Two-Sample Inference for Gaussian-Smoothed Wasserstein Costs with Finite Moments

Gaussian smoothing has emerged as an effective technique for reducing the sample complexity of optimal transport. In this paper, we study the two-sample plug-in estimator of the Gaussian-smoothed Wasserstein cost $T_p^{(\sigma )}(\mu ,\nu )=W_p(\mu \ast {\gamma }_{\sigma },\nu \ast {\gamma }_{\sigma }{)}^p$ on ${\mathbb{R}}^d$. For fixed smoothing and finite polynomial moments $M_{q_{\mu }}(\mu )<\mathrm{\infty }$, $M_{q_{\nu }}(\nu )<\mathrm{\infty }$, with $q_{\mu },q_{\nu }>p$, we establish upper bounds in probability of order ${\rho }_{q_{\mu },p,d}(m)+{\rho }_{q_{\nu },p,d}(n)$. Here ${\rho }_{q,p,d}(N)=N^{-(q-p)\mathrm{/}(q+d)}$ for $p<q<d+2p$, $N^{-1\mathrm{/}2}\log N$ at $q=d+2p$, and $N^{-1\mathrm{/}2}$ for $q>d+2p$. This order also holds in expectation under $q_{\mu },q_{\nu }\ge 2p$. When the smoothed population distance is positive, the cost bound yields this rate for the distance itself. For $p>1$ and $q_{\mu },q_{\nu }>d+2p$, we also derive a first-order expansion, a separated two-sample central limit theorem, and a sample-splitting variance estimator.