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

Peter Kevei, David M. Mason

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\geq 1…