The self-normalized Donsker theorem revisited

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 ${\mathrm{\ell }}_p$-norms are given.