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

AbstractLet X = {Xn}n≥1 and Y = {Yn}n≥1 be two independent random sequences. We obtain rates of convergence to the normal law of randomly weighted self-normalized sums $$\psi _n \left( {X,Y} \right) = \sum\nolimits_{i = 1}^n {{{X_i Y_i } \mathord{\left/ {\vphantom {{X_i Y_i } {V_n , V_n }}} \right. \kern-\nulldelimiterspace} {V_n , V_n }}} = \sqrt {Y_1^2 + \cdots + Y_n^2 } .$$. 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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