A note on the normal approximation error for randomly weighted self-normalized sums

TL;DR

This paper investigates the rate at which randomly weighted self-normalized sums converge to a normal distribution, with implications for statistics like Student's t and correlation coefficients.

Contribution

It provides explicit convergence rates for the normal approximation of self-normalized sums involving independent random sequences.

Findings

Established convergence rates for self-normalized sums to the normal law.

Applied results to Student's t-statistic and empirical correlation coefficient.

Demonstrated the relevance of these rates in statistical inference.

Abstract

Let $\bX=\{X_n\}_{n\geq 1}$ and $\bY=\{Y_n\}_{n\geq 1}$ be two independent random sequences. We obtain rates of convergence to the normal law of randomly weighted self-normalized sums $$ \psi_n(\bX,\bY)=\sum_{i=1}^nX_iY_i/V_n,\quad V_n=\sqrt{Y_1^2+...+Y_n^2}. $$ These rates are seen to hold for the convergence of a number of important statistics, such as for instance Student's $t$-statistic or the empirical correlation coefficient.