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.

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Pages: 70-81

Publication Date: April 16, 2005

DOI: 10.1214/ECP.v10-1135

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