On the Breiman conjecture

Authors:
Péter Kevei and David M. Mason

Journal:
Proc. Amer. Math. Soc. **144** (2016), 4043-4053

MSC (2010):
Primary 60F05

DOI:
https://doi.org/10.1090/proc/13024

Published electronically:
February 17, 2016

MathSciNet review:
3513559

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 for $F\in \mathcal {F}$ in a specified class of distributions $\mathcal {F}$, whenever $\sum X_{i}Y_{i}/\sum Y_{i}$ converges in distribution to a nondegenerate limit 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 $1<\alpha <2$.

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Additional Information

**Péter Kevei**

Affiliation:
Center for Mathematical Sciences, Technische Universität München, Boltzmann- straße 3, 85748 Garching, Germany

MR Author ID:
834278

Email:
peter.kevei@tum.de

**David M. Mason**

Affiliation:
Department of Applied Economics and Statistics, University of Delaware, 213 Townsend Hall, Newark, Delaware 19716

MR Author ID:
120985

Email:
davidm@udel.edu

Received by editor(s):
August 6, 2015

Received by editor(s) in revised form:
October 28, 2015

Published electronically:
February 17, 2016

Additional Notes:
The research of the first author was funded by a postdoctoral fellowship of the Alexander von Humboldt Foundation.

Communicated by:
Mark M. Meerschaert

Article copyright:
© Copyright 2016
American Mathematical Society