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

TL;DR

This paper investigates the asymptotic behavior of self-normalized sums involving independent random variables, establishing conditions under which their limit laws are continuous, specifically when the weights are in the centered Feller class.

Contribution

It characterizes the limiting distribution of self-normalized sums with random weights, showing they are continuous if and only if the weights belong to the centered Feller class.

Findings

Limit laws of self-normalized sums are continuous under the centered Feller class condition.

The results apply to sums with non-negative i.i.d. weights and independent summands.

Provides a necessary and sufficient condition for the continuity of subsequential limit laws.

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.