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 $T_p^{(\sigma)}(\mu,\nu)=W_p(\mu*\gamma_\sigma,\nu*\gamma_\sigma)^p$ on $\R^d$. For fixed smoothing and finite polynomial moments $M_{q_\mu}(\mu)<\infty$, $M_{q_\nu}(\nu)<\infty$, 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 $\rho_{q,p,d}(N)=N^{-(q-p)/(q+d)}$ for $p<q<d+2p$, $N^{-1/2}\log N$ at $q=d+2p$, and $N^{-1/2}$ for $q>d+2p$. This order also holds in expectation under $q_\mu,q_\nu\ge2p$. 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.