We provide upper bounds of the expected Wasserstein distance between a
probability measure and its empirical version, generalizing recent results for
finite dimensional Euclidean spaces and bounded functional spaces. Such a
generalization can cover Euclidean spaces with large dimensionality, with the
optimal dependence on the dimensionality. Our method also covers the important
case of Gaussian processes in separable Hilbert spaces, with rate-optimal upper
bounds for functional data distributions whose coordinates decay geometrically
or polynomially. Moreover, our bounds of the expected value can be combined
with mean-concentration results to yield improved exponential tail probability
bounds for the Wasserstein error of empirical measures under Bernstein-type or
log Sobolev-type conditions.

本文提出一种泛化近期针对有限维欧几里得空间和有界函数空间的结果的，衡量概率测度和其经验版本之间期望 Wasserstein 距离的上界方法，并将其推广到具有大维度的欧几里得空间及分离的 Hilbert 空间中的 Gaussian process。此外，结合均值集中结果，给出了 Bernstein 型或 log Sobolev 型条件下，经验测度的 Wasserstein 误差的改进指数尾部概率界。

无界函数空间中 Wasserstein 距离下的经验测度的收敛与集中

Convergence and Concentration of Empirical Measures under Wasserstein  Distance in Unbounded Functional Spaces

Concentration inequalities quantify the deviation of a random variable from a
fixed value. In spite of numerous applications, such as opinion surveys or
ecological counting procedures, few concentration results are known for the
setting of sampling without replacement from a finite population. Until now,
the best general concentration inequality has been a Hoeffding inequality due
to Serfling [Ann. Statist. 2 (1974) 39-48]. In this paper, we first improve on
the fundamental result of Serfling [Ann. Statist. 2 (1974) 39-48], and further
extend it to obtain a Bernstein concentration bound for sampling without
replacement. We then derive an empirical version of our bound that does not
require the variance to be known to the user.

本文提出了一种改进的采样无替换的分布浓度不等式并且扩展至 Bernstein 浓度边界，同时还导出了一个经验版本的边界不需要用户知道方差。