On the Breiman conjecture
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- by Péter Kevei and David M. Mason PDF
- Proc. Amer. Math. Soc. 144 (2016), 4043-4053 Request permission
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 $1<\alpha <2$.References
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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
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 4043-4053
- MSC (2010): Primary 60F05
- DOI: https://doi.org/10.1090/proc/13024
- MathSciNet review: 3513559