The asymptotic distribution of randomly weighted sums and self-normalized sums

Abstract

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.