  Given a collection of multidimensional pairs $\{(X_i,Y_i):1 \leq i\leq n\}$,
we study the problem of projecting the associated suitably smoothed empirical
measure onto the space of martingale couplings (i.e. distributions satisfying
$\mathbb{E}[Y|X]=X$) using the adapted Wasserstein distance. We call the
resulting distance the smoothed empirical martingale projection distance
(SE-MPD), for which we obtain an explicit characterization. We also show that
the space of martingale couplings remains invariant under the smoothing
operation. We study the asymptotic limit of the SE-MPD, which converges at a
parametric rate as the sample size increases if the pairs are either i.i.d. or
satisfy appropriate mixing assumptions. Additional finite-sample results are
also investigated. Using these results, we introduce a novel consistent
martingale coupling hypothesis test, which we apply to test the existence of
arbitrage opportunities in recently introduced neural network-based generative
models for asset pricing calibration.
