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The asymptotic distribution of randomly weighted sums and self-normalized sums. (English) Zbl 1245.60028
Summary: We consider the self-normalized sums T n = i=1 n X i Y i / i=1 n Y i , where {Y i :i1} are non-negative i.i.d. random variables, and {X i :i1} are i.i.d. random variables, independent of {Y i :i1}. The main result of the paper is that each subsequential limit law of T n is continuous for any non-degenerate X 1 with finite expectation, if and only if Y 1 is in the centered Feller class.
MSC:
60F05Central limit and other weak theorems
60E07Infinitely divisible distributions; stable distributions