Greater contribution in a sum of independent random variables

Question

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

Answer 1

Heyde 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.

Answer 2

The paper by Heyde discussed in Iosif Pinelis' excellent answer concerns a more general situation where the summands are not stable but rather there is convergence to a stable law after normalization. The case where the summands themselves are stable is easier, since one can work directly with the tail estimate $P(X_1>R) \le CR^{-\alpha}$. This goes back to Levy (1925), see e.g. Feller volume 2 or [1]. To rule out the specific scenario you described where $n^\nu$ variables have magnitude $n^\mu$, a union bound suffices, since $$(1-\nu)n^\nu<\alpha \mu n^{\nu} \,.$$

[1] https://link.springer.com/content/pdf/10.1023/A:1009908026279.pdf