How to prove that is a consistent estimator? Let $\hat{\pi}^N$ be an AW-consistent estimator of $\pi$ (i.e., $\hat{\pi}^N$ is a strongly consistent estimator of $\pi$ under adapted (or called nested) Wasserstein distance $AW(\pi, \hat{\pi}^N)\to 0 $ a.s.).
How to prove that $W(\hat{\pi}^N)$ is a consistent estimator of $W(\pi)$?
 A: Take $\pi^N$ with $AW(\pi^N, \pi) \leq \frac{1}{N}$, where we denote by $\mu^N$ and $\nu^N$ the marginals of $\pi^N$.
Note that by the backward induction for $AW$ (cf. here), it holds
$$
AW(\pi, \pi^N) = \inf_{\kappa_1 \in \Pi(\mu, \mu^N)} d_{X_1}(x_1, y_1) + W_1(\pi_{x_1}, \pi^N_{y_1}) \kappa_1(dx_1, dy_1),
$$
and thus we can choose $\kappa_1^N \in \Pi(\mu, \mu^N)$ such that
$$
\int W_1(\pi_{x_1}, \pi^N_{y_1}) \kappa_1^N(dx_1, dy_1) \leq \frac{1}{N},
$$
and further clearly the second marginal converges, i.e., $W_1(\nu^N, \nu) \leq \frac{1}{N}$.
By applying twice the triangle inequality, we get
\begin{align}
\int W_1(\pi_{x_1}, \nu) \mu(dx_1) - \int W_1(\pi^N_{y_1}, \nu^N) \mu^N(dy_1) &= \int W_1(\pi_{x_1}, \nu) - W_1(\pi^N_{y_1}, \nu^N) \kappa_1(dx_1, dy_1) \\
&\leq \int W_1(\nu, \nu^N) + W_1(\pi_{x_1}, \pi_{y_1}^N) \kappa_1^N(dx_1, dy_1) \\
&\leq \frac{2}{N}
\end{align}
and vice versa.
Since the denominator of $W(\pi)$ should be strictly larger than zero (and converges as well since $W_1(\nu^N, \nu)$ goes to zero), we get that the estimator is consistent.
