Related papers: Berry-Esseen smoothing inequality for the Wasserst…
We consider a random walk $S_k$ with i.i.d. steps on a compact group equipped with a bi-invariant metric. We prove quantitative ergodic theorems for the sum $\sum_{k=1}^N f(S_k)$ with H\"older continuous test functions $f$, including the…
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…
We prove general upper estimates for the distance between two Borel probability measures in Wasserstein metric in terms of the Fourier transforms of the measures. We work in compact manifolds including the torus, the Euclidean unit sphere,…
In this note, we generalise a Bourgain's construction of finitely-supported symmetric measures whose Furstenberg measure has a smooth density from the case of $\mathrm{SL}_2(\mathbb{R})$ to that of a general simple Lie group. The proof is…
We investigate contraction of the Wasserstein distances on $\mathbb{R}^d$ under Gaussian smoothing. It is well known that the heat semigroup is exponentially contractive with respect to the Wasserstein distances on manifolds of positive…
We investigate properties of some extensions of a class of Fourier-based probability metrics, originally introduced to study convergence to equilibrium for the solution to the spatially homogeneous Boltzmann equation. At difference with the…
The Wasserstein distance between probability measures on compact spaces provides a natural invariant quantitative measure of equidistribution, which is partly similar to the classical discrepancy appearing in Erd\"os-Tur\'an type…
The Wasserstein distance between two probability measures on a metric space is a measure of closeness with applications in statistics, probability, and machine learning. In this work, we consider the fundamental question of how quickly the…
The asymptotic behaviour of empirical measures has been studied extensively. In this paper, we consider empirical measures of given subordinated processes on complete (not necessarily compact) and connected Riemannian manifolds with…
The object of study in this paper is the expected $2$-Wasserstein distance between the empirical measures of several point processes and their respective limit. For this, the main tool developed is a smoothing procedure in Euclidean spaces…
The Wasserstein distance quantifies the distance between two probability measures on a metric space. We prove an analogue of the Berry-Esseen inequality for the Wasserstein distance on a finite area hyperbolic surface. This inequality…
We estimate the rate of convergence for the Kantorovich (or Wasserstein) distance between empirical measures of i.i.d. random variables associated with the Laguerre model of order $\alpha$ on $(0,\infty)^N$ and their common law, which is…
We estimate rates of convergence for empirical measures associated with the subordinated fractional Brownian motion to the uniform distribution on the flat torus under the Wasserstein distance $\mathbb{W}_p$ for all $p\geq1$. In particular,…
In this paper we introduce some recent progresses on the convergence rate in Wasserstein distance for empirical measures of Markov processes. For diffusion processes on compact manifolds possibly with reflecting or killing boundary…
Data represented by probability measures arise as empirical distributions, posterior distributions, and feature-based representations of complex objects. We study heterogeneity in a population of probability measures through the expected…
Estimating a $d$-dimensional distribution $\mu$ by the empirical measure $\hat{\mu}_n$ of its samples is an important task in probability theory, statistics and machine learning. It is well known that $\mathbb{E}[\mathcal{W}_p(\hat{\mu}_n,…
In this manuscript we investigate the equivalence of Fourier-based metrics on discrete state spaces with the well-known Wasserstein distances. While the use of Fourier-based metrics in continuous state spaces is ubiquitous since its…
The primary choice to summarize a finite collection of random objects is by using measures of central tendency, such as mean and median. In the field of optimal transport, the Wasserstein barycenter corresponds to the Fr\'{e}chet or…
Fix an irrational number $\alpha$. Let $X_1,X_2,\cdots$ be independent, identically distributed, integer-valued random variables with characteristic function $\varphi$, and let $S_n=\sum_{i=1}^n X_i$ be the partial sums. Consider the random…
We study the reknown deconvolution problem of recovering a distribution function from independent replicates (signal) additively contaminated with random errors (noise), whose distribution is known. We investigate whether a Bayesian…