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

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Abstract

Let X = {X n } n≥1 and Y = {Y n } 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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Authors and Affiliations

  1. Département de mathématique, Université libre de Bruxelles (ULB), Bd. Triomphe, CP210, Brussels, Belgium

    Siegfried Hörmann

  2. Faculté des Sciences, de la Technologie et de la Communication, Université du Luxembourg, Campus Kirchberg, Mathematics Research Unit, BLG, 6 rue Richard Coudenhove-Kalergi, L-1359, Luxembourg, Grand Duchy of Luxembourg

    Yvik Swan

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  1. Siegfried Hörmann
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  2. Yvik Swan
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Correspondence to Siegfried Hörmann.

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Communicated by István Berkes

Research supported by the Banque Nationale de Belgique and the Communauté française de Belgique — Actions de Recherche Concertées.

Research supported by a Mandat de Chargé de Recherche from the Fonds National de la Recherche Scientifique, Communauté française de Belgique.

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Hörmann, S., Swan, Y. A note on the normal approximation error for randomly weighted self-normalized sums. Period Math Hung 67, 143–154 (2013). https://doi.org/10.1007/s10998-013-4789-8

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