Related papers: The Kummer distribution in free probability, and i…
The equivalence of the characteristic function approach and the probabilistic approach to monotone and boolean convolutions is proven for non-compactly supported probability measures. A probabilistically motivated definition of the…
It has been shown by Voiculescu that important classes of square independent random matrices are asymptotically free, where freeness is a noncommutative analog of classical independence. Recently, we introduced the concept of matricial…
This article provides some characterizations of extended COM-Poisson distribution: conditional distribution given the sum, functional operator characterization (Stein identity). We also give some conditions such that the extended…
We completely characterize $\Delta$- and local subexponentialities of positive-half compound Poisson distributions and extend the characterization on two-sided distributions. Moreover, $\Delta$-subexponentiality of infinitely divisible…
A simple exposition of the rarely discussed fact that a set of free boson fields describing different, i.e. kinematically different particle types can be quantized with mutual anticommutation relations is given by the explicit construction…
Based on a new analytical approach to the definition of additive free convolution on probability measures on the real line we prove free analogs of limit theorems for sums for non-identically distributed random variables in classical…
We propose results of the investigation of properties of the random sums of random variables. We consider the case, where the number of summands is the first moment of an event occurrence. An integral equation is presented that determines…
For a general free L\'evy process, we prove the existence of its higher variation processes as limits in distribution, and identify the limits in terms of the L\'evy-It\^o representation of the original process. For a general free compound…
The Poisson distribution is the probability distribution of the number of independent events in a given period of time. Although the Poisson distribution appears ubiquitously in various stochastic dynamics of gene expression, both as…
In this report we discuss some results of non--commutative (quantum) probability theory relating the various notions of statistical independence and the associated quantum central limit theorems to different aspects of mathematics and…
Lukacs type characterization of Marchenko--Pastur distribution in free probability is studied here. We prove that for free $\mathbb{X}$ and $\mathbb{Y}$ when conditional moments of order $1$ and $-1$ of…
We provide an unifying polynomial expression giving moments in terms of cumulants, and viceversa, holding in the classical, boolean and free setting. This is done by using a symbolic treatment of Abel polynomials. As a by-product, we show…
Let A be a unital $C^*$-algebra, given together with a specified state $\phi:A \to C$. Consider two selfadjoint elements a,b of A, which are free with respect to $\phi$ (in the sense of the free probability theory of Voiculescu). Let us…
Most of the characterizations of probability distributions are based on properties of functions of possibly independent random variables. We investigate characterizations of probability distributions through properties of minima or maxima…
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…
We study the joint distribution of the set of all marginals of a random Wishart matrix acting on a tensor product Hilbert space. We compute the limiting free mixed cumulants of the marginals, and we show that in the balanced asymptotical…
We introduce a new class of large structured random matrices characterized by four fundamental properties which we discuss. We prove that this class is stable under matrix-valued and pointwise non-linear operations. We then formulate an…
We study asymptotic expansions in free probability. In a class of classical limit theorems Edgeworth expansion can be obtained via a general approach using sequences of "influence" functions of individual random elements described by…
The time evolution of a bounded quantum system is considered in the framework of the orthogonal, unitary and symplectic circular ensembles of random matrix theory. For an $N$ dimensional Hilbert space we prove that in the large $N$ limit…
In this article we give a purely noncommutative criterion for the characterization of two-state normal distribution. We prove that families of two-state normal distribution can be described by relations which is similar to the conditional…