Publication 25-CNA-024

Convergence of Empirical Measures for I.I.D. Samples in $W^{-\alpha, p}$

Gautam Iyer
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
gautam@math.cmu.edu

Raghavendra Venkatraman
Department of Mathematics
University of Utah
Salt Lake City, UT
raghav@math.utah.edu

Abstract: Given $N$ i.i.d. samples from a probability measure $\mu$ on $\mathbb{R}^d$, 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 $\alpha p \geq (p - 1)d)$, we show $E\,\bigl\| \mu_{N} - \mu \bigr\|_{W^{-\alpha,p}}^{\,p} \leq \dfrac{C_{d}}{N^{p/2}}$ for an explicit dimensional constant $C_{d}$, and obtain a Gaussian tail bound. When $0 \leq \alpha p \leqslant d(p - 1)$, we prove a similar result for Gaussian regularizations.

Get the paper in its entirety as  25-CNA-024.pdf

---

«   Back to CNA Publications