# Greater contribution in a sum of independent random variables

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

• There is a typo in the quote: The arithmetic mean has the same distribution as $n^{-1+1/\alpha}X_1$, note the minus sign. Apr 1 at 16:40

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.
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 $$(1-\nu)n^\nu<\alpha \mu n^{\nu} \,.$$