Modern Stochastics: Theory and Applications

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 $\mathit{D}\mathrm{(}[0\mathrm{,}1]\mathrm{)}$. 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 ${\ell }_{\mathit{p}}$-norms are given.