preprint

Inserted: 19 may 2026

Year: 2026

Abstract:

We establish quantitative rates of convergence for the empirical estimation of probability measures by means of the Maximum Mean Discrepancy (MMD) with power kernel $K_q(x,y) = -
x-y
^q$, $q \in (0,2)$. The resulting discrepancy is the classical energy distance $$\mathcal E_q²(μ, ω) = -\frac{1}{2}\iint_{\mathbb{R}^d \times \mathbb{R}^d}
x-y
^q \, d(μ- ω)(x)\, d(μ- ω)(y),$$ and we ask how fast the best $N$-point empirical approximation $\inf_{μ_N \in \mathcal{P}^N}\mathcal{E}_q(μ_N,ω)$ decays as $N \to \infty$. Given a probability measure $ω$ on $\mathbb{R}^d$ satisfying an Ahlfors regularity condition of exponent $β$, we prove that the sharp two-sided bound $$\mathcal E_q(μ_N, ω) \asymp N^{-\frac{1}{2}\left(1 + \frac{q}β\right)}$$ holds both for the worst-case empirical measure $μ_N$ (lower bound, holding for every configuration of $N$ points) and for an optimally chosen empirical measure $μ_N$ (upper bound). This complements the qualitative consistency result of Fornasier and Hütter \cite{fornasier2014consistency}, who proved narrow convergence of the minimizers of $\mathcal E_q^2(\cdot, ω)$ over empirical measures without quantitative rates.