Convergence of adapted empirical measures on Rd

Abstract

We consider empirical measures of ${\mathbb{R}}^{\mathit{d}}$-valued stochastic process in finite discrete-time. We show that the adapted empirical measure introduced in the recent work (Ann. Appl. Probab. 32 (2022) 529–550) by Backhoff et al. in compact spaces can be defined analogously on ${\mathbb{R}}^{\mathit{d}}$, and that it converges almost surely to the underlying measure under the adapted Wasserstein distance. Moreover, we quantitatively analyze the convergence of the adapted Wasserstein distance between those two measures. We establish convergence rates of the expected error as well as the deviation error under different moment conditions. Under suitable integrability and kernel assumptions, we recover the optimal convergence rates of both expected error and deviation error. Furthermore, we propose a modification of the adapted empirical measure with projection on a nonuniform grid, which obtains the same convergence rate but under weaker assumptions.