Monotone Independence Research Articles

Infinitesimal operators and the distribution of anticommutators and commutators

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 analyzed random matrix models are captured by this framework. We show that there is a simple way of finding non-commutative distributions involving infinitesimal operators and apply this to the commutator and anticommutator. We show the joint infinitesimal distribution of an operator and an infinitesimal idempotent gives us the Boolean cumulants of the given operator. We also show that Boolean cumulants can be expressed as infinitesimal moments thus giving matrix models which exhibit asymptotic Boolean independence and monotone independence.

Cumulants, spreadability and the Campbell–Baker–Hausdorff series

We define spreadability systems as a generalization of exchangeability systems in order to unify various notions of independence and cumulants known in noncommutative probability. In particular, our theory covers monotone independence and monotone cumulants which do not satisfy exchangeability. To this end we study generalized zeta and Möbius functions in the context of the incidence algebra of the semilattice of ordered set partitions and prove an appropriate variant of Faà di Bruno’s theorem. With the aid of this machinery we show that our cumulants cover most of the previously known cumulants. Due to noncommutativity of independence the behaviour of these cumulants with respect to independent random variables is more complicated than in the exchangeable case and the appearance of Goldberg coefficients exhibits the role of the Campbell–Baker–Hausdorff series in this context. Moreover, we exhibit an interpretation of the Campbell–Baker–Hausdorff series as a sum of cumulants in a particular spreadability system, thus providing a new derivation of the Goldberg coefficients.

On operator-valued infinitesimal Boolean and monotone independence

We introduce the notion of operator-valued infinitesimal (OVI) Boolean independence and OVI monotone independence. Then we show that OVI Boolean (respectively, monotone) independence is equivalent to the operator-valued (OV) Boolean (respectively, monotone) independence over an algebra of [Formula: see text] upper triangular matrices. Moreover, we derive formulas to obtain the OVI Boolean (respectively, monotone) additive convolution by reducing it to the OV case. We also define OVI Boolean and monotone cumulants and study their basic properties. Moreover, for each notion of OVI independence, we construct the corresponding OVI Central Limit Theorem. The relations among free, Boolean and monotone cumulants are extended to this setting. Besides, in the Boolean case we deduce that the vanishing of mixed cumulants is still equivalent to independence, and use this to connect scalar-valued with matrix-valued infinitesimal Boolean independence.

Relations between convolutions and transforms in operator-valued free probability

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 additive convolutions, we will focus on two other important additive convolutions, which are orthogonal and subordination additive convolutions. We show that the operator-valued subordination functions, which come from the free additive convolutions or the operator-valued free convolution powers, are reciprocal Cauchy transforms of operator-valued random variables which are uniquely determined up to Voiculescu's fully matricial function theory. In the end, we study relations between certain convolutions and transforms in C⁎-operator-valued probability.

Joint monotone and boolean numerical and spectral radii of d-tuples of operators

We study joint numerical and spectral radii defined for d-tuples of bounded operators on a Hilbert space and related to noncommutative notions of independence. The definitions are in analogy with the ones of Popescu, where his formulations turned out to be related with free creation operators, and in this way related to the free independence of Voiculescu. In our study the definitions are related with either weakly monotone creation operators, and thus associated with the monotone independence of Muraki, or with boolean creation operators, and hence related with the boolean independence.

On the empirical validity of axioms in unstructured bargaining

We report experimental results and test axiomatic models of unstructured bargaining by checking the empirical relevance of the underlying axioms. Our data support strong efficiency, symmetry, independence of irrelevant alternatives and monotonicity, and reject scale invariance. Individual rationality and midpoint domination are violated by a significant fraction of agreements that implement equal division in highly unequal circumstances. Two well-known bargaining solutions that satisfy the confirmed properties explain the observed agreements reasonably well. The most frequent agreements in our sample are the ones suggested by the equal-division solution. In terms of the average Euclidean distance, the theoretical solution that best explains the data is the deal-me-out solution (Sutton, 1986; Binmore et al., 1989, 1991). Popular solutions that satisfy scale invariance, individual rationality, and midpoint domination, as the well-known Nash or Kalai-Smorodinsky bargaining solutions, perform poorly in the laboratory.

An operad of non-commutative independences defined by trees

We study certain notions of $N$-ary non-commutative independence, which generalize free, Boolean, and monotone independence. For every rooted subtree $\mathcal{T}$ of an $N$-regular rooted tree, we define the $\mathcal{T}$-free product of $N$ non-com

The Strong Consistency of Neutral and Monotonic Binary Social Decision Rules

The purpose of this paper is to investigate the strong consistency of neutral and monotonic binary social decision rules. Individuals are assumed to satisfy von Neumann-Morgenstern axioms of individual rationality. The main result of the paper shows that there does not exist any neutral and monotonic non-null non-dictatorial binary social decision rule which is strongly consistent. The relationship between restricted preferences and the existence of strong equilibria is also investigated. It is shown that for every non-dictatorial social decision function satisfying the conditions of independence of irrelevant alternatives, neutrality, monotonicity and weak Pareto-criterion there exists a profile of individual orderings satisfying value-restriction corresponding to which there is no strong equilibrium.

V-monotone independence

We introduce and study a new notion of non-commutative independence, called V-monotone independence, which can be viewed as an extension of the monotone independence of Muraki. We investigate the combinatorics of mixed moments of V-monotone independent ra

Conditionally monotone independence and the associated products of graphs

We reduce the conditionally monotone (c-monotone) independence of Hasebe to tensor independence on suitably constructed larger algebras. For that purpose, we use the approach developed for a reduction of similar type for boolean, free and monotone independences. 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.

Bi-monotonic Independence for Pairs of Algebras

In this article, the notion of bi-monotonic independence is introduced as an extension of monotonic independence to the two-faced framework for a family of pairs of algebras in a non-commutative space. The associated cumulants are defined, and a moment-cumulant formula is derived in the bi-monotonic setting. In general, the bi-monotonic product of states is not a state and the bi-monotonic convolution of probability measures on the plane is not a probability measure. This provides an additional example of how positivity need not be preserved under conditional bi-free convolutions.

Free probability for purely discrete eigenvalues of random matrices

In this paper, we study random matrix models which are obtained as a non-commutative polynomial in random matrix variables of two kinds: (a) a first kind which have a discrete spectrum in the limit, (b) a second kind which have a joint limiting distribution in Voiculescu's sense and are globally rotationally invariant. We assume that each monomial constituting this polynomial contains at least one variable of type (a), and show that this random matrix model has a set of eigenvalues that almost surely converges to a deterministic set of numbers that is either finite or accumulating to only zero in the large dimension limit. For this purpose we define a framework (cyclic monotone independence) for analyzing discrete spectra and develop the moment method for the eigenvalues of compact (and in particular Schatten class) operators. We give several explicit calculations of discrete eigenvalues of our model.

Extended de Finetti theorems for boolean independence and monotone independence

We construct several new spaces of quantum sequences and their quantum families of maps in the sense of Sołtan. The noncommutative distributional symmetries associated with these quantum maps are noncommutative versions of spreadability and partial exchangeability. Then, we study simple relations between these symmetries. 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 integers, rather than unilateral sequences. At the end of the paper, we will show Ryll-Nardzewski type theorems for monotone independence and boolean independence: Roughly speaking, an infinite bilateral sequence of random variables is monotonically (boolean) spreadable if and only if the variables are identically distributed and monotone (boolean) with respect to the conditional expectation onto its tail algebra. For an infinite sequence of noncommutative random variables, boolean spreadability is equivalent to boolean exchangeability.

On operator-valued monotone independence

AbstractWe 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 the central limit theorem for the 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.

On operator-valued monotone independence

AbstractWe 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 the central limit theorem for the 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.

An axiomatic approach in minimum cost spanning tree problems with groups

We study minimum cost spanning tree problems with groups, where agents are located in different villages, cities, etc. The groups are formed by agents living in the same village. In Bergantinos and Gomez-Rua (Economic Theory 43:227–262, 2010) we define the rule F as the Owen value of the irreducible game with groups and we prove that F generalizes the folk rule of minimum cost spanning tree problems. Bergantinos and Vidal-Puga (Journal of Economic Theory 137:326–352, 2007a) give two characterizations of the folk rule. In the first one they characterize it as the unique rule satisfying cost monotonicity, population monotonicity and equal share of extra costs. In the second characterization of the folk rule they replace cost monotonicity by independence of irrelevant trees and population monotonicity by separability. In this paper we extend such characterizations to our setting. Some of the properties are the same (cost monotonicity and independence of irrelevant trees) and the other need to be adapted. In general, we do it by claiming the property twice: once among the groups and the other among the agents inside the same group.

Cooperative bargaining: Independence and monotonicity imply disagreement

A unique bargaining solution satisfies restricted monotonicity, independence of irrelevant alternatives, independence of equivalent utility representations, and symmetry—the disagreement solution. It is also the unique bargaining solution that satisfies independence of irrelevant alternatives, continuity, and strong disagreement monotonicity.

Conditionally Monotone Independence II: Multiplicative Convolutions and Infinite Divisibility

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.

Comparison of the Mantel test and alternative approaches for detecting complex multivariate relationships in the spatial analysis of genetic data

The Mantel test is widely used to test the linear or monotonic independence of the elements in two distance matrices. It is one of the few appropriate tests when the hypothesis under study can only be formulated in terms of distances; this is often the case with genetic data. In particular, the Mantel test has been widely used to test for spatial relationship between genetic data and spatial layout of the sampling locations. We describe the domain of application of the Mantel test and derived forms. Formula development demonstrates that the sum-of-squares (SS) partitioned in Mantel tests and regression on distance matrices differs from the SS partitioned in linear correlation, regression and canonical analysis. Numerical simulations show that in tests of significance of the relationship between simple variables and multivariate data tables, the power of linear correlation, regression and canonical analysis is far greater than that of the Mantel test and derived forms, meaning that the former methods are much more likely than the latter to detect a relationship when one is present in the data. Examples of difference in power are given for the detection of spatial gradients. Furthermore, the Mantel test does not correctly estimate the proportion of the original data variation explained by spatial structures. The Mantel test should not be used as a general method for the investigation of linear relationships or spatial structures in univariate or multivariate data. Its use should be restricted to tests of hypotheses that can only be formulated in terms of distances.

A SIMPLE PROOF FOR MONOTONE CLT

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.