The Matsumoto-Yor property in free probability via subordination and Boolean cumulants

The Matsumoto–Yor property on trees

Viewing the Matsumoto–Yor property as a bivariate property with respect to the simple tree with two vertices and one edge, we extend it to a p-variate property with respect to any tree with p vertices. The converse of the Matsumoto–Yor property, which characterizes the product of a gamma and a generalized inverse Gaussian distribution, is extended to characterize the product of a gamma and p 1 ...

Kummer and gamma laws through independences on trees - Another parallel with the Matsumoto-Yor property

On the computation of classical, boolean and free cumulants

This paper introduces a simple and computationally efficient algorithm for conversion formulae between moments and cumulants. The algorithm provides just one formula for classical, boolean and free cumulants. This is realized by using a suitable polynomial representation of Abel polynomials. The algorithm relies on the classical umbral calculus, a symbolic language introduced by Rota and Taylor...

Subordination Problems Related to Free Probability

Our research project is in the area of noncommutative probability. Noncommutative probability emerged in the early ’80s as a very powerful tool for the study of finite operator algebras. The fundamental idea is to view a pair (A, τ), where A is a unital algebra over the complex numbers (usually endowed with a suitable norm topology), and τ is a linear, unit preserving, functional on A, as a non...

Differential subordination and argumental property

For analytic functions f (z) in the open unit disk E and convex functions g(z) in E, has proved one theorem which is a generalization of the result by K. Sakaguchi [K. Sakaguchi, On a certain univalent mapping, J. Math. Soc. Japan 11 (1959) 72–75]. The object of the present paper is to generalize the theorem due to Pommerenke.