Related papers: Asymptotics for Optimal Empirical Quantization of …
The history based formalism known as Quantum Measure Theory (QMT) generalizes the concept of probability-measure so as to incorporate quantum interference. The resulting \textit{quantum measure} $\mu$ is defined for arbitrary events (sets…
The asymptotic behaviour of empirical measures has plenty of studies. However, the research on conditional empirical measures is limited. Being the development of Wang \cite{eW1}, under the quadratic Wasserstein distance, we investigate the…
In the first part of this two-part article, we have introduced and analyzed a multidimensional model, called the 'general tension-reduction' (GTR) model, able to describe general quantum-like measurements with an arbitrary number of…
We present a way to use Stein's method in order to bound the Wasserstein distance of order $2$ between two measures $\nu$ and $\mu$ supported on $\mathbb{R}^d$ such that $\mu$ is the reversible measure of a diffusion process. In order to…
We are interested in the Wasserstein distance between two probability measures on $\R^n$ sharing the same copula $C$. The image of the probability measure $dC$ by the vectors of pseudo-inverses of marginal distributions is a natural…
The aim of this article is to write the $p$-Wasserstein metric $W_p$ with the $p$-norm, $p\in [1,\infty)$, on $\R^d$ in terms of copula. In particular for the case of one-dimensional distributions, we get that the copula employed to get the…
We establish quantitative rates of convergence for the empirical estimation of probability measures by means of the Maximum Mean Discrepancy (MMD) with power kernel $K_q(x,y) = -|x-y|^q$, $q \in (0,2)$. The resulting discrepancy is the…
Symmetry is a cornerstone of much of mathematics, and many probability distributions possess symmetries characterized by their invariance to a collection of group actions. Thus, many mathematical and statistical methods rely on such…
Measurement incompatibility stipulates the existence of quantum measurements that cannot be carried out simultaneously on single systems. We show that the set of input-output probabilities obtained from d-dimensional classical systems…
Quantum metrology is a general term for methods to precisely estimate the value of an unknown parameter by actively using quantum resources. In particular, some classes of entangled states can be used to significantly suppress the…
We establish a general framework to study the rate of convergence of a Euler type approximation scheme with decreasing time steps to the invariant measure, for a general class of stochastic systems. The error is measured in general…
We introduce a weak notion of barycenter of a probability measure $\mu$ on a metric measure space $(X, d, {\bf m})$, with the metric $d$ and reference measure ${\bf m}$. Under the assumption that optimal transport plans are given by…
We discuss two ways to construct standard probability measures, called push-down measures, from internal probability measures. We show that the Wasserstein distance between an internal probability measure and its push-down measure is…
Asymptotic methods for hypothesis testing in high-dimensional data usually require the dimension of the observations to increase to infinity, often with an additional condition on its rate of increase compared to the sample size. On the…
For probability distributions on $\mathbb{R}^n$, we study the optimal sample size N = N(n,p) that suffices to uniformly approximate the pth moments of all one-dimensional marginals. Under the assumption that the marginals have bounded 4p…
We consider the problem of finding the infimum, over probability measures being in a ball defined by Wasserstein distance, of the expected value of a bounded Lipschitz random variable on $\mathbf{R}^d$. We show that if the $\sigma-$algebra…
We show how to determine the maximum and minimum possible values of one measure of entropy for a given value of another measure of entropy. These maximum and minimum values are obtained for two standard forms of probability distribution (or…
We consider the problem of discriminating between two different states of a finite quantum system in the setting of large numbers of copies, and find a closed form expression for the asymptotic exponential rate at which the specified error…
We derive normal approximation bounds in the Wasserstein distance for sums of weighted U-statistics, based on a general distance bound for functionals of independent random variables of arbitrary distributions. Those bounds are applied to…
In many contemporary statistical and machine learning methods, one needs to optimize an objective function that depends on the discrepancy between two probability distributions. The discrepancy can be referred to as a metric for…