  The adapted Wasserstein distance controls the calibration errors of optimal
values in various stochastic optimization problems, pricing and hedging
problems, optimal stopping problems, etc. Motivated by approximating the true
underlying distribution by empirical data, we consider empirical measures of
$\mathbb{R}^d$-valued stochastic process in finite discrete-time. It is known
that the empirical measures do not converge under the adapted Wasserstein
distance. To address this issue, we consider convolutions of Gaussian kernels
and empirical measures as an alternative, which we refer to the
Gaussian-smoothed empirical measures. By setting the bandwidths of Gaussian
kernels depending on the number of samples, we prove the convergence of the
Gaussian-smoothed empirical measures to the true underlying measure in terms of
mean, deviation, and almost sure convergence. Although Gaussian-smoothed
empirical measures converge to the true underlying measure and can potentially
enlarge data, they are not discrete measures and therefore not applicable in
practice. Therefore, we combine Gaussian-smoothed empirical measures and the
adapted empirical measures in \cite{acciaio2022convergence} to introduce the
adapted smoothed empirical measures, which are discrete substitutes of the
smoothed empirical measures. We establish the polynomial mean convergence rate,
the exponential deviation convergence rate and the almost sure convergence of
the adapted smoothed empirical measures.
