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

Siegfried Hoermann; Yvik Swan

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arXiv · 1109.5812

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

Siegfried Hoermann·Yvik Swan

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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 $$ ψ_n(\bX,\bY)=\sum_{i=1}^nX_iY_i/V_n,\quad V_n=\sqrt{Y_1^2+...+Y_n^2}. $$ 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.

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Siegfried Hoermann, Yvik Swan. 2011-09-27. A note on the normal approximation error for randomly weighted self-normalized sums. https://arxiv.org/abs/1109.5812

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