Annals of Probability

Donsker's theorem for self-normalized partial sums processes

Miklós Csörgő, Barbara Szyszkowicz, and Qiying Wu

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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 $V_n^2=\sum_{j=1}^{n}X_j^2$.

Article information

Ann. Probab., Volume 31, Number 3 (2003), 1228-1240.

First available in Project Euclid: 12 June 2003

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Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60F17: Functional limit theorems; invariance principles
Secondary: 62E20: Asymptotic distribution theory

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


Csörgő, Miklós; Szyszkowicz, Barbara; Wu, Qiying. Donsker's theorem for self-normalized partial sums processes. Ann. Probab. 31 (2003), no. 3, 1228--1240. doi:10.1214/aop/1055425777.

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