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July 2003 Donsker's theorem for self-normalized partial sums processes
Miklós Csörgő, Barbara Szyszkowicz, Qiying Wu
Ann. Probab. 31(3): 1228-1240 (July 2003). DOI: 10.1214/aop/1055425777


Let $X, X_1, X_2,\ldots$ be a sequence of nondegenerate i.i.d. random variables with zero means. In this paper we show that a self-normalized version of Donsker's theorem holds only under the assumption that X belongs to the domain of attraction of the normal law. A thus resulting extension of the arc sine law is also discussed. We also establish that a weak invariance principle holds true for self-normalized, self-randomized partial sums processes of independent random variables that are assumed to be symmetric around mean zero, if and only if $\max_{1\le j\le n}|X_j|/V_n\to_P 0$, as $n\to\infty$, where $V_n^2=\sum_{j=1}^{n}X_j^2$.


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Miklós Csörgő. Barbara Szyszkowicz. Qiying Wu. "Donsker's theorem for self-normalized partial sums processes." Ann. Probab. 31 (3) 1228 - 1240, July 2003.


Published: July 2003
First available in Project Euclid: 12 June 2003

zbMATH: 1045.60020
MathSciNet: MR1988470
Digital Object Identifier: 10.1214/aop/1055425777

Primary: 60F05 , 60F17
Secondary: 62E20

Keywords: arc sine law. , Donsker's theorem , self-normalized sums

Rights: Copyright © 2003 Institute of Mathematical Statistics


Vol.31 • No. 3 • July 2003
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