Related papers: Polynomial convolutions and (finite) free probabil…
The free metaplectic transformation (FMT) is widely used in many fields such as filter design, pattern recognition, image processing and optics. In order to obtain a more concise and intuitive convolution form, this paper studies two kinds…
We consider the three finite free convolutions for polynomials studied in a recent paper by Marcus, Spielman, and Srivastava. Each can be described either by direct explicit formulae or in terms of operations on randomly rotated matrices.…
We extend the notions of finite free convolution and finite free cumulants to the setting of formal power series by introducing their natural analogues, namely $t$-deformed convolution and $t$-deformed cumulants. In this framework, we…
We extend the free convolution of Brown measures of $R$-diagonal elements introduced by K\"{o}sters and Tikhomirov [Probab. Math. Statist. 38 (2018), no. 2, 359--384] to fractional powers. We then show how this fractional free convolution…
A new class of distributional transformations is introduced, characterized by equations relating function weighted expectations of test functions on a given distribution to expectations of the transformed distribution on the test function's…
We introduce new homomorphisms relative to additive convolutions and max-convolutions in free, boolean and classical cases. Crucial roles are played by the limit distributions for free multiplicative law of large numbers.
We determine the distributional behavior for products of free random variables in a general infinitesimal triangular array. In the case of positive variables, the main theorem extends a result proved earlier for arrays with identically…
We use techniques from finite free probability to analyze matrix processes related to eigenvalues, singular values, and generalized singular values of random matrices. The models we use are quite basic and the analysis consists entirely of…
This paper investigates the use of different transformations for improving the randomness of sequences. In particular, convolutional codes are used for increasing the size of a given sequence and then a random mapping function is used for…
We extend the relation between random matrices and free probability theory from the level of expectations to the level of all correlation functions (which are classical cumulants of traces of products of the matrices). We introduce the…
We prove an inverse relation and a family of convolution formulas involving partial Bell polynomials. Known and some presumably new combinatorial identities of convolution type are discussed. Our approach relies on an interesting…
We define an extension of the polynomial calculus on a W*-probability space by introducing an abstract algebra which contains polynomials. This extension allows us to define transition operators for additive and multiplicative free…
We seek random versions of some classical theorems on complex approximation by polynomials and rational functions, as well as investigate properties of random compact sets in connection to complex approximation.
This paper describes the quality of convergence to an infinitely divisible law relative to free multiplicative convolution. We show that convergence in distribution for products of identically distributed and infinitesimal free random…
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,…
This talk is organized as follows: First we explain some basic concepts in non-commutative probability theory in the frame of operator algebras. In Section 2, we discuss related topics in von Neumann algebras. Sections 3 and 4 contain some…
Recently, in a work that grew out of their exploration of interlacing polynomials, Marcus, Spielman and Srivastava and then Marcus studied certain combinatorial polynomial convolutions. These convolutions preserve real-rootedness and…
This is an introduction to some of the most probabilistic aspects of free probability theory.
Using the technique developed in approximation theory, we construct examples of exponential families of infinitely divisible laws which can be viewed as deformations of the normal, gamma, and Poisson exponential families. Replacing the…
Given two polynomials $p(x), q(x)$ of degree $d$, we give a combinatorial formula for the finite free cumulants of $p(x)\boxtimes_d q(x)$. We show that this formula admits a topological expansion in terms of non-crossing multi-annular…