For α∈(0, 1) an α-trimming, P∗, of a probability P is a new probability obtained by re-weighting the probability of any Borel set, B, according to a positive weight function, f≤1/(1−α), in the way P∗(B)=∫Bf(x)P(dx). If P, Q are probability measures on euclidean space, we consider the problem of obtaining the best L2-Wasserstein approximation between: (a) a fixed probability and trimmed versions of the other; (b) trimmed versions of both probabilities. These best trimmed approximations naturally...

We study the universality of the local eigenvalue statistics of Gaussian divisible Hermitian Wigner matrices. These random matrices are obtained by adding an independent GUE matrix to an Hermitian random matrix with independent elements, a Wigner matrix. We prove that Tracy–Widom universality holds at the edge in this class of random matrices under the optimal moment condition that there is a uniform bound on the fourth moment of the matrix elements. Furthermore, we show that universality holds...

We prove two universality results for random tensors of arbitrary rank $D$. We first prove that a random tensor whose entries are $N^D$ independent, identically distributed, complex random variables converges in distribution in the large $N$ limit to the same limit as the distributional limit of a Gaussian tensor model. This generalizes the universality of random matrices to random tensors. We then prove a second, stronger, universality result. Under the weaker assumption that the joint probability distribution...

We consider complex sample covariance matrices MN = (1/N)YY* where Y is a N × p random matrix with i.i.d. entries Yij, 1 ≤ i ≤ N, 1 ≤ j ≤ p, with distribution F. Under some regularity and decay assumptions on F, we prove universality of some local eigenvalue statistics in the bulk of the spectrum in the limit where N → ∞ and limN→∞ p/N = γ for any real number γ ∈ (0, ∞).

There has been much success in describing the limiting spatial fluctuations of growth models in the Kardar–Parisi–Zhang (KPZ) universality class. A proper rescaling of time should introduce a non-trivial temporal dimension to these limiting fluctuations. In one-dimension, the KPZ class has the dynamical scaling exponent z = 3/2, that means one should find a universal space–time limiting process under the scaling of time as tT, space like t2/3X and fluctuations like t1/3 as t → ∞. In this paper we...

The Choquet-Deny theorem and Deny’s theorem are extended to the vector-valued case. They are applied to give a simple nonprobabilistic proof of the vector-valued renewal theorem, which is used to study the $L^p$-dimension, the $L^p$-density and the Fourier transformation of vector-valued self-similar measures. The results answer some questions raised by Strichartz.

Nous étudions un théorème de Skorohod pour des mesures vectorielles à valeurs ${ℝ}^d$. En notant $X\left(\mathbb{P}\right)$ la mesure image de $\mathbb{P}$ par la variable aléatoire $X,$ nous donnons des classes de mesures $\mathbb{P}$ et éventuel-lement de variables telles que, si la suite $\left\{X_n\left(\mathbb{P}\right)\right\}$ converge étroitement, il existe une suite $\left\{{\phi }_n\right\},{\phi }_n\left(\mathbb{P}\right)=X_n\left(\mathbb{P}\right)$ qui converge en mesure, éventuel-lement p.s.Le problème de Monge est abordé comme application. Soit $\left|\mathbb{P}\right|$ la mesure variation de $\mathbb{P}$, pour un couple $(\mathbb{P},\mathbb{Q})$ et une fonction coût $c,$ le problème de Monge est l’existence d’une fonction...

We give a new proof, relying on polynomial inequalities and some aspects of potential theory, of large deviation results for ensembles of random hermitian matrices.

Let $(X,d_X)$, $(\Omega ,d_{\Omega })$ be complete separable metric spaces. Denote by (X) the space of probability measures on X, by $W_p$ the p-Wasserstein metric with some p ∈ [1,∞), and by ${}_p\left(X\right)$ the space of probability measures on X with finite Wasserstein distance from any point measure. Let $f:\Omega {\to }_p\left(X\right)$, $\omega \mapsto f_{\omega }$, be a Borel map such that f is a contraction from $(\Omega ,d_{\Omega })$ into ${(}_p\left(X\right),W_p)$. Let ν₁,ν₂ be probability measures on Ω with $W_p(\nu ₁,\nu ₂)$ finite. On X we consider the subordinated measures ${\mu }_i={\int }_{\Omega }{f}_{\omega }d{\nu }_i\left(\omega \right)$. Then $W_p(\mu ₁,\mu ₂)\le W_p(\nu ₁,\nu ₂)$. As an application we show that the solution measures ${\varrho }_{\alpha }\left(t\right)$ to the partial...

We discuss a conjecture about comparability of weak and strong moments of log-concave random vectors and show the conjectured inequality for unconditional vectors in normed spaces with a bounded cotype constant.