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 $n^{\mu(\alpha)}$ $\left( \text{with } \mu(\alpha) < \frac{1}{\alpha} \text{ and } \mu(\alpha)+\nu(\alpha) = \frac{1}{\alpha} \right)$?



Greater contribution in a sum of independent random variables
[Heyde](https://www.zbmath.org/?q=an%3A0182.22903) showed the following:



> 
> Let $X\_1,X\_2,\dots$ be a sequence of iid random variables such that, for $S\_n:=\sum\_1^n X\_i$,
> $S\_n/B\_n$ converges in distribution to a stable law, whose support is the whole real line and which has index $\alpha\ne1$.
> Then for any sequence $(x\_n)$ going to $\infty$,
> $$P(|S\_n|>x\_nB\_n)\sim nP(|X\_1|>x\_nB\_n) \\ 
> \sim P(\max\_{1\le i\le n}|X\_i|>x\_nB\_n).\tag{$\*$}$$
> 
> 
> 



> 
> *MR Reviewer's H. Kesten's remark*: Without much trouble one could have proved the (slightly sharper) one-sided form of ($∗$) obtained by dropping the absolute values around $S\_n$ and $X\_i$.
> 
> 
> 


It is seen from ($∗$) or, better, from its proof, that the overwhelming contribution to a large deviation of the sum $S\_n$ is provided, with an overwhelming probability, by the largest of summands $X\_i$. If you have access to MathSciNet, you can also see the many references to Heyde's paper, with further developments.


