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.




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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>



Reference for multivariate generalised CLT
$\newcommand\al\alpha\newcommand\R{\mathbb R}$The domains of attraction to multidimensional stable distributions are characterized by Rvačeva's [Theorems 4.1 (p. 194, for $\al=2$) and 4.2 (p. 196, for $\al<2$)](https://books.google.com/books?id=5d__hmE9q34C&pg=PA183&lpg=PA183&dq=%22On%20domains%20of%20attraction%20of%20multi-dimensional%20distributions%22&source=bl&ots=ilMkRwGxz8&sig=ACfU3U0pqpq33nNTpI8Ya0Jkx7Lny0uKYg&hl=en&sa=X&ved=2ahUKEwjQ25uDgZbrAhUTU80KHWDoA0kQ6AEwBnoECAoQAQ#v=onepage&q=%22On%20domains%20of%20attraction%20of%20multi-dimensional%20distributions%22&f=false).


The domains of strict attraction to multidimensional strictly stable distributions are characterized by Shimura's [Theorems 3.1 (p. 354, for $\al=1$), 3.3 (p. 356, for $\al\in(0,1)$), 3.4 (p. 356, for $\al\in(1,2)$), and 3.5 (p. 356, for $\al=2$)](https://www.jstage.jst.go.jp/article/math1924/16/2/16_2_351/_pdf).


