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

  • $\begingroup$ 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. $\endgroup$ Apr 1 at 16:40

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.


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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.