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

Let $\text{bX}=\{X_n{\}}_{n\ge 1}$ and $\text{bY}=\{Y_n{\}}_{n\ge 1}$ be two independent random sequences. We obtain rates of convergence to the normal law of randomly weighted self-normalized sums $${\psi }_n(\text{bX},\text{bY})=\sum _{i=1}^n{X}_i{Y}_i\mathrm{/}{V}_n,V_n=\sqrt{Y_1^2+\mathrm{.}\mathrm{.}\mathrm{.}+Y_n^2}\mathrm{.}$$ 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.