When Does a Randomly Weighted Self-normalized Sum Converge in Distribution?
نویسندگان
چکیده
منابع مشابه
When Does a Randomly Weighted Self–normalized Sum Converge in Distribution?
We determine exactly when a certain randomly weighted self–normalized sum converges in distribution, partially verifying a 1965 conjecture of Leo Breiman, and 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...
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We consider the self-normalized sums Tn = ∑n i=1 XiYi/ ∑n i=1 Yi, where {Yi : i ≥ 1} are non-negative i.i.d. random variables, and {Xi : i ≥ 1} are i.i.d. random variables, independent of {Yi : i ≥ 1}. The main result of the paper is that each subsequential limit law of Tn is continuous for any non-degenerate X1 with finite expectation, if and only if Y1 is in the centered Feller class.
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ژورنال
عنوان ژورنال: Electronic Communications in Probability
سال: 2005
ISSN: 1083-589X
DOI: 10.1214/ecp.v10-1135