CONVERGENCE OF ADAPTED EMPIRICAL MEASURES ON Rd

• 成果类型: Article
• 署名作者: Acciaio, Beatrice; Hou, Songyan
• 署名单位: Swiss Federal Institutes of Technology Domain; ETH Zurich
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/24-AAP2082
• 发表日期: 2024
• 页码: 4799-4835
• 关键词: wasserstein distance; Occupation measures; causal transport; approximation; inequalities; constant

摘要

nite discrete-time. We show that the adapted empirical measure introduced in in compact spaces can be defined analogously on Rd, 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.