Related papers: Eta-diagonal distributions and infinite divisibili…

On the space of (non-commutative) distributions of k-tuples of selfadjoint elements in a $C^*$-probability space $D_c(k)$, one has an operation $\freeplus$ of free additive convolution, and one can consider the subspace $D_c^{inf-div}$ of…

This paper is devoted to studying $R$-diagonal and $\eta$-diagonal pairs of random variables. We generalize circular elements to the bi-free setting, defining bi-circular element pairs of random variables, which provide examples of…

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,…

In this paper additive bi-free convolution is defined for general Borel probability measures, and the limiting distributions for sums of bi-free pairs of selfadjoint commuting random variables in an infinitesimal triangular array are…

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…

Let M denote the space of Borel probability measures on the real line. For every nonnegative t we consider the transformation $\mathbb B_t : M \to M$ defined for any given element in M by taking succesively the the (1+t) power with respect…

Initiated by a result of Gorin and Marcus [Int. Math. Res. Not., (3):883--913, 2020] and an observation of Steinerberger [Proc. Amer. Math. Soc., 147(11):4733--4744, 2019], there has been a recent growing body of literature connecting…

We study the class $\mathcal{M}_{\mathrm{ratio}}$ of those probability distributions for which the free $R$-transforms are rational functions. This class is closed under the additive free convolution, additive free powers and under the…

Let $\{\xi_1,\xi_2,\ldots\}$ be a sequence of independent random variables (not necessarily identically distributed), and $\eta$ be a counting random variable independent of this sequence. We obtain sufficient conditions on…

Let $\boxplus$, $\boxtimes$ and $\uplus$ be the free additive, free multiplicative, and boolean additive convolutions, respectively. For a probability measure $\mu$ on $[0,\infty)$ with finite second moment, we find the scaling limit of…

In the free probability theory of Voiculescu two of the most frequently used *-distributions are those of a Haar unitary and of a circular element. We define an $R$-diagonal pair as a generalization of these distributions by the requirement…

We consider the framework of an operator-valued noncommutative probability space over a unital C*-algebra B. We show how for a B-valued distribution \mu one can define convolution powers with respect to free additive convolution and with…

We use the theory of fully matricial, or non-commutative, functions to investigate infinite divisibility and limit theorems in operator-valued non-commutative probability. Our main result is an operator-valued analogue of the Bercovici-Pata…

We study the multiplicative convolution for c-monotone independence. This convolution unifies the monotone, Boolean and orthogonal multiplicative convolutions. We characterize convolution semigroups for the c-monotone multiplicative…

We give an explicit realization of the $\eta$-convolution power of an $A$-valued distribution, as defined earlier by Anshelevich, Belinschi, Fevrier and Nica. If $\eta:A\to A$ is completely positive and $\eta\geq\operatorname{id}$, we give…

We prove that the classical normal distribution is infinitely divisible with respect to the free additive convolution. We study the Voiculescu transform first by giving a survey of its combinatorial implications and then analytically,…

This article, which is substantially motivated by the previous joint work with J. McKay [8], establishes the analytic analogues of the relations we found free probability has with Witt vectors. Therefore, we first present a novel analytic…

This article addresses an equidistribution problem concerning the zeros of systems of random holomorphic sections of positive line bundles on compact K\"{a}hler manifolds and random polynomials on $\mathbb{C}^{m}$ in the setting of the…

Bercovici and Pata showed that the correspondence between classically, freely, and Boolean infinitely divisible distributions holds on the level of limit theorems. We extend this correspondence also to distributions infinitely divisible…

We introduce a finite version of free probability and show the link between recent results using polynomial convolutions and the traditional theory of free probability. One tool for accomplishing this is a seemingly new transformation that…