Abstract
We establish some deviation inequalities, moment bounds and almost sure results for the Wasserstein distance of order $p\in [1, \infty )$ between the empirical measure of independent and identically distributed ${\mathbb R}^d$-valued random variables and the common distribution of the variables. We only assume the existence of a (strong or weak) moment of order $rp$ for some $r>1$, and we discuss the optimality of the bounds.
Highlights
Introduction and notationsWe begin with some notations, that will be used all along the paper
Let X denote a random variable with distribution μ
We prove deviation inequalities, moment inequalities and almost sure results for the quantity Wp(μn, μ), when X has a weak or strong moment of order rp for r > 1
Summary
We begin with some notations, that will be used all along the paper. Let X1, . . . , Xn be n independent and identically distributed (i.i.d.) random variables with values in Rd (d ≥ 1), with common distribution μ. Since the rates depend on the dimension d, it is easy to see that they cannot be optimal for all measures: for instance the rates will be faster as announced if the measure μ is supported on a linear subspace of Rd with dimension strictly less than d This is not the end of the story, and the problem can be formulated in the general context of metric spaces (X, δ). Bach and Weed [2] obtain sharper results by generalizing some ideas going back to Dudley ([14], case p = 1) They introduce the notion of Wasserstein dimension d∗p(μ) of the measure μ, and prove that np/sE(Wpp(μn, μ)) converges to 0 for any s > d∗p(μ) (with sharp lower bounds in most cases). We shall use the notation f (n, μ, x) g(n, μ, x), which means that there exists a positive constant C, not depending on n, μ, x such that f (n, μ, x) ≤ Cg(n, μ, x) for all positive integer n and all positive real x
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
