The multivariate generalised central limit theorem with their domains of attraction was given by [Rvačeva](https://books.google.com.hk/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&redir_esc=y#v=onepage&q&f=false) (see also this [post](https://mathoverflow.net/questions/368932/reference-for-multivariate-generalised-clt)). The original paper is not very accessible on the internet, and neither can I find accessible books or papers stating both the statement and the proof. What I can find so far is its original statement:



> 
> $X$ is an $\mathbb R^d$ random vector, $\alpha<2$ and $h$ is a spectral measure. The following statements are equivalent:
> 
> 
> (1) $X$ is in the domain of attraction of a stable distribution $S(\alpha,h,0)$;
> 
> 
> (2) For any $h$-continuous set $A,B$ with $h(A)+h(B)\ne 0$ and positive real number $k$, $$\frac{\mathbb P(|X|>kR,X/|X|\in A)}{\mathbb P(|X|>R,X/|X|\in B)}\rightarrow k^{-\alpha}\frac{h(A)}{h(B)}$$ when $R$ goes to infinity.
> 
> 
> 


However, the second statement does not depend on the scaling of $h$, i.e. it holds for $\gamma h$ with any $\gamma>0$. This means (in the case $\alpha\ne 1$) it only classifies the equivalence classes of random variables, the elements of which differ only by a scaling. In the $\alpha=1$ case scaling corresponds with an extra shift.


My question is, is this really what the theorem says (as the original statement is inaccurate about not mentioning the equivalence class)? Or is there any way to also classify the scaling?



Stable law and the domains of attraction
1. As stated in the [linked post](https://mathoverflow.net/a/368974/36721), the domain of attraction to multidimensional stable distributions with for $\alpha<2$ was characterized by Rvačeva's [Theorem 4.2 (p. 196)](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). You can see the statement of this theorem by Rvačeva there (click on Preview).
2. You misread Rvačeva's Theorem 4.2 quite a bit: It says that a certain condition -- which looks a bit like your condition (2) but is in fact very different from your condition (2) -- (in conjunction with a certain other condition) is equivalent to the **distribution of $X$ being in the domain of attraction** to a multidimensional stable distribution (of course, **not itself being a stable law** with spectral measure $h$).


As for the scaling, that becomes quite simple once your condition (1) is corrected according to Point 2 above: as $X$ being **in the domain of attraction to $S(\alpha,h,0)$**. (Why have you not corrected (1) appropriately yet?) [Indeed](https://en.wikipedia.org/wiki/Multivariate_stable_distribution), if $Y\sim S(\alpha,h,0)$ with $\alpha\in(0,1)\cup(1,2)$, then for any real $t>0$ we have $t^{1/\alpha}Y\sim S(\alpha,th,0)$; for $\alpha=1$ an additional shift will in general be needed.


Therefore, if $X$ is in the domain of attraction to $S(\alpha,h,0)$, then $X$ is in the domain of attraction to $S(\alpha,th,0)$ for every real $t>0$. Indeed, the condition that $X$ is in the domain of attraction to $S(\alpha,h,0)$ means that
$$\frac1{B\_n}\,\sum\_1^n X\_i-a\_n\to Y\sim S(\alpha,h,0),$$
where the $X\_i$'s are iid copies of $X$, the $B\_n$'s are some positive real numbers, the $a\_n$'s are some real numbers, and the convergence is in distribution. But then, obviously, for each $\alpha\in(0,1)\cup(1,2)$ and each real $t>0$,
$$\frac1{B\_n/t^{1/\alpha}}\,\sum\_1^n X\_i-t^{1/\alpha}a\_n\to t^{1/\alpha}Y\sim S(\alpha,th,0),$$
so that $X$ is in the domain of attraction to $S(\alpha,th,0)$ as well; for $\alpha=1$ an additional shift will in general be needed.


This resolves your problem with the scaling.


