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When does a randomly weighted self-normalized sum converge in distribution? (English) Zbl 1112.60014

Summary: We determine exactly when a certain randomly weighted, self-normalized sum converges in distribution, partially verifying a 1965 conjecture of L. Breiman [Teor. Veroyatn. Primen. 10, 351–360 (1965); translation in Theory Probab. Appl. 10, 323–331 (1965; Zbl 0147.37004)]. We, then, apply our results to characterize the asymptotic distribution of relative sums and to provide a short proof of a 1973 conjecture of B. F. Logan, C. L. Mallows, S. O. Rice and L. A. Shepp [Ann. Probab. 1, 788–809 (1973; Zbl 0272.60016)] on the asymptotic distribution of self-normalized sums in the case of symmetry.

MSC:

60F05 Central limit and other weak theorems
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