We know that: \begin{equation} \sum_{i=1}^n X_i \stackrel{d}{=} cX+d, \end{equation} when $X_i, X$ are i.i.d. random variables which come from any $\alpha$-stable distribution, $c, d \in \mathbb{R}$. Does the relation also occurs when $ n \rightarrow \infty$?
To be specific, let $X_i, X$- as above. Do we have series of $a_n \in \mathbb{R}^+$ and $b_n\in \mathbb{R}$ such that:
\begin{equation} \lim_{n \rightarrow \infty} \sum_{i=1}^n a_n(X_i -b_n)\stackrel{d}{=}X \end{equation}