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 formula. As an application, one can obtain an easy proof of Central Limit Theorem for operator-valued case. Moreover, we prove a generalization of Muraki's formula for the sum of independent random variables and a relation between generating functions of moments and cumulants.
We introduce the notion of operator-valued infinitesimal (OVI) independence for the Boolean and monotone cases. Then show that OVI Boolean (resp. monotone) independence is equivalent to the operator-valued Boolean (resp. monotone) independence over an algebra of $2\times 2$ upper triangular matrices. Moreover, we derive formulas to obtain the OVI Boolean (resp. monotone) additive convolution by reducing it to the operator-valued case. We also define OVI Boolean and monotone cumulants and study its basic properties. Moreover, for each notion of OVI independence, we construct the corresponding O…
The paper gives an operator algebras model for the conditional monotone independence, introduced by T. Hasebe. The construction is used to prove an embedding result for the N. Muraki's monotone product of C*-algebras. Also, the formulas from the definition of conditional monotone independence are used to define the monotone product of maps which is shown to preserve complete positivity, a similar to the results from the case of free products.
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 monotone and Boolean products, and moreover, the orthogonal product. Then we define the associated cumulants and calculate the limit distributions in central limit theorem and Poisson's law of small numbers. We also prove a combinatorial moment-cumulant formula using monotone partitions. We investigate some other topics such as infinite divisibility for the addit…
We reduce the conditionally monotone (c-monotone) independence of Hasebe to tensor independence. For that purpose, we use the approach developed for the reduction of boolean, free and monotone independences to tensor independence. We apply the tensor product realization of c-monotone random variables to introduce the c-comb (loop) product of birooted graphs, a generalization of the comb (loop) product of rooted graphs, and we show that it is related to the c-monotone additive (multiplicative) convolution of distributions.
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 convolution on the unit circle. We also prove that an infinitely divisible distribution can always be embedded in a convolution semigroup. We furthermore discuss the (non)-uniqueness of such embeddings including the monotone case. Finally connections to the multiplicative Boolean convolution are discussed.
We construct several new spaces of quantum sequences and their quantum families of maps in sense of Sołtan. Then, we introduce noncommutative distributional symmetries associated with these quantum maps and study simple relations between them. We will focus on studying two kinds of noncommutative distributional symmetries: monotone spreadability and boolean spreadability. We provide an example of a spreadable sequence of random variables for which the usual unilateral shift is an unbounded map. As a result, it is natural to study bilateral sequences of random objects, which are indexed by inte…
Many kinds of independence have been defined in non-commutative probability theory. Natural independence is an important class of independence; this class consists of five independences (tensor, free, Boolean, monotone and anti-monotone ones). In the present paper, a unified treatment of joint cumulants is introduced for natural independence. The way we define joint cumulants enables us not only to find the monotone joint cumulants but also to give a new characterization of joint cumulants for other kinds of natural independence, i.e., tensor, free and Boolean independences. We also investigat…
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 and cumulant functions. It is demonstrated that conditionally bi-free independence with amalgamation is equivalent to the vanishing of mixed operator-valued bi-free and conditionally bi-free cumulants. Furthermore, an operator-valued conditionally bi-free partial $\mathcal{R}$-transform is constructed and various operator-valued conditionally bi-free limit t…
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 systems of Hilbert modules we show that monotone conditional independence arises naturally in dilation theory.
We define a new independence in non-commutative probability, called $α$-freeness, with respect to a triplet of states. This concept unifies several independences in non-commutative probability, in particular, free, monotone, antimonotone and Boolean ones as well as conditionally free, conditionally monotone and conditionally antimonotone independences. Moreover, the associative law of $α$-freeness is transferred to the other independences. As a consequence, $α$-free cumulants unify the cumulants for free, monotone, antimonotone and Boolean independences. The central limit theorem for $α$-freen…
We show how to reduce free independence to tensor independence in the strong sense. We construct a suitable unital *-algebra of closed operators `affiliated' with a given unital *-algebra and call the associated closure `monotone'. Then we prove that monotone closed operators of the form $$ X'= \sum_{k=1}^{\infty}X(k)\bar{\otimes} p_{k}, X''=\sum_{k=1}^{\infty} p_{k}\bar{\otimes}X(k) $$ are free with respect to a tensor product state, where $X(k)$ are tensor independent copies of a random variable $X$ and $(p_{k})$ is a sequence of orthogonal projections. For unital free *-algebras, we constru…
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 Lévy distance. As applications, we obtain quantitative bounds for the corresponding CLTs, provide a quantitative "fourth moment theorem" for monotone independent random variables including the operator-valued case, and generalize the results by Hao and Popa on matrices with Boolean entries. Our approach relies on a Lindeberg method that we develop for sums of Boolean/monotone indepen…
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 $\mathcal{T}$-free product of $N$ non-commutative probability spaces and we define the $\mathcal{T}$-free additive convolution of $N$ non-commutative laws. These $N$-ary convolution operations form a topological symmetric operad which includes the free, Boolean, monotone, and anti-monotone convolutions, as well as the orthogonal and subordination convolutions. Using the ope…
We construct pairs of algebras with mixed independence relations by using truncations of reduced free products of algebras. For example, we construct free-Boolean pairs of algebras and free-monotone pairs of algebras. We also introduce free-Boolean cumulants and show that free-Boolean independence is equivalent to the vanishing of mixed cumulants.
In a 1999 paper, Bercovici and Pata showed that a natural bijection between the classically, free and Boolean infinitely divisible measures held at the level of limit theorems of triangular arrays. This result was extended to include monotone convolution by the authors. In recent years, operator-valued versions of free, Boolean and monotone probability have also been developed. Belinschi, Popa and Vinnikov showed that the Bercovici-Pata bijection holds for the operator-valued versions of free and Boolean probability. In this article, we extend the bijection to include monotone probability theo…
In a central lemma we characterize "generating functions" of certain functors on the category of algebraic non-commutative probability spaces. Special families of such generating functions correspond to "unital, associative universal products" on this category, which again define a notion of non-commutative stochastic independence. Using the central lemma, we can prove the existence of cumulants and of "cumulant Lie-algebras" for a wide class of independences. These include the five independences (tensor, free, Boolean, monotone, anti-monotone) appearing in N. Murakis classification, c-free in…
The notion of monotonic independence, introduced by N. Muraki, is considered in a more general frame, similar to the construction of operator-valued free probability. The paper presents constructions for maps with similar properties to the H and K transforms from the literature, semi inner-product bimodule analogues for the monotone and weakly monotone product of Hilbert spaces, an ad-hoc version of the Central Limit Theorem, an operator-valued arsine distribution as well as a connection to operator-valued conditional freeness.
In the case of monotone independence, the transparent understanding of the mechanism to validate the central limit theorem (CLT) has been lacking, in sharp contrast to commutative, free and Boolean cases. We have succeeded in clarifying it by making use of simple combinatorial structure of peakless pair partitions.
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 easily study their associated convolutions via Voiculescu's fully matricial function theory. Based the matricial extension property, we show that many results can be generalized to multi-variable cases. Besides free, Boolean and monotone independence convolutions, we will focus on two important convolutions, which are orthogonal and subordination additive convol…
We show that the monotonic independence introduced by Muraki can also be used to define a multiplicative convolution. We also find a method for the calculation of this convolution based on an appropriate form of the Cauchy transform. We discuss infinite divisibility in the multiplicative monotonic context as well.
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. Furthermore, we prove an operator valued analogue of Voiculescu's asymptotic freeness theorem. By replacing conditional independence with Boolean independence, we show that the limit laws of the random matrices are Operator-valued Bernoulli laws.
We adapt the theory of chordal Loewner chains to the operator-valued matricial upper-half plane over a $C^*$-algebra $\mathcal{A}$. We define an $\mathcal{A}$-valued chordal Loewner chain as a subordination chain of analytic self-maps of the $\mathcal{A}$-valued upper half-plane, such that each $F_t$ is the reciprocal Cauchy transform of an $\mathcal{A}$-valued law $μ_t$, such that the mean and variance of $μ_t$ are continuous functions of $t$. We relate $\mathcal{A}$-valued Loewner chains to processes with $\mathcal{A}$-valued free or monotone independent independent increments just as was do…
We study of the connection between operator valued central limits for monotone, Boolean and free probability theory, which we shall call the arcsine, Bernoulli and semicircle distributions, respectively. In scalar-valued non-commutative probability these measures are known to satisfy certain arithmetic relations with respect to Boolean and free convolutions. We show that generally the corresponding operator-valued distributions satisfy the same relations only when we consider them in the fully matricial sense introduced by Voiculescu. In addition, we provide a combinatorial description in term…
In the present paper we define the notion of generalized cumulants which gives a universal framework for commutative, free, Boolean, and especially, monotone probability theories. The uniqueness of generalized cumulants holds for each independence, and hence, generalized cumulants are equal to the usual cumulants in the commutative, free and Boolean cases. The way we define (generalized) cumulants needs neither partition lattices nor generating functions and then will give a new viewpoint to cumulants. We define ``monotone cumulants'' in the sense of generalized cumulants and we obtain quite s…
The paper presents several combinatorial properties of the boolean cumulants. A corollary is a new proof of the multiplicative property of the boolean cumulant series that can be easily adapted for the case of boolean independence with amalgamation over an algebra.
It has been shown by Voiculescu and Biane that the analytic subordination property holds for free additive and multiplicative convolutions. In this paper, we present an operatorial approach to subordination for free multiplicative convolutions. This study is based on the concepts of `freeness with subordination', or `s-free independence', and `orthogonal independence', introduced recently in the context of free additive convolutions. In particular, we introduce and study the associated multiplicative convolutions and construct related operators, called `subordination operators' and `subordinat…
