On the mean speed of convergence of empirical and occupation measures in Wasserstein distance

Abstract

In this work, we provide non-asymptotic bounds for the average speed of convergence of the empirical measure in the law of large numbers, in Wasserstein distance. We also consider occupation measures of ergodic Markov chains. One motivation is the approximation of a probability measure by finitely supported measures (the quantization problem). It is found that rates for empirical or occupation measures match or are close to previously known optimal quantization rates in several cases. This is notably highlighted in the example of infinite-dimensional Gaussian measures.1.1. Main result and first consequences.

Constructive quantization: Approximation by empirical measures

“…Closely related problems are the bipartite matching problem [DY95] and the traveling salesman problem [BHH59]. Interestingly, there has been progress on this class of problems in several aspects [BB11,BG11] parallel to our research.…”

“…It improves the asymptotic estimates of [HK94] which considered the case p = 2 and was of non-optimal order. Interestingly, it is also possible to give estimates for E[ρ p (µ,μ N )] for compactly supported measures µ in general metric spaces based on covering numbers [BG11].…”

Inference for Empirical Wasserstein Distances on Finite Spaces

“…() considering for μ 0 the uniform distribution on the unit square, followed by Talagrand (, ) for the uniform distribution in higher dimensions and Horowitz and Karandikar () giving bounds on mean rates of convergence. Boissard and Gouic () and Fournier and Guillin () gave general deviation inequalities for the empirical Wasserstein distance on metric spaces. For a discussion in the light of our distributional limit results see Section .…”

Some recent developments in functional inequalities

“…We now quote one important application from [BLG12]. We tackle the case where E is a separable Banach space with norm · , and µ a centered Gaussian random variable with values in E. The couple (E, µ) is called a (separable) Gaussian Banach space.…”

“…Section 3 contains the contribution of Emmanuel Boissard based on his paper [BLG12] (joint with Thomas Le Gouic) and deals with the question of finding a good approximation of a continuous distribution by a discrete one. The quality of the approximation is measured with respect to the Wasserstein distances W p .…”