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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
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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
Email: peter.kevei@tum.de

David M. Mason
Affiliation: Department of Applied Economics and Statistics, University of Delaware, 213 Townsend Hall, Newark, Delaware 19716
Email: davidm@udel.edu

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