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The self-normalized Donsker theorem revisited
Volume 4, Issue 3 (2017), pp. 189–198
Peter Parczewski  

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https://doi.org/10.15559/17-VMSTA82
Pub. online: 18 September 2017      Type: Research Article      Open accessOpen Access

Received
18 May 2017
Revised
9 August 2017
Accepted
9 August 2017
Published
18 September 2017

Abstract

We extend the Poincaré–Borel lemma to a weak approximation of a Brownian motion via simple functionals of uniform distributions on n-spheres in the Skorokhod space $D([0,1])$. This approach is used to simplify the proof of the self-normalized Donsker theorem in Csörgő et al. (2003). Some notes on spheres with respect to $\ell _{p}$-norms are given.

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Keywords
Poincaré–Borel lemma Brownian motion Donsker theorem self-normalized sums

MSC2010
60F05 60F17

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