Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Empirical measures and random walks on compact spaces in the quadratic
  Wasserstein metric   Estimating the rate of convergence of the empirical measure of an i.i.d.
sample to the reference measure is a classical problem in probability theory.
Extending recent results of Ambrosio, Stra and Trevisan on 2-dimensional
manifolds, in this paper we prove sharp asymptotic and nonasymptotic upper
bounds for the mean rate in the quadratic Wasserstein metric $W_2$ on a
$d$-dimensional compact Riemannian manifold. Under a smoothness assumption on
the reference measure, our bounds match the classical rate in the optimal
matching problem on the unit cube due to Ajtai, Koml\'os, Tusn\'ady and
Talagrand. The i.i.d. condition is relaxed to stationary samples with a mixing
condition. As an example of a nonstationary sample, we also consider the
empirical measure of a random walk on a compact Lie group. Surprisingly, on
semisimple groups random walks attain almost optimal rates even without a
spectral gap assumption. The proofs are based on Fourier analysis, and in
particular on a Berry-Esseen smoothing inequality for $W_2$ on compact
manifolds, a result of independent interest with a wide range of applications.

7
The categories of this paper are: hep-th
19
The categories of this paper are: eess
30
The categories of this paper are: q-alg
37
The categories of this paper are: supr-con
16
The categories of this paper are: stat
10
The categories of this paper are: q-bio
14
The categories of this paper are: math-ph
26
The categories of this paper are: bayes-an
35
The categories of this paper are: mtrl-th
32
The categories of this paper are: atom-ph
29
The categories of this paper are: funct-an
1
The categories of this paper are: cs
13
The categories of this paper are: quant-ph
31
The categories of this paper are: ao-sci
3
The categories of this paper are: physics
36
The categories of this paper are: cmp-lg