Convergence Rates for the Isotrope Discrepancy

Abstract

For each sequence of independent and identically distributed $\mathbb{R}^k$-valued random variables, $k \geqq 3$, with distribution $\mu$ defined on some probability space $(\Omega, \mathscr{F}, \mathbb{P})$, let $$D_n(\omega, \mu) \equiv \sup_C |\mu_n^\omega(C) - \mu(C)|,\quad n \in \mathbb{N}, \omega \in \Omega,$$ be the so-called isotrope discrepancy (at stage $n$), where $\mu_n^\omega$ denotes the $n$th empirical distribution pertaining to $\omega$ and where the supremum is taken over the class of all convex measurable sets $C \subset \mathbb{R}^k$. It is proved that almost everywhere and in the mean $D_n(\bullet)$ converges to zero as $n^{-2/(k+1)}$ (up to a logarithmic factor), provided $\mu$ is absolutely continuous with a bounded density function of compact support.