The self-normalized Donsker theorem revisited

TL;DR

This paper extends the Poincaré--Borel lemma to approximate Brownian motion using uniform distributions on n-spheres, simplifying the proof of the self-normalized Donsker theorem in the Skorokhod space.

Contribution

It introduces a new approach to approximate Brownian motion and simplifies the proof of the self-normalized Donsker theorem using sphere-based functionals.

Findings

Extended Poincaré--Borel lemma for Brownian motion approximation

Simplified proof of the self-normalized Donsker theorem

Notes on spheres with respect to p-norms

Abstract

We extend the Poincar\'{e}--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 Cs\"{o}rg\H{o} et al. (2003). Some notes on spheres with respect to $\ell_p$-norms are given.