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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 \(T_{n}=\sum_{i=1}^{n}X_{i}Y_{i}/\sum_{i=1}^{n}Y_{i}\), where \(\{ Y_{i} : i\geq 1 \}\) are non-negative i.i.d. random variables, and \(\{ X_{i} : i\geq 1 \} \) are i.i.d. random variables, independent of \(\{ Y_{i} : i \geq 1 \}\). 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:

60F05 Central limit and other weak theorems
60E07 Infinitely divisible distributions; stable distributions
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