Related papers: On the support of the free additive convolution

Let k be a positive integer and let D_k denote the space of joint distributions for k-tuples of selfadjoint elements in C*-probability space. The paper studies the concept of "subordination distribution of \mu \boxplus \nu with respect to…

A consequence of de Finetti's representation theorem is that for every infinite sequence of exchangeable 0-1 random variables $(X_k)_{k\geq1}$, there exists a probability measure $\mu$ on the Borel sets of $[0,1]$ such that $\bar X_n =…

Debbah and Ryan have recently proved a result about the limit empirical singular distribution of the sum of two rectangular random matrices whose dimensions tend to infinity. In this paper, we reformulate it in terms of the rectangular free…

Consider a square random matrix with independent and identically distributed entries of mean zero and unit variance. We show that as the dimension tends to infinity, the spectral radius is equivalent to the square root of the dimension in…

Rue and Held (2005) proposed a method for efficiently computing the Gaussian likelihood for stationary Markov random field models, when the data locations fall on a complete regular grid, and the model has no additive error term. The…

In this note we present an algorithm to obtain a uniform lower bound on Hausdorff dimension of the stationary measure of an affine iterated function scheme with similarities, the best known example of which is Bernoulli convolution. The…

Asymptotic expansions are derived for associated Legendre functions of degree $\nu$ and order $\mu$, where one or the other of the parameters is large. The expansions are uniformly valid for unbounded real and complex values of the argument…

We prove a Central Limit Theorem for probability measures defined via the variation of the sum-of-digits function, in base $b\ge 2$. For $r\ge 0$ and $d \in \mathbb{Z}$, we consider $\mu^{(r)}(d)$ as the density of integers $n\in…

Let ${\mathcal C}= \bigcup_{i=1}^n C_i \subseteq \mathbb{P}^2$ be a collection of smooth rational plane curves. We prove that the addition-deletion operation used in the study of hyperplane arrangements has an extension which works for a…

We study the problem of free denoising. For free selfadjoint random variables $a,b$, where we interpret $a$ as a signal and $b$ as noise, we find $E(a|a+b)$. To that end, we study a probability measure $\mu^{( \mathrm{ov} )}_{a,a+b}$ on…

We generalize a theorem of Bellow and Calder\'on concerning the a.e. convergence of the convolution powers $\ds \mu^nf(x)=\sum_{k}\mu^n(k)f(T^k x)$ where $T$ is a measure preserving transformation of a probability space and $\mu$ is a…

We introduce a general result relating "short averages" of a multiplicative function to "long averages" which are well understood. This result has several consequences. First, for the M\"obius function we show that there are cancellations…

Free probability analogs of the basics of extreme-value theory are obtained, based on Ando's spectral order. This includes classification of freely max-stable laws and their domains of attraction, using ``free extremal convolutions'' on the…

Given a subset $A \subseteq \{0,1\}^n$, let $\mu(A)$ be the maximal ratio between $\ell_4$ and $\ell_2$ norms of a function whose Fourier support is a subset of $A$. We make some simple observations about the connections between $\mu(A)$…

We obtain asymptotic expansions for local probabilities of partial sums for uniformly bounded independent but not necessarily identically distributed integer-valued random variables. The expansions involve products of polynomials and…

We study the line ensembles of non-crossing Brownian bridges above a hard wall, each tilted by the area of the region below it with geometrically growing pre-factors. This model, which mimics the level lines of the $(2+1)$D SOS model above…

We consider a new class of non-Hermitian random matrices, namely the ones which have the form of sums of freely independent terms involving unitary matrices. To deal with them, we exploit the recently developed quaternion technique. After…

Beiglb\"ock and Juillet ("On a problem of optimal transport under marginal martingale constraints") introduced the left-curtain martingale coupling of probability measures $\mu$ and $\nu$, and proved that, when the initial law $\mu$ is…

For each positive number $\alpha$ we study the analog $\nu_alpha$ in free probability of the classical Gamma distribution with parameter $\alpha$. We prove that $\nu_\alpha$ is absolutely continuous and establish the main properties of the…

Edgeworth-type expansions for convolutions of probability densities and powers of the characteristic functions with non-uniform error terms are established for i.i.d. random variables with finite (fractional) moments of order $s \geq 2$,…