On the Breiman conjecture

TL;DR

This paper investigates the conditions under which a normalized sum involving i.i.d. variables converges in distribution, showing that the underlying distribution must belong to the domain of attraction of a stable law with index less than 1.

Contribution

It proves that convergence of certain sums implies the underlying distribution belongs to the domain of attraction of a stable law with index less than 1, extending Breiman's conjecture.

Findings

Convergence in distribution implies $G$ is in the domain of attraction of a stable law with index < 1.

The class $$ includes variables with finite second moments and those in the domain of attraction of stable laws with index between 1 and 2.

The result confirms a specific case of Breiman's conjecture for a class of distributions.

Abstract

Let $Y_{1},Y_{2},\ldots $ be positive, nondegenerate, i.i.d. $G$ random variables, and independently let $X_{1},X_{2},\ldots $ be i.i.d. $F$ random variables. In this note we show that whenever $\sum X_{i}Y_{i}/\sum Y_{i}$ converges in distribution to nondegenerate limit for some $F\in \mathcal{F}$, in a specified class of distributions $\mathcal{F}$, then $G$ necessarily belongs to the domain of attraction of a stable law with index less than 1. The class $\mathcal{F}$ contains those nondegenerate $X$ with a finite second moment and those $X$ in the domain of attraction of a stable law with index $1<\alpha <2$.