In section 2.4 (Summation of strictly stable random variables), page 54, of the book "chance and stability", Zolotarev, Uchaikin there is the following consideration :

The general relation of equivalence for sums S n of independent identically distributed strictly stable r.v.s $Y_i$ is of the form:

$\sum_{i=1}^n Y_i(\alpha; \beta) \sim n^{\frac{1}{\alpha}} Y(\alpha; \beta)$

As was noticed by Feller, these results have important and unexpected consequences. Let us consider, for example, a stable distribution with $\alpha<1$ The arithmetic mean $(X_1 + \dots + X_n )/n$ has the same distribution as $X_1$ $n^{-1+\frac{1}{α}}$.Meanwhile, the factor $n^{− 1+1/ α}$ tends to infinity as n grows. Without pursuing the rigor, we can say that the average of n variables $X_k$ turns out considerably greater than any fixed summand $X_k$ .This is possible only in the case where the maximum term

$M_n = max \{ X_1 , \dots, X_n \}$

grows extremely fast and gives the greatest contribution to the sum $S_n$. The more detailed analysis confirms this speculation.

How can be the last statement be proved in a more rigorous way? How can one exclude, for example, that the greater contribution to the sum $S_n$ comes from a subgroup of $n^{\nu(\alpha)}$ elements, each one of order $n^{\mu(\alpha)}$ $\left( \text{with } \mu(\alpha) < \frac{1}{\alpha} \text{ and } \mu(\alpha)+\nu(\alpha) = \frac{1}{\alpha} \right)$?