-multivariate-generalised-clt?noredirect=1#comment7800922_3786947