Noncommutative Random Variables Research Articles

⁎-freeness in finite tensor products

In this paper, we consider the following question and variants thereof: given D:=(a1;i⊗⋯⊗aK;i:i∈I), a collection of elementary tensor non-commutative random variables in the tensor product of probability spaces (A1⊗⋯⊗AK,φ1⊗⋯⊗φK), when is D ⁎-free? (See Section 1.2 for a precise formulation of this problem.)Settling whether or not freeness occurs in tensor products is a recurring problem in operator algebras, and the following two examples provide a natural motivation for the above question:(A)If (a1;i:i∈I) is a ⁎-free family of Haar unitary variables and ak,i are arbitrary unitary variables for k≥2, then the ⁎-freeness persists at the level of the tensor product D.(B)A converse of (A) holds true if all variables ak;i are group-like elements (see Corollary 1.7 of Proposition 1.6).It is therefore natural to seek to understand the extent to which such simple characterizations hold true in more general cases. While our results fall short of a complete characterization, we make notable steps toward identifying necessary and sufficient conditions for the freeness of D. For example, we show that under evident assumptions, if more than one family (ak,i:i∈I) contains non-unitary variables, then the tensor family fails to be ⁎-free (see Theorem 1.8 (1)).

Strong convergence for reduced free products

Using an inequality due to Ricard and Xu, we give a different proof of Paul Skoufranis’s recent result showing that the strong convergence of possibly non-commutative random variables [Formula: see text] is stable under reduced free product with a fixed non-commutative random variable [Formula: see text]. In fact we obtain a more general fact: assuming that the families [Formula: see text] and [Formula: see text] are ∗-free as well as their limits (in moments) [Formula: see text] and [Formula: see text], the strong convergences [Formula: see text] and [Formula: see text] imply that of [Formula: see text] to [Formula: see text]. Phrased in more striking language: the reduced free product is “continuous” with respect to strong convergence. The analogue for weak convergence (i.e. convergence of all moments) is obvious. Our approach extends to the amalgamated free product, left open by Skoufranis.

Two-faced families of non-commutative random variables having bi-free infinitely divisible distributions

We study two-faced families of non-commutative random variables having bi-free (additive) infinitely divisible distributions. We prove a limit theorem of the sums of bi-free two-faced families of random variables within a triangular array. As a corollary of our limit theorem, we get Voiculescu’s bi-free central limit theorem. Using the full Fock space operator model, we show that a two-faced pair of random variables has a bi-free (additive) infinitely divisible distribution if and only if its distribution is the limit distribution in our limit theorem. Finally, we characterize the bi-free (additive) infinite divisibility of the distribution of a two-faced pair of random variables in terms of bi-free Levy processes.

Elements of Quantum Finance Models

Both academic research and practical application of mathematical finance have been extremely fruitful since the seminal work of Black-Scholes-Merton in the early 1970s. In this framework, the prices of financial assets are modeled as stochastic processes in probability spaces inside which the machinery of stochastic calculus is a powerful tool. The fundamental asset pricing theorem states that the absence of arbitrage opportunities in a market is equivalent to the existence of a probability measure, equivalent to the objective probability, under which the discounted prices of the assets become local martingales. This linkage between finance on the one hand and probability theory on the other is the key to the success of mathematical finance. In this note, we show that it is possible to extend a classical probability model to a quantum probability model. Classical stochastic calculus is replaced by its quantum counterpart on a Boson Fock space. In particular, we show that the fundamental asset pricing theorem remains valid in this non-commutative setting. By its very nature of coping with non-commutative random variables however, many essential elements of classic mathematical finance but in quantum form have not been recovered in this note. A couple that have been established are not in their full generality. For instance, existence of non-trivial self-financing portfolios and preserving of self-financing property under change of numeraire can only be reclaimed subject to technical conditions. This writing puts together a few helpful results for the purpose of future studies of this potentially new research subject.

Multidimensional limit theorems for homogeneous sums: A survey and a general transfer principle

The aim of the present paper is to establish the multidimensional counterpart of the \textit{fourth moment criterion} for homogeneous sums in independent leptokurtic and mesokurtic random variables (that is, having positive and zero fourth cumulant, respectively), recently established in \cite{NPPS} in both the classical and in the free setting. As a consequence, the transfer principle for the Central limit Theorem between Wiener and Wigner chaos can be extended to a multidimensional transfer principle between vectors of homogeneous sums in independent commutative random variables with zero third moment and with non-negative fourth cumulant, and homogeneous sums in freely independent non-commutative random variables with non-negative fourth cumulant.

On Two-faced Families of Non-commutative Random Variables

AbstractWe demonstrate that the notions of bi-free independence and combinatorial-bi-free independence of two-faced families are equivalent using a diagrammatic view of bi-non-crossing partitions. These diagrams produce an operator model on a Fock space suitable for representing any two-faced family of non-commutative random variables. Furthermore, using a Kreweras complement on bi-non-crossing partitions we establish the expected formulas for the multiplicative convolution of a bi-free pair of two-faced families.

Free Probability for Pairs of Faces IV: Bi-free Extremes in the Plane

We compute the bi-free max-convolution which is the operation on bivariate distribution functions corresponding to the max-operation with respect to the spectral order on bi-free bipartite two-faced pairs of Hermitian non-commutative random variables. With the corresponding definitions of bi-free max-stable and max-infinitely divisible laws, their determination becomes in this way a classical analysis question.

A noncommutative de Finetti theorem for boolean independence

We introduce a family of quantum semigroups and their natural coactions on noncommutative polynomials. We present three invariance conditions, associated with these coactions, for the joint distribution of sequences of selfadjoint noncommutative random variables. For one of the invariance conditions, we prove that the joint distribution of an infinite sequence of noncommutative random variables satisfies it is equivalent to the fact that the sequence of the random variables are identically distributed and boolean independent with respect to the conditional expectation onto its tail algebra. This is a boolean analogue of de Finetti theorem on exchangeable sequences. In the end of the paper, we will discuss the other two invariance conditions which lead to some trivial results.

Two-parameter non-commutative Central Limit Theorem

En 1992, Speicher a montré que les mesures de probabilités jouant le rôle des lois gaussiennes dans les différentes théories des probabilités non-commutatives (probabilités fermioniques, probabilités libres à la Voiculescu, probabilités $q$-déformées à la Bożejko et Speicher) apparaissent toutes comme limites d’un Théorème de la limite centrale généralisé. Ceci concerne des suites de variables aléatoires non-commutatives (éléments d’une $\ast$-algèbre munie d’un état) choisies dans un ensemble d’éléments qui commutent ou anti-commutent deux-à-deux, avec les distributions limites respectives déterminées par la valeur moyenne des coefficients de commutation. Dans ce papier, nous dérivons une forme plus générale du Théorème de la limite centrale où les coefficients de commutation deux-à-deux sont des nombres réels arbitraires. Les statistiques gaussiennes classiques dépendent maintenant d’un second paramètre comme résultat du contrôle du premier et du second moment des coefficients de commutation. Une application donne le modèle de matrices aléatoires pour les statistiques $(q,t)$-gaussiennes, pour lesquelles il a été montré récemment qu’elles ont des profondes connexions avec les algèbres d’opérateurs, les fonctions spéciales, les polynômes orthogonaux, la physique mathématique et la théorie des matrices aléatoires.

Tail algebras of quantum exchangeable random variables

We show that any countably generated von Neumann algebra with specified normal faithful state can arise as the tail algebra of a quantum exchangeable sequence of noncommutative random variables. We also characterize the cases when the state corresponds to a limit of convex combinations of free products states.

Free Probability for Pairs of Faces I

We consider a notion of bi-freeness for systems of non-commutative random variables with two faces, one of left variables and another of right variables. This includes bi-free convolution operations, bi-free cumulants, the bi-free central limit, and bi-freeness with amalgamation over an algebra B.

Moments of two noncommutative random variables in terms of their joint quantum operators

The computation of the generating function of the joint moments of two non- commutative gaussiano-poissonian random variables, from the commutators of their quantum operators, is presented first. An example of such random variables is also included in the paper.

Free monotone transport

By solving a free analog of the Monge-Ampere equation, we prove a non-commutative analog of Brenier’s monotone transport theorem: if an n-tuple of self-adjoint non-commutative random variables Z 1,…,Z n satisfies a regularity condition (its conjugate variables ξ 1,…,ξ n should be analytic in Z 1,…,Z n and ξ j should be close to Z j in a certain analytic norm), then there exist invertible non-commutative functions F j of an n-tuple of semicircular variables S 1,…,S n , so that Z j =F j (S 1,…,S n ). Moreover, F j can be chosen to be monotone, in the sense that and g is a non-commutative function with a positive definite Hessian. In particular, we can deduce that C ∗(Z 1,…,Z n )≅C ∗(S 1,…,S n ) and $W^{*}(Z_{1},\dots,Z_{n})\cong L(\mathbb{F}(n))$ . Thus our condition is a useful way to recognize when an n-tuple of operators generate a free group factor. We obtain as a consequence that the q-deformed free group factors $\varGamma_{q}(\mathbb{R}^{n})$ are isomorphic (for sufficiently small q, with bound depending on n) to free group factors. We also partially prove a conjecture of Voiculescu by showing that free Gibbs states which are small perturbations of a semicircle law generate free group factors. Lastly, we show that entrywise monotone transport maps for certain Gibbs measure on matrices are well-approximated by the matricial transport maps given by free monotone transport.

WEAK LAW OF LARGE NUMBERS FOR WEIGHTED SUMS IN NONCOMMUTATIVE LORENTZ SPACE

In this paper, we prove the weak law of large numbers for weighted sums of noncommutative random variables in noncommutative Lorentz space under weaker conditions than the conditions in [7].

Characterizations of Some Free Random Variables by Properties of Conditional Moments of Third Degree Polynomials

We investigate Laha-Lukacs properties of noncommutative random variables (processes). We prove that some families of free Meixner distributions can be characterized by the conditional moments of polynomial functions of degree 3. We also show that this fact has consequences in describing some free Levy processes. The proof relies on a combinatorial identity. At the end of this paper we show that this result can be extended to a q-Gausian variable.

De Finetti theorems for easy quantum groups

We study sequences of noncommutative random variables which are invariant under "quantum transformations" coming from an orthogonal quantum group satisfying the "easiness" condition axiomatized in our previous paper. For 10 easy quantum groups, we obtain de Finetti type theorems characterizing the joint distribution of any infinite quantum invariant sequence. In particular, we give a new and unified proof of the classical results of de Finetti and Freedman for the easy groups S_n, O_n, which is based on the combinatorial theory of cumulants. We also recover the free de Finetti theorem of K\"ostler and Speicher, and the characterization of operator-valued free semicircular families due to Curran. We consider also finite sequences, and prove an approximation result in the spirit of Diaconis and Freedman.

Freeness of linear and quadratic forms in von Neumann algebras

We characterize the semicircular distribution by freeness of linear and quadratic forms in noncommutative random variables from tracial W ⁎ -probability spaces with relaxed moment conditions.

Quantum invariant families of matrices in free probability

We consider (self-adjoint) families of infinite matrices of noncommutative random variables such that the joint distribution of their entries is invariant under conjugation by a free quantum group. For the free orthogonal and hyperoctahedral groups, we obtain complete characterizations of the invariant families in terms of an operator-valued R-cyclicity condition. This is a surprising contrast with the Aldous–Hoover characterization of jointly exchangeable arrays.

A characterization of freeness by invariance under quantum spreading

We construct spaces of quantum increasing sequences, which give quantum families of maps in the sense of Sołtan. We then introduce a notion of quantum spreadability for a sequence of noncommutative random variables, by requiring their joint distribution to be invariant under taking quantum subsequences. Our main result is a free analogue of a theorem of Ryll–Nardzewski: for an infinite sequence of noncommutative random variables, quantum spreadability is equivalent to free independence and identical distribution with respect to a conditional expectation.

Quantum rotatability

Recently, Köstler and Speicher showed that de Finetti’s theorem on exchangeable sequences has a free analogue if one replaces exchangeability by the stronger condition of invariance of the joint distribution under quantum permutations. In this paper we study sequences of noncommutative random variables whose joint distribution is invariant under quantum orthogonal transformations. We prove a free analogue of Freedman’s characterization of conditionally independent Gaussian families; namely, the joint distribution of an infinite sequence of self-adjoint random variables is invariant under quantum orthogonal transformations if and only if the variables form an operator-valued free centered semicircular family with common variance. Similarly, we show that the joint distribution of an infinite sequence of random variables is invariant under quantum unitary transformations if and only if the variables form an operator-valued free centered circular family with common variance. We provide an example to show that, as in the classical case, these results fail for finite sequences. We then give an approximation for how far the distribution of a finite quantum orthogonally invariant sequence is from the distribution of an operator-valued free centered semicircular family with common variance.