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 $Y_i:i\ge 1$ 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$iscontinuousforanynon-degenerate$X_1$withfiniteexpectation,ifandonlyif$Y_1$ is in the centered Feller class.