Related papers: Large deviations of empirical zero point measures …

This is a continuation of a project on large deviations for the empirical measures of zeros of random holomorphic sections of random line bundles over a Riemann surface X. In a previous article with O. Zeitouni (arXiv:0904.4271), we proved…

We extend results of Zeitouni-Zelditch on large deviations principles for zeros of Gaussian random polynomials $s$ in one complex variable to certain non-Gaussian ensembles that we call $P(\phi)_2$ random polynomials. The probability…

We study determinantal random point processes on a compact complex manifold X associated to an Hermitian metric on a line bundle over X and a probability measure on X. Physically, this setup describes a free fermion gas on X subject to a…

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…

For a regular compact set $K$ in $C^m$ and a measure $\mu$ on $K$ satisfying the Bernstein-Markov inequality, we consider the ensemble $P_N$ of polynomials of degree $N$, endowed with the Gaussian probability measure induced by $L^2(\mu)$.…

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…

In the first part we study critical points of random polynomials. We choose two deterministic sequences of complex numbers,whose empirical measures converge to the same probability measure in complex plane. We make a sequence of polynomials…

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…

Consider an empirical measure $\mathbb{P}_n$ induced by $n$ iid samples from a $d$-dimensional $K$-subgaussian distribution $\mathbb{P}$ and let $\gamma = N(0,\sigma^2 I_d)$ be the isotropic Gaussian measure. We study the speed of…

We introduce Gaussian-type measures on the manifold of all metrics with a fixed volume form on a compact Riemannian manifold of dimension $\geq 3$. For this random model we compute the characteristic function for the $L^2$ (Ebin) distance…

For a random vector X in R^n, we obtain bounds on the size of a sample, for which the empirical p-th moments of linear functionals are close to the exact ones uniformly on an n-dimensional convex body K. We prove an estimate for a general…

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…

Consider random polynomials of the form $G_n = \sum_{i=0}^n \xi_i p_i$, where the $\xi_i$ are i.i.d.\ non-degenerate complex random variables, and $\{p_i\}$ is a sequence of orthonormal polynomials with respect to a regular measure $\tau$…

In this article we study the limiting empirical measure of zeros of higher derivatives for sequences of random polynomials. We show that these measures agree with the limiting empirical measure of zeros of corresponding random polynomials.…

We consider a randomly forced Ginzburg-Landau equation on an unbounded domain. The forcing is smooth and homogeneous in space and white noise in time. We prove existence and smoothness of solutions, existence of an invariant measure for the…

Let $X:=(X_t)_{t\geq 0}$ be an ergodic Markov process on $\real^d$, and $p>0$. We derive upper bounds of the $p$-Wasserstein distance between the invariant measure and the empirical measures of the Markov process $X$. For this we assume,…

This paper studies convergence of empirical measures smoothed by a Gaussian kernel. Specifically, consider approximating $P\ast\mathcal{N}_\sigma$, for $\mathcal{N}_\sigma\triangleq\mathcal{N}(0,\sigma^2 \mathrm{I}_d)$, by…

The entropy of an ergodic finite-alphabet process can be computed from a single typical sample path x_1^n using the entropy of the k-block empirical probability and letting k grow with $n$ roughly like log n. We further assume that the…

Given $N$ i.i.d. samples from a probability measure $\mu$ on $\mathbf{R}^d$, we study the rate of convergence of the empirical measure $\mu_N \to \mu$ in the negative Sobolev space $W^{-\alpha, p}$. When $W^{-\alpha, p}$ contains point…

Given a compactly supported probability measure on a Riemannian manifold, we study the asymptotic speed at which it can be approximated (in Wasserstein distance of any exponent p) by finitely supported measure. This question has been…