When Does a Randomly Weighted Self-normalized Sum Converge in Distribution?
David M. Mason, University of Delaware, USA
Joel Zinn, Texas A&M University, USA
Abstract
We determine exactly when a certain randomly weighted,
self--normalized sum converges in distribution, partially verifying a 1965
conjecture of Leo Breiman. We, then, apply our results to characterize the
asymptotic distribution of relative sums and to provide a short proof of a
1973 conjecture of Logan, Mallows, Rice and Shepp on the asymptotic
distribution of self--normalized sums in the case of symmetry.
Full text: PDF | PostScript
Copyright for articles published in this journal is retained by the authors, with first publication rights granted to the journal. By virtue of their appearance in this open access journal, articles are free to use, with proper attribution, in educational and other non-commercial settings.
The authors of papers published in EJP/ECP retain the copyright. We ask for the permission to use the material in any form. We also require that the initial publication in EJP or ECP is acknowledged in any future publication of the same article.
Before a paper is published in the Electronic Journal of Probability or Electronic Communications in Probability we must receive a hard-copy of the copyright form. Please mail it to
Philippe Carmona
Laboratoire Jean Leray UMR 6629
Universite de Nantes,
2, Rue de la Houssinière BP 92208
F-44322 Nantes Cédex 03
France
You can also send it by FAX: (33|0) 2 51 12 59 12 to the attention of Philippe Carmona.
The preferred way is to send a scanned (jpeg or pdf) copy of the signed copyright form to the managing editor Philippe Carmona at ejpecpme@math.univ-nantes.fr.
If a paper has several authors, the corresponding author signs the copyright form
on behalf of all the authors.