Kevei, Peter; Mason, David M. The asymptotic distribution of randomly weighted sums and self-normalized sums. (English) Zbl 1245.60028 Electron. J. Probab. 17, Paper No. 46, 21 p. (2012). 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. Cited in 3 Documents MSC: 60F05 Central limit and other weak theorems 60E07 Infinitely divisible distributions; stable distributions Keywords:self-normalized sums; Feller class; stable distributions PDFBibTeX XMLCite \textit{P. Kevei} and \textit{D. M. Mason}, Electron. J. Probab. 17, Paper No. 46, 21 p. (2012; Zbl 1245.60028) Full Text: DOI arXiv