Convergence of Empirical Measures for i.i.d. samples in $W^{-{\alpha}, p}$

Abstract

Given $N$ i.i.d. samples from a probability measure $\mu$ on $\mathbf{R}^d$, we study the rate of convergence of the empirical measure $\mu_N \to \mu$ in the negative Sobolev space $W^{-\alpha, p}$. When $W^{-\alpha, p}$ contains point measures (i.e. when $\alpha p > (p-1)d$), we show $\mathbf{E} \| \mu_N - \mu \|_{W^{-\alpha, p}}^p \leq C_d / N^{p/2}$ for an explicit dimensional constant $C_d$, and obtain a Gaussian tail bound. When $0 < \alpha p \leq d(p-1)$, we prove a similar result for Gaussian regularizations.