  A number of researchers have independently introduced topologies on the set
of laws of stochastic processes that extend the usual weak topology. Depending
on the respective scientific background this was motivated by applications and
connections to various areas (e.g. Plug-Pichler - stochastic programming,
Hellwig - game theory, Aldous - stability of optimal stopping, Hoover-Keisler -
model theory). Remarkably, all these seemingly independent approaches define
the same adapted weak topology in finite discrete time. Our first main result
is to construct an adapted variant of the empirical measure that consistently
estimates the laws of stochastic processes in full generality. A natural
compatible metric for the weak adapted topology is the given by an adapted
refinement of the Wasserstein distance, as established in the seminal works of
Pflug-Pichler. Specifically, the adapted Wasserstein distance allows to control
the error in stochastic optimization problems, pricing and hedging problems,
optimal stopping problems, etc. in a Lipschitz fashion. The second main result
of this article yields quantitative bounds for the convergence of the adapted
empirical measure with respect to adapted Wasserstein distance. Surprisingly,
we obtain virtually the same optimal rates and concentration results that are
known for the classical empirical measure wrt. Wasserstein distance.
