The self-normalized Donsker theorem revisited

Peter Parczewski

doi:10.15559/17-vmsta82

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.

Highlights

Let Sn−1(d) = {x ∈ Rn : x = d} be the (n − 1)-sphere with radius d, where · denotes the Euclidean norm
The celebrated Poincaré–Borel lemma is a Gaussian distribution by projections of the the classical uniform mreesauslut roenothneSanp−p1r(o√xinm)aatisonn tends to uniform infinity: measure
Following the historical notes in [6, Section 6] on the earliest reference to this result by Émile Borel, we acquire the usual practice to speak about the Poincaré–Borel lemma

Summary

Introduction

Following the historical notes in [6, Section 6] on the earliest reference to this result by Émile Borel, we acquire the usual practice to speak about the Poincaré–Borel lemma. Among other fields, this convergence stimulated the development of the infinitedimensional functional analysis (cf [12]) as well as the concentration of measure theory (cf [10, Section 1.1]). This convergence stimulated the development of the infinitedimensional functional analysis (cf [12]) as well as the concentration of measure theory (cf [10, Section 1.1]) It inspired to consider connections of the Wiener measure and the uniform measure on an infinite-dimensional sphere [21].

Poincaré–Borel lemma for Brownian motion

The self-normalized Donsker theorem