The Asymptotic Distribution of Randomly Weighted Sums and Self-normalized Sums

We consider the self-normalized sums $T_n={\sum }_{i=1}^n{X}_i{Y}_i\mathrm{/}{\sum }_{i=1}^n{Y}_i$, where $Y_i:i\ge 1$ are non-negative i.i.d. random variables, and $X_i:i\ge 1$ are i.i.d. random variables, independent of $Y_i:i\ge 1$. 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.