Related papers: On operator-valued infinitesimal Boolean and monot…
We consider the notions of operator-valued infinitesimal (OVI) free independence, OVI Boolean independence, and OVI monotone independence. For each notion of OVI independence, we introduce the corresponding infinitesimal transforms, and…
We consider the infinitesimal freeness in the operator-valued framework, and we show that the operator-valued infinitesimal (OVI) free independence is equivalent to the operator-valued free independence over an algebra of $2\times 2$ upper…
We investigate operator-valued monotone independence, a noncommutative version of independence for conditional expectation. First we introduce operator-valued monotone cumulants to clarify the whole theory and show the moment-cumulant…
The present paper introduces a modified version of cyclic-monotone independence which originally arose in the context of random matrices, and also introduces its natural analogy called cyclic-Boolean independence. We investigate formulas…
We introduce a class of independence relations, which include free, Boolean and monotone independence, in operator valued probability. We show that this class of independence relations have a matricial extension property so that we can…
We define a product of algebraic probability spaces equipped with two states. This product is called a conditionally monotone product. This product is a new example of independence in non-commutative probability theory and unifies the…
Unlike classical and free independence, the boolean and monotone notions of independence lack of the property of independent constants. In the scalar case, this leads to restrictions for the central limit theorems, as observed by F.…
In an infinitesimal probability space we consider operators which are infinitesimally free and one of which is infinitesimal, in that all its moments vanish. Many previously analysed random matrix models are captured by this framework. We…
We introduce a new kind of free independence, called real infinitesimal freeness. We show that independent orthogonally invariant with infinitesimal laws are asymptotically real infinitesimally free. We introduce new cumulants, called real…
In this paper, we introduce the notion of conditionally bi-free independence in an amalgamated setting. We define operator-valued conditionally bi-multiplicative pairs of functions and construct operator-valued conditionally bi-free moment…
We provide Berry-Esseen bounds for sums of operator-valued Boolean and monotone independent variables, in terms of the first moments of the summands. Our bounds are on the level of Cauchy transforms as well as the L\'evy distance. As…
In this paper, the notion of bi-Boolean independence for non-unital pairs of algebras is introduced thereby extending the notion of Boolean independence to pairs of algebras. The notion of B-$(\ell, r)$-cumulants is defined via a bi-Boolean…
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…
Motivated by the recent work on asymptotic independence relations for random matrices with non-commutative entries, we investigate the limit distribution and independence relations for large matrices with identically distributed and Boolean…
We show that the limit laws of random matrices, whose entries are conditionally independent operator valued random variables having equal second moments proportional to the size of the matrices, are operator valued semicircular laws.…
Starting from elementary considerations about independence and Markov processes in classical probability we arrive at the new concept of conditional monotone independence (or operator-valued monotone independence). With the help of product…
In this paper, we introduce the notion of free-free-Boolean independence relation for triples of algebras. We define free-free-Boolean cumulants ans show that the vanishing of mixed cumulants is equivalent to free-free-Boolean independence.…
We study $N$-ary non-commutative notions of independence, which are given by trees and which generalize free, Boolean, and monotone independence. For every rooted subtree $\mathcal{T}$ of the $N$-regular tree, we define the…
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 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…