Convergence of the adapted empirical measure for mixing observations
The adapted Wasserstein distance $\mathcal{AW}$ is a modification of the classical Wasserstein metric, that provides robust and dynamically consistent comparisons of laws of stochastic processes, and has proved particularly useful in the analysis of stochastic control problems, model uncertainty, and mathematical finance. In applications, the law of a stochastic process $μ$ is not directly observed, and has to be inferred from a finite number of samples. As the empirical measure is not $\mathcal{AW}$-consistent, Backhoff, Bartl, Beiglböck and Wiesel introduced the adapted empirical measure $\widehatμ^N$, a suitable modification, and proved its $\mathcal{AW}$-consistency when observations are i.i.d. In this paper we study $\mathcal{AW}$-convergence of the adapted empirical measure $\widehatμ^N$ to the population distribution $μ$, for observations satisfying a generalization of the $η$-mixing condition introduced by Kontorovich and Ramanan. We establish moment bounds and sub-exponential concentration inequalities for $\mathcal{AW}(μ,\widehatμ^N)$, and prove consistency of $\widehatμ^N$. In addition, we extend the Bounded Differences inequality of Kontorovich and Ramanan for $η$-mixing observations to uncountable spaces, a result that may be of independent interest. Numerical simulations illustrating our theory are also provided.