Electronic Communications in Probability

When Does a Randomly Weighted Self-normalized Sum Converge in Distribution?

David Mason and Joel Zinn

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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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Electron. Commun. Probab., Volume 10 (2005), paper no. 8, 70-81.

Accepted: 16 April 2005
First available in Project Euclid: 4 June 2016

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Mason, David; Zinn, Joel. When Does a Randomly Weighted Self-normalized Sum Converge in Distribution?. Electron. Commun. Probab. 10 (2005), paper no. 8, 70--81. doi:10.1214/ECP.v10-1135. https://projecteuclid.org/euclid.ecp/1465058073

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