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 $\mathbf{E}\Vert {\mu }_N-\mu {\Vert }_{W^{-\alpha ,p}}^p\le C_d/N^{p/2}$ for an explicit dimensional constant $C_d$, and obtain a Gaussian tail bound. When $0<\alpha p\le d(p-1)$, we prove a similar result for Gaussian regularizations.

21 pages, 0 figures