论文标题
$ \ mathbb {r}^{d} $上适应的经验措施的收敛
Convergence of Adapted Empirical Measures on $\mathbb{R}^{d}$
论文作者
论文摘要
我们考虑$ \ r^{d} $的经验度量 - 在有限的离散时间中有价值的随机过程。我们表明，Backhoff等人的最新工作{Backhoff2022222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222估算的经验度量。在紧凑的空间中可以在$ \ r^{d} $上类似地定义，并且它几乎可以肯定地收敛到适应的瓦斯坦斯坦距离下的基础措施。此外，我们定量分析了这两种措施之间改编的Wasserstein \添加{距离}的收敛性。我们建立了预期误差的收敛速率以及在不同力矩条件下的偏差误差。 \添加{在适当的集成性和内核假设下，我们恢复了预期误差和偏差误差的最佳收敛速率。}此外，我们提出了在非均匀网格上使用\ add {propportive {propportional {prosption {prosption {prosption}的修改，在非均匀的网格上，该量度的收敛速度相同，但在较弱的假设下。
We consider empirical measures of $\R^{d}$-valued stochastic process in finite discrete-time. We show that the adapted empirical measure introduced in the recent work \cite{backhoff2022estimating} by Backhoff et al. in compact spaces can be defined analogously on $\R^{d}$, 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 \add{distance} between those two measures. We establish convergence rates of the expected error as well as the deviation error under different moment conditions. \add{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 \add{projection} on a non-uniform grid, which obtains the same convergence rate but under weaker assumptions.
