Related papers: How to prove that some Bernoulli convolution has t…

For a polynomial with palindromic coefficients, unimodality is equivalent to having a nonnegative $g$-vector. A sufficient condition for unimodality is having a nonnegative $\gamma$-vector, though one can have negative entries in the…

We present conditions that allow us to pass from the convergence of probability measures in distribution to the uniform convergence of the associated quantile functions. Under these conditions, one can in particular pass from the asymptotic…

We study the angles between the eigenvectors of a random $n\times n$ complex matrix $M$ with density $\propto \mathrm{e}^{-n\operatorname{Tr}V(M^*M)}$ and $x\mapsto V(x^2)$ convex. We prove that for unit eigenvectors…

Fix a positive integer $d$ and let $(G_n)_{n\geq1}$ be a sequence of finite abelian groups with orders tending to infinity. For each $n \geq 1$, let $C_n$ be a uniformly random $G_n$-circulant matrix with entries in $\{0,1\}$ and exactly…

Let $\xi$ be a non-constant real-valued random variable with finite support, and let $M_{n}(\xi)$ denote an $n\times n$ random matrix with entries that are independent copies of $\xi$. For $\xi$ which is not uniform on its support, we show…

We improve and expand in two directions the theory of norms on complex matrices induced by random vectors. We first provide a simple proof of the classification of weakly unitarily invariant norms on the Hermitian matrices. We use this to…

By Zeckendorf's Theorem, every positive integer is uniquely written as a sum of non-adjacent terms of the Fibonacci sequence, and its converse states that if a sequence in the positive integers has this property, it must be the Fibonacci…

We consider an independent and identically distributed (i.i.d.) random dynamical system of simple linear transformations on the unit interval $T_{\beta}(x)=\beta x$ (mod $1$), $x\in[0,1]$, $\beta>0$, which are the so-called…

Let $X$ be a real $(\beta=1)$ or complex $(\beta=2)$ Ginibre ensemble. Let $\{\sigma_i\}_{1\le i\le n}$ be the eigenvalues of $X,$ and $Z_n$ be some rescaled version of $\max_i \Re \sigma_i.$ It was proved that $Z_n$ converges weakly to the…

Pickrell has fully characterized the unitarily invariant probability measures on infinite Hermitian matrices, and an alternative proof of this classification has been found by Olshanski and Vershik. Borodin and Olshanski deduced from this…

We discuss a general method to construct correlated binomial distributions by imposing several consistent relations on the joint probability function. We obtain self-consistency relations for the conditional correlations and conditional…

The Bernoulli convolution $\nu_\lambda$ with parameter $\lambda\in(0,1)$ is the probability measure supported on $\mathbf{R}$ that is the law of the random variable $\sum\pm\lambda^n$, where the $\pm$ are independent fair coin-tosses. We…

We establish some identities in law for the convolution of a beta prime distribution with itself, involving the square root of beta distributions. The proof of these identities relies on transformations on generalized hypergeometric series…

Any Borel probability measure supported on a Cantor set of zero Lebesgue measure on the real line possesses a discrete inverse measure. We study the validity of the multifractal formalism for the inverse measures of random weak Gibbs…

Random permutations with distribution conditionally uniform given the set of record values can be generated in a unified way, coherently for all values of $n$. Our central example is a two-parameter family of random permutations that are…

Inspired by the idea of Bernoulli decomposition, we give a simple proof for a generalization of Hal\'asz anti--concentration result about random sum of vectores in $\mathbb{R}^d$. From our results, we can give one upper bound for the…

Let $\nu_\lambda^p$ be the distribution of the random series $\sum_{n=1}^\infty i_n \lambda^n$, where $i_n$ is a sequence of i.i.d. random variables taking the values 0,1 with probabilities $p,1-p$. These measures are the well-known…

Starting from an n-by-n matrix of zeros, choose uniformly random zero entries and change them to ones, one-at-a-time, until the matrix becomes invertible. We show that with probability tending to one as n tends to infinity, this occurs at…

A random vector ${\bf X}$ is weakly stable iff for all $a,b \in \mathbb{R}$ there exists a random variable $\Theta$ such that $a{\bf X} + b {\bf X}' \stackrel{d}{=} {\bf X} \Theta$, where $X'$ is an independent copy of $X$ and $\Theta$ is…

We study the convergence properties of a collapsed Gibbs sampler for Bayesian vector autoregressions with predictors, or exogenous variables. The Markov chain generated by our algorithm is shown to be geometrically ergodic regardless of…