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We consider a finite collection of independent Hermitian heavy-tailed random matrices of growing dimension. Our model includes the L\'evy matrices proposed by Bouchaud and Cizeau, as well as sparse random matrices with O(1) non-zero entries…

Bivariate count data arise in several different disciplines (epidemiology, marketing, sports statistics, etc., to name but a few) and the bivariate Poisson distribution which is a generalization of the Poisson distribution plays an…

Consider a sequence of polynomials of bounded degree evaluated in independent Gaussian, Gamma or Beta random variables. We show that, if this sequence converges in law to a nonconstant distribution, then (i) the limit distribution is…

In a previous paper (called "Rectangular random matrices. Related covolution"), we defined, for $\lambda \in [0,1]$, the rectangular free convolution with ratio $\lambda$. Here, we investigate the related notion of infinite divisiblity,…

To a higher Landau Level corresponds a generalization of the Poisson distribution arising from generalized coherent states. In this paper, we write down the atomic decomposition of this probability measure and expressed its weights through…

We study periodic points and finitely supported invariant measures for continuous semigroup actions. Introducing suitable notions of periodicity in both topological and measure-theoretical contexts, we analyze the space of invariant Borel…

We prove that Poisson measures are invariant under (random) intensity preserving transformations whose finite difference gradient satisfies a cyclic vanishing condition. The proof relies on moment identities of independent interest for…

A L\'evy noise on $\mathbb{R}^d$ assigns a random real "mass" $\Pi(B)$ to each Borel subset $B$ of $\mathbb{R}^d$ with finite Lebesgue measure. The distribution of $\Pi(B)$ only depends on the Lebesgue measure of $B$, and if $B_1, ..., B_n$…

Using techniques from Poisson approximation, we prove explicit error bounds on the number of permutations that avoid any pattern. Most generally, we bound the total variation distance between the joint distribution of pattern occurrences…

We study monotone and convex stochastic orders for processes with independent increments. Our contributions are twofold: First, we relate stochastic orders of the Poisson component to orders of their (generalized) L\'evy measures. The…

We provide an inequality which is a useful tool in studying both large deviation results and limit theorems for sums of random fields with "negligible" small values. In particular, the inequality covers cases of stable limits for random…

We study a class of Hermitian random matrices which includes and generalizes Wigner matrices, heavy-tailed random matrices, and sparse random matrices such as the adjacency matrices of Erdos-Renyi random graphs with p ~ 1/N. Our NxN random…

We investigate the random continuous trees called L\'evy trees, which are obtained as scaling limits of discrete Galton-Watson trees. We give a mathematically precise definition of these random trees as random variables taking values in the…

We examine how the measure and the number of vertices of the convex hull of a random sample of $n$ points from an arbitrary probability measure in $\mathbf{R}^d$ relates to the wet part of that measure. This extends classical results for…

Let $Y=\sum_{k\ge 1} 1_{A_k}$ be an infinite sum of the indicators of independent events. We investigate a precise (as opposed to logarithmic) first-order asymptotic behavior of the tail probabilities $\mathbb{P}\{Y\ge n\}$ and the point…

The proposal and study of dependent prior processes has been a major research focus in the recent Bayesian nonparametric literature. In this paper, we introduce a flexible class of dependent nonparametric priors, investigate their…

In this paper, we consider certain $\sigma$-finite measures which can be interpreted as the output of a linear filter. We assume that these measures have regularly varying tails and study whether the input to the linear filter must have…

In this paper, we establish the precise asymptotic behaviors of the tail probability and the transition density of a large class of isotropic L\'evy processes when the scaling order is between 0 and 2 including 2. We also obtain the precise…

The non-asymptotic tail bounds of random variables play crucial roles in probability, statistics, and machine learning. Despite much success in developing upper bounds on tail probability in literature, the lower bounds on tail…

There are many ways of measuring and modeling tail-dependence in random vectors: from the general framework of multivariate regular variation and the flexible class of max-stable vectors down to simple and concise summary measures like the…