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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 this paper, we develop the notion of free-Boolean independence in an amalgamation setting. We construct free-Boolean cumulants and show that the vanishing of mixed free-Boolean cumulants is equivalent to our free-Boolean independence…
In this paper we give an analytic interpretation of free convolution of type B, introduced by Biane, Goodman and Nica, and provide a new formula for its computation. This formula allows us to show that free additive convolution of type B is…
We study the conditional distribution of zeros of a Gaussian system of random polynomials (and more generally, holomorphic sections), given that the polynomials or sections vanish at a point p (or a fixed finite set of points). The…
The free Meixner laws arise as the distributions of orthogonal polynomials with constant-coefficient recursions. We show that these are the laws of the free pairs of random variables which have linear regressions and quadratic conditional…
In this work we study conditional monotone cumulants and additive convolution in the shuffle-algebraic approach to non-commutative probability. We describe c-monotone cumulants as an infinitesimal character and identify the c-monotone…
Motivated by the $(q,\gamma)$-cumulants, introduced by Xu [arXiv:2303.13812] to study $\beta$-deformed singular values of random matrices, we define the $(n,d)$-rectangular cumulants for polynomials of degree $d$ and prove several…
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…
The Appell-type polynomial family corresponding to the simplest non-commutative derivative operator turns out to be connected with the Boolean probability theory, the simplest of the three universal non-commutative probability theories (the…
In a recent article A.M. Kagan and G.J.Sz\'ekely introduced a notion of Q-independent and Q-identical distributed random variables. We give a complete description of polynomials which appear in these definitions.
Motivated by recent work of Au, C{\'e}bron, Dahlqvist, Gabriel, and Male, we study regularity properties of the distribution of a sum of two selfad-joint random variables in a tracial noncommutative probability space which are free over a…
The number of square-free integers in $x$ consecutive values of any polynomial $f$ is conjectured to be $c_fx$, where the constant $c_f$ depends only on the polynomial $f$. This has been proven for degrees less or equal to 3. Granville was…
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 develope the framework of transitional conditional independence. For this we introduce transition probability spaces and transitional random variables. These constructions will generalize, strengthen and unify previous notions of…
We present a simplified explanation of why free fractional convolution corresponds to the differentiation of polynomials, by finding how the finite free cumulants of a polynomial behave under differentiation. This approach allows us to…
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…
Polynomial chaos expansions (PCE) are well-suited to quantifying uncertainty in models parameterized by independent random variables. The assumption of independence leads to simple strategies for evaluating PCE coefficients. In contrast,…
Polynomials are common algebraic structures, which are often used to approximate functions including probability distributions. This paper proposes to directly define polynomial distributions in order to describe stochastic properties of…
We define a new independence in non-commutative probability, called $\alpha$-freeness, with respect to a triplet of states. This concept unifies several independences in non-commutative probability, in particular, free, monotone,…
We introduce a finite version of free probability for rectangular matrices that amounts to operations on singular values of polynomials. We show that we can replicate the transforms from free probability, and that asymptotically there is…