Strong central limit theorem for isotropic random walks in \({\mathbb{R}^d}\)
Let \(d\geq2\), and let \(\sigma_d\) and \(\mu_d\) be the uniform distributions on the unit sphere and unit ball in \(\mathbb R^d\) respectively. Consider the associated normalized sums \(\tilde S_n:=n^{-1/2}\cdot \sum_{k=1}^n X_k\) for iid random variables \(X_k\) on \(\mathbb R^d\) with the distributions \(\sigma_d\) or \(\mu_d\) . The authors prove that the associated Lebesgue densities have a strong upper bound of the form \(c_d\cdot e^{-d|x|^2/2}\) independent of \(n\) for some constant \(c_d\) for \(\sigma_d\); for \(\mu_d\) and powers of \(\sigma_d\) and \(\mu_d\), a corresponding upper bound is constructed. These upper bounds lead to a strong version of a CLT for \(E(f(\tilde S_n))\) for \(n\to\infty\) for functions \(f\) which are integrable w.r.t. \(c_d\cdot e^{-d|x|^2/2}dx\). Moreover, the bounds have applications to strong hypercontractivity inequalities and strong logarithmic Sobolev inequalities for these isotropic random walks.
scientific article; zbMATH DE number 5968711
scientific article