on the matsumoto yor property in free probability abstract we study the matsumoto yor property in free probability we prove that the limiting empirical eigenvalue distribution of the gig matrices and the marchenko pastur distribution have the free matsumoto yor property finally we characterize these distributions by regression properties in free probability

asymptotic expansions in the clt in free probability we prove edgeworth type expansions for distribution functions of sums of free random variables under minimal moment conditions the proofs are based on the analytic definition of free convolution

free martingale polynomials abstract in this paper we investigate the properties of free sheffer systems which are certain families of martingale polynomials with respect to the free levy processes first we classify such families that consist of orthogonal polynomials these are the free analogs of the meixner systems next we show that the fluctuations around free convolution semigroups have as principal directions the polynomials whose derivatives are martingale polynomials finally we indicate how rota s finite operator calculus can be modified for the free context

the complex wishart distribution and the symmetric group let v be the space of r r hermitian matrices and let omega be the cone of the positive definite ones we say that the random variable s taking its values in overline omega has the complex wishart distribution gamma p sigma if mathbb e exp tr theta s det i r sigma theta p where sigma and sigma 1 theta are in omega and where p 1 2 r 1 or p r 1 in this paper we compute all moments of s and s 1 the techniques involve in particular the use of the irreducible characters of the symmetric group

laplace transforms which are negative powers of quadratic polynomials we find the distributions in r n for the independent random variables x and y such that e x x y a x y and e g x x y bq x y where q runs through the set of all quadratic forms on r n orthogonal to a given quadratic form v the essential part of this class is provided by distributions with laplace transforms 1 2 c s v s p that we describe completely obtaining a generalization of a gindikin theorem this leads to the classification of natural exponential families with the variance function of type 1 pm m m m v where m v is the symmetric matrix associated to the quadratic form v and m m is a real function these natural exponential families extend the classical wishart distributions on lorentz cones already considered by jensen and later on by faraut and koranyi

the characterizations of a semicircle law by the certain freeness in a c probability space in usual probability theory various characterizations of the gaussian law have been obtained for instance independence of the sample mean and the sample variance of independently identically distributed random variables characterizes the gaussian law and the property of remaining independent under rotations characterizes the gaussian random variables in this paper we consider the free analogue of such a kind of characterizations replacing independence by freeness we show that freeness of the certain pair of the linear form and the quadratic form in freely identically distributed noncommutative random variables which covers the case for the sample mean and the sample variance characterizes the semicircle law moreover we give the alternative proof for nica s result that the property of remaining free under rotations characterizes a semicircular system our proof is more direct and straightforward one

on the limit distributions of random matrices with independent or free entries we state limit distribution results for random matrices with independent or free entries also addressing when we get freeness in the limit and semicircular and circular limits the results generalize some already known results about asymptotic freeness of large random matrices but our goal is to get a more optimal flavour on these results when having matrices with identically distributed entries we show that freeness in the limit is natural when we have free entries but unnatural when the entries are independent and restricted to the case of circular limits

freely independent random variables with non atomic distributions 2015 american mathematical society we examine the distributions of non commutative polynomials of non atomic freely independent random variables in particular we obtain an analogue of the strong atiyah conjecture for free groups thus proving that the measure of each atom of any n n matricial polynomial of nonatomic freely independent random variables is an integer multiple of n 1 in addition we show that the cauchy transform of the distribution of any matricial polynomial of freely independent semicircular variables is algebraic and thus the polynomial has a distribution that is real analytic except at a finite number of points

a one parameter family of transforms linearizing convolution laws for probability distributions we study a family of transforms depending on a parameterq 0 1 which interpolate in an algebraic framework between a relative namely iz log iz of the logarithm of the fourier transform for probability distributions and its free analogue constructed by d voiculescu 16 17 the classical case corresponds toq 1 and the free one toq 0

characterization of wishart laplace distributions via jordan algebra homomorphisms abstract for a real hermitian or quaternion normal random matrix y with mean zero necessary and sufficient conditions for a quadratic form q y to have a wishart laplace distribution the distribution of the difference of two independent central wishart w p m i random matrices are given in terms of a certain jordan algebra homomorphism further it is shown that q k y is independent laplace wishart if and only if in addition to the aforementioned conditions the images k of the moore penrose inverse of are mutually orthogonal k l 0 for k l

stable laws and domains of attraction in free probability theory in this paper we determine the distributional behavior of sums of free in the sense of voiculescu identically distributed inflnitesimal random variables the theory is shown to parallel the classical theory of independent random variables though the limit laws are usually quite difierent our work subsumes all previously known instances of weak convergence of sums of free identically distributed random variables in particular we determine the domains of attraction of stable distributions in the free theory these freely stable distributions are studied in detail in the appendix where their unimodality and duality properties are demonstrated

superconvergence to the central limit and failure of the cramer theorem for free random variables we show that convergence of the semicircle law in the free central limit theorem for bounded random variables is much better than expected thus the distributions which tend to the semicircle become absolutely continuous in finite time and the densities converge in a very strong sense we also show that the semicircle law is the free convolution of laws which are not semicircular thus proving that cramer s classical result for the normal distribution does not have a free counterpart

proof of a conjecture of m l eaton on the characteristic function of the wishart distribution let m geq 2 be a positive integer i m be the m times m identity matrix and sigma and a be symmetric m times m matrices where sigma is positive definite by proving that the function phi alpha a i m 2ia sigma alpha is a characteristic function only if alpha in 0 frac 1 2 1 frac 3 2 ldots m 2 2 cup lbrack m 1 2 infty we establish a conjecture of eaton a similar result is established for the rank 1 noncentral wishart distribution and is conjecture to also be valid for any greater rank

logarithmic convexity for discrete harmonic function and the approximation of the cauchy problem for poisson s equation logarithmic convexity type continuous dependence results for discrete harmonic functions defined as solutions of the standard co piecewise linear approximation to laplace s equation are proved using this result error estimates for a regularization method for approximating the cauchy problem for poisson s equation on a rectangle are obtained numerical results are presented 1 introduction this paper will examine numerical methods for approximating the following cauchy problem for poisson s equation let q 0 1 x 0 1 let f denote the boundary of q let i denote the open segment of f lying on the x axis and let i denote the open segment of f along y 1 then given functions f gj and g2 and positive parameters m e1 and 2 we seek a function u that satisfies

a novel tensor distribution model for the diffusion weighted mr signal diffusion mri is a non invasive imaging technique that allows the measurement of water molecule diffusion through tissue in vivo the directional features of water diffusion allow one to infer the connectivity patterns prevalent in tissue and possibly track changes in this connectivity over time for various clinical applications in this paper we present a novel statistical model for diffusion weighted mr signal attenuation which postulates that the water molecule diffusion can be characterized by a continuous mixture of diffusion tensors an interesting observation is that this continuous mixture and the mr signal attenuation are related through the laplace transform of a probability distribution over symmetric positive definite matrices we then show that when the mixing distribution is a wishart distribution the resulting closed form of the laplace transform leads to a rigaut type asymptotic fractal expression which has been phenomenologically used in the past to explain the mr signal decay but never with a rigorous mathematical justification until now our model not only includes the traditional diffusion tensor model as a special instance in the limiting case but also can be adjusted to describe complex tissue structure involving multiple fiber populations using this new model in conjunction with a spherical deconvolution approach we present an efficient scheme for estimating the water molecule displacement probability functions on a voxel by voxel basis experimental results on both simulations and real data are presented to demonstrate the robustness and accuracy of the proposed algorithms

quadratic and inverse regressions for wishart distributions if u and v are independent random variables which are gamma distributed with the same scale parameter then there exist a and b in r such that e u u v a u v and e u 2 u v b u v 2 this in fact is characteristic of gamma distributions our paper extends this property to the wishart distributions in a suitable way by replacing the real number u 2 by a pair of quadratic functions of the symmetric matrix u this leads to a new characterization of the wishart distributions and to a shorter proof of the 1962 characterization given by olkin and rubin similarly if e u 1 exists there exists c in r such that e u 1 u v c u v 1 wesolowski has proved that this also is characteristic of gamma distributions we extend it to the wishart distributions finally things are explained in the modern framework of symmetric cones and simple euclidean jordan algebras

gaussian fluctuations of characters of symmetric groups and of young diagrams we study asymptotics of reducible representations of the symmetric groups s q for large q we decompose such a representation as a sum of irreducible components or alternatively young diagrams and we ask what is the character of a randomly chosen component or what is the shape of a randomly chosen young diagram our main result is that for a large class of representations the fluctuations of characters and fluctuations of the shape of the young diagrams are asymptotically gaussian in this way we generalize kerov s central limit theorem the considered class consists of representations for which the characters almost factorize and this class includes for example the left regular representation plancherel measure irreducible representations and tensor representations this class is also closed under induction restriction outer product and tensor product of representations our main tool in the proof is the method of genus expansion well known from the random matrix theory

commutators of free random variables let a be a unital c algebra given together with a specified state a c consider two selfadjoint elements a b of a which are free with respect to in the sense of the free probability theory of voiculescu let us denote c i ab ba where the i in front of the commutator is introduced to make c selfadjoint in this paper we show how the spectral distribution of c can be calculated from the spectral distributions of a and b some properties of the corresponding operation on probability measures are also discussed the methods we use are combinatorial based on the description of freeness in terms of non crossing partitions an important ingredient is the notion of r diagonal pair introduced and studied in our previous paper 12

representations of symmetric groups and free probability abstract we consider representations of symmetric groupssqfor largeq we give the asymptotic behaviour of the characters when the corresponding young diagrams rescaled by a factorq 1 2 converge to some prescribed shape this behaviour can be expressed in terms of the free cumulants for a probability measure associated with the limit shape of the diagram we also show that the basic operations of representation theory like taking tensor products restriction or induction have a limiting behavior which can be described using the free probability theory of d voiculescu

chordal loewner families and univalent cauchy transforms we study chordal loewner families in the upper half plane and show that they have a parametric representation we show one that to every chordal loewner family there corresponds a unique measurable family of probability measures on the real line and two that to every measurable family of probability measures on the real line there corresponds a unique chordal loewner family in both cases the correspondence is being given by solving the chordal loewner equation we use this to show that any probability measure on the real line with finite variance and mean zero has univalent cauchy transform if and only if it belongs to some chordal loewner family if the probability measure has compact support we give two further necessary and sufficient conditions for the univalence of the cauchy transform the first in terms of the transfinite diameter of the complement of the image domain of the reciprocal cauchy transform and the second in terms of moment inequalities corresponding to the grunsky inequalities

the lukacs olkin rubin characterization of wishart distributions on symmetric cones we characterize the wishart distributions on a symmetric cone c if c 0 this has been done by lukacs in 1955 if c is the cone of positive definite symmetric matrices this has been done by olkin and rubin in 1962 we both shorten and extend the olkin rubin proof sometimes obscure by using three modern ideas i try to avoid artificial coordinates in differential geometry ii the variance function of a natural exponential family f characterizes f iii symmetric matrices are a particular example of a euclidean simple jordan algebra

cumulants for random matrices as convolutions on the symmetric group in this paper we show that free cumulants can be naturally seen as the limiting value of cumulants of matrices we define these objects as functions on the symmetric group by some convolution relations involving the generalized moments we state that some characteristic properties of the free cumulants already hold for these cumulants

r transforms of free joint distributions and non crossing partitions abstract we introduce an extension of the notion of r transform defined by d voiculescu to free joint distributions i e normalized linear functionals on algebras of non commutative polynomials in several indeterminates we point out that the r transform has good behavior with respect to the operations of free product and free convolution of joint distributions we prove that the explicit computation of the inverse r transform of a joint distribution is done via a formula of summation over the lattice of non crossing partitions which shows that the r transform is the operator theoretic counterpart of the free cumulants considered in the combinatorial approach of r speicher moreover in connection to the equivariance of the multidimensional r transform under rotations we show that a natural free analogue of a classical result about rotations of independent random variables is holding

free products of c algebras small spatial free products of c algebras are constructed under certain conditions they have properties similar to those proved by paschke and salinas for the algebras c gi g2 where g1 g2 are discrete groups the freeproduct analogs of noncommutative bernoulli shifts are discussed 0 introduction let k be a field consider the category of unital algebras over k it is well known that this category admits coproducts free products of algebras 2 heuristically the free product of algebras is the algebra generated by them with no relations except for the identification of unit elements if k c the complex numbers and we consider unital algebras we can easily define a operation on the free products let a b be unital c algebras and a b their free product which is a unital algebra the question arises in what ways may one define a pre c norm on a b that extends the norms on a and b guided by analogy with tensor products we expect to have a choice among many pre c norms giving rise to many c free products of a and b one natural norm is 1c i i sup 1it c ih st representation of a b the representations of a b are in 1 1 correspondence with pairs of representations of a and b which act on the same hilbert space let a b be the completion of a b in this norm it is easy to see that this construction defines a coproduct in the category of c algebras and that a b is the biggest free product of a and b analogous to the biggest tensor product a 0 b if g 1 g2 are discrete groups we obtain c g1 c g2 c gi g2 where g1 g2 is the free product group and this is analogous to the relation c g1 0 c g2 c g1 x g2 this paper is motivated by the question is there a smallest c product a b in analogy to the smallest tensor product a b satisfying a relation cr gi g2 cr gi cr g2 received by the editors january 28 1981 and in revised form may 19 1981 1980 mathematics subject classification primary 46l05 secondary 46l55 1 982 american mathematical society 0002 9947 82 00001022 04 25

on a class of free levy laws related to a regression problem abstract the free meixner laws arise as the distributions of orthogonal polynomials with constant coefficient recursions we show that these are the laws of the free pairs of random variables which have linear regressions and quadratic conditional variances when conditioned with respect to their sum we apply this result to describe free levy processes with quadratic conditional variances and to prove a converse implication related to asymptotic freeness of random wishart matrices

free stein kernels and an improvement of the free logarithmic sobolev inequality we introduce a free version of the stein kernel relative to a semicircular law we use it to obtain a free counterpart of the hsi inequality of ledoux peccatti and nourdin which is an improvement of the free logarithmic sobolev inequality of biane and speicher as well as a rate of convergence in the multivariate entropic free central limit theorem we also compute the free stein kernels for several relevant families of self adjoint operators

test set generation with a large number of unspecified bits using static and dynamic techniques this work presents two new methods for the generation of test sets with a small number of specified bits such type of test sets have been proven beneficial to a large number of test related applications such as deterministic bist low power testing and test set enrichment the first technique is static since it considers an initial test set which attempts to relax via test replacement with tests of similar coverage but with fewer specified bits the second technique is dynamic it generates a test set from a zero base using a hierarchical fault compatibility algorithm both methods are applicable to any enumerative fault method linear to the circuit size the experiments performed using the stuck at fault model demonstrate the superiority of the proposed methods over comparable existing techniques in reducing the total number of specified bits per generated test set the applicability of the generated relaxed test sets is demonstrated for one out of the many possible applications that of deterministic test set embedding a general framework that integrates the proposed relaxation methods in two popular lfsr based test set embedding schemes full and partial reseeding along with a systematic exploration of related parameters is proposed the obtained results show significant reductions in seed storage requirements

quantum bernstein s theorem and the hyperoctahedral quantum group we study an extension of bernstein theorem to the setting of quantum groups for a d tuple of free identically distributed random variables we consider a problem of preservation of freeness under the transformations from the free orthogonal quantum group for a subset not contained in the hyperoctahedral quantum group we prove that preservation of freeness characterizes wigner s semicircle law we show that freeness is always preserved if the quantum subset is contained in the hyperoctahedral quantum group we provide examples of quantum subsets which show that our result is an extension of results known in the literature we study an extension of bernstein theorem to the setting of quantum groups for a d tuple of free identically distributed random variables we consider a problem of preservation of freeness under the transformations from the free orthogonal quantum group for a subset not contained in the hyperoctahedral quantum group we prove that preservation of freeness characterizes wigner s semicircle law we show that freeness is always preserved if the quantum subset is contained in the hyperoctahedral quantum group we provide examples of quantum subsets which show that our result is an extension of results known in the literature

continuous family of invariant subspaces for r diagonal operators we show that every r diagonal operator x has a continuous family of invariant subspaces relative to the von neumann algebra generated by x this allows us to find the brown measure of x and to find a new conceptual proof that voiculescu s s transform is multiplicative our considerations base on a new concept of r diagonality with amalgamation for which we give several equivalent characterizations

a fourier view on the r transform and related asymptotics of spherical integrals abstract we estimate the asymptotics of spherical integrals of real symmetric or hermitian matrices when the rank of one matrix is much smaller than its dimension we show that it is given in terms of the r transform of the spectral measure of the full rank matrix and give a new proof of the fact that the r transform is additive under free convolution these asymptotics also extend to the case where one matrix has rank one but complex eigenvalue a result related with the analyticity of the corresponding spherical integrals

addition of certain non commuting random variables soit g le groupe libre non commutatif sur deux generateurs et soit u j j 1 2 les unites dans l 2 g correspondant a la translation a gauche par les generateurs et on considere l 2 g la fonction g g e soient x j des operateurs de la forme x j j u j les operateurs x j peuvent etre consideres comme des variables aleatoires de moments ou de facon equivalente avec des distributions donnees par les fonctionnelles analytiques j ou j f avec ces conventions la distribution de x 1 x 2 depend seulement des distributions de x 1 et x 2 on explique cette relation

cumulants in noncommutative probability theory ii we continue the investigation of noncommutative cumulants in this paper various characterizations of generalized gaussian random variables are proved

the q t gaussian process the q t fock space fq t h introduced in this paper is a deformation of the q fock space of bozejko and speicher the corresponding creation and annihilation operators now satisfy the commutation relation aq t f aq t g qaq t g aq t f f g htn turn mathjax off a defining relation of the chakrabarti jagannathan deformed quantum oscillator algebra the moments of the deformed gaussian element sq t h aq t h aq t h are encoded by the joint statistics of crossings and nestings in pair partitions the q 0 t specialization yields a natural single parameter deformation of the full boltzmann fock space of free probability with the corresponding semicircular measure variously encoded via the rogers ramanujan continued fraction the t airy function the t catalan numbers of carlitz riordan and the first order statistics of the reduced wigner process

brown s spectral distribution measure for r diagonal elements in finite von neumann algebras abstract in 1983 l g brown introduced a spectral distribution measure for non normal elements in a finite von neumann algebra m with respect to a fixed normal faithful tracial state in this paper we compute brown s spectral distribution measure in case t has a polar decomposition t uh where u is a haar unitary and u and h are free when ker t 0 this is equivalent to that t t is an r diagonal pair in the sense of nica and speicher the measure t is expressed explicitly in terms of the s transform of the distribution t t of the positive operator t t in case t is a circular element i e t x1 ix2 2 where x1 x2 is a free semicircular system then sp t d the closed unit disk and t has constant density 1 on d

design of optimal scan tree based on compact test patterns for test time reduction scan tree architecture has been proposed to reduce the test application time of full scan chain by placing multiple scan cells in parallel most existing techniques rely on non compact test pattern sets to construct the scan tree however they produce inefficient scan tree when highly compact test sets with few don t cares are used in this paper the depth of the scan tree based on approximate compatibility relation for completely specified test data set is analyzed probabilistically by modeling its construction as a vertex coloring problem the upper bound of edges per vertex is computed and demonstrated to be a prime factor that limits the efficiency of scan tree construction based on both compatible and approximately compatible test data between two flip flops inverse compatibility and aggressive approximate compatibility are then proposed to increase the edges per vertex for vertex coloring the q sd connection between two adjacent scan cells is exploited to implement the inverse compatibility with no cost or timing impact to maintain the fault coverage the missing faults under the tree scan mode can be detected by switching the same base architecture into the linear scan mode with negligible hardware overhead as shown by the experimental results on iscas89 iscas99 and lgsynth93 benchmark circuits on average the scan tree generated by our method reduces the test time of the full scan chain by 56 65 percent and that of the scan tree designed by the approximate compatibility method by 39 18 percent under the same compact test sets

recovering nonlinear terms in an inverse boundary value problem for laplace s equation a stability estimate stationary thermography can be used for investigating the functional form of a nonlinear cooling law that describes heat exchanges through an inaccessible part of the boundary of a conductor in this paper we obtain a logarithmic stability estimate for the associated nonlinear inverse problem this stability estimate is obtained from the convergence and sensitivity analysis of a finite difference method for the numerical solution of the cauchy problem for laplace s equation based on the stormer verlet scheme

conditional moments of q meixner processes we show that stochastic processes with linear conditional expectations and quadratic conditional variances are markov and their transition probabilities are related to a three parameter family of orthogonal polynomials which generalize the meixner polynomials special cases of these processes are known to arise from the non commutative generalizations of the levy processes

static test compaction for full scan circuits based on combinational test sets and nonscan input sequences and a lower bound on the number of tests a new class of static compaction procedures is described that generate test sets with reduced test application times for scan circuits the proposed class of procedures combines the advantages of two earlier static compaction procedures one that tends to generate large numbers of tests with a short primary input sequence included in every test and one that tends to generate small numbers of tests with a long primary input sequence included in one of the tests a procedure of the proposed class starts from an initial test set that has a large number of tests and long primary input sequences and it selects a subset of the tests and subsequences of their primary input sequences it thus has the flexibility of finding an appropriate balance between the number of tests and the lengths of the primary input sequences in order to minimize the test application time several ways of computing the primary input sequences for the initial test set are considered the most compact test sets are obtained when a test sequence for the nonscan circuit is available and this sequence is used as part of every test in the initial test set however it is shown that high levels of compaction can also be achieved without the overhead of test generation for the nonscan circuit specifically we show that the industry practice of holding a primary input vector constant between scan operations can be accommodated we estimate the ability of the procedure to achieve optimum test sets by computing a lower bound on the number of tests and demonstrating that the procedure achieves or approaches this lower bound

the linearization of the central limit operator in free probability theory we interpret the central limit theorem as a fixed point theorem for a certain operator and consider the problem of linearizing this operator in classical as well as in free probability theory vdn92 we consider two methods giving such a linearization and interpret the result as a weak form of the clt in the classical case the analysis involves dilation operators in the free case more general composition operators appear

dual lukacs regressions for non commutative variables abstract dual lukacs type characterizations of random variables in free probability are studied here first we develop a freeness property satisfied by lukacs type transformations of free poisson and free binomial non commutative variables which are free second we give a characterization of non commutative free poisson and free binomial variables by properties of first two conditional moments which mimic lukacs type assumptions known from classical probability more precisely our result is a non commutative version of the following result known in classical probability if u v are independent real random variables such that e v 1 u u v and e v 2 1 u 2 u v are non random then v has a gamma distribution and u has a beta distribution

stability and regularization of a discrete approximation to the cauchy problem for laplace s equation the standard five point difference approximation to the cauchy problem for laplace s equation satisfies stability estimates and hence turns out to be a well posed problem when a certain boundedness requirement is fulfilled the estimates are of logarithmic convexity type herewith a regularization method will be proposed and associated error bounds can be derived moreover the error between the given continuous cauchy problem and the difference approximation obtained via a suitable minimization problem can be estimated by a discretization and a regularization term

on the construction of free random variables abstract we investigate some ways to obtain free families of random variables from an initial free family in a tracial noncommutative probability space the random variables in this initial free family are multiplied on the left and the right by certain operators from an algebra free from the entire family aspects of freeness related to this algebra are also considered

sample variance in free probability abstract let x 1 x 2 x n denote i i d centered standard normal random variables then the law of the sample variance q n i 1 n x i x 2 is the 2 distribution with n 1 degrees of freedom it is an open problem in classical probability to characterize all distributions with this property and in particular whether it characterizes the normal law in this paper we present a solution of the free analogue of this question and show that the only distributions whose free sample variance is distributed according to a free 2 distribution are the semicircle law and more generally so called odd laws by which we mean laws with vanishing higher order even cumulants in the way of proof we derive an explicit formula for the free cumulants of q n which shows that indeed the odd cumulants do not contribute and which exhibits an interesting connection to the concept of r cyclicity

on operator valued bi free distributions abstract in this paper operator valued bi free distributions are investigated given a subalgebra d of a unital algebra b it is established that a two faced family z is bi free from b b op over d if and only if certain conditions relating the b valued and d valued bi free cumulants of z are satisfied using this we verify that a two faced family of matrices is r cyclic if and only if they are bi free from the scalar matrices over the scalar diagonal matrices furthermore the operator valued bi free partial r s and t transforms are constructed new proofs of results from free probability are developed in order to facilitate many of these bi free results

a noncommutative extended de finetti theorem abstract the extended de finetti theorem characterizes exchangeable infinite sequences of random variables as conditionally i i d and shows that the apparently weaker distributional symmetry of spreadability is equivalent to exchangeability our main result is a noncommutative version of this theorem in contrast to the classical result of ryll nardzewski exchangeability turns out to be stronger than spreadability for infinite sequences of noncommutative random variables out of our investigations emerges noncommutative conditional independence in terms of a von neumann algebraic structure closely related to popa s notion of commuting squares and kummerer s generalized bernoulli shifts our main result is applicable to classical probability quantum probability in particular free probability braid group representations and jones subfactors

on the multiplication of free n tuples of noncommutative random variables let a1 a n b 1 b n be random variables in a noncommutative probability space such that a1 a n is free from b1 b n we show how the joint distribution of the n tuple a1b1 a n b n can be described in terms of the joint distributions of a1 a n a nd b 1 b n by using the combinatorics of the n dimensional r transform we point out a few applications that can be easily derived from our result concerning the left and right translation with a semicircular element see sections 1 6 1 10 and the compression with a projection see sections 1 11 1 14 of an n tuple of noncommutative random variables a different approach to two of these applications is presented by dan voiculescu in an appendix to the paper introduction the theory of free random variables was developed in a se quence of papers of d voiculescu see 17 or the recent survey in 16 as an instrument for approaching free products of operator algebras its particular aspect addressed in the present paper is the one concerning the addition and multiplica tion of free random variables as shown by voiculescu in 13 14 a powerful method in the study of these operations is the use of transforms that convert them respectively into addition and multiplication of complex analytic func tions or in an algebraic framework of formal power series in an indeterminate z the precise definitions of these transforms called r transform for the addition problem and s transform for the multiplication problem will be reviewed in the sections 1 2 1 3 below in the present paper we are pursuing a combination of two ideas that have appeared recently in the study of the r and s transforms the first idea is that the connection between the r and the s transform is closer than one might suspect at first glance a way of making this precise was pointed out in our paper 6 in the form of the equation s r for a distribution with nonvan ishing mean and where is a combinatorial object with a precise significance the fourier transform for multiplicative functions on noncrossing partitions a byproduct of our result in 6 is the remark that the multiplication of free

convolution operations arising from vandermonde matrices different types of convolution operations involving large vandermonde matrices are considered the convolutions parallel those of large gaussian matrices and additive and multiplicative free convolution and include additive and multiplicative convolution of vandermonde matrices and deterministic diagonal matrices and cases where two independent vandermonde matrices are involved it is also shown that the convergence of any combination of vandermonde matrices is almost sure the convolutions are divided into two types those which depend on the phase distribution of the vandermonde matrices and those which depend only on the spectra of the matrices a general criterion is presented to find which type applies for any given convolution a simulation is presented verifying the results implementations of the presented convolutions are provided and discussed the implementation is based on the technique of fourier motzkin elimination and is quite general as it can be applied to virtually any combination of vandermonde matrices connections with related matrices such as toeplitz and hankel matrices are also discussed

an independence property for the product of gig and gamma laws matsumoto and yor have recently discovered an interesting transformation which preserves a bivariate probability measure which is a product of the generalized inverse gaussian gig and gamma distributions this paper is devoted to a detailed study of this phenomenon let x and y be two independent positive random variables we prove theorem 4 1 that u x y 1 and v x 1 x y 1 are independent if and only if there exists p a b 0 such that y is gamma distributed with shape parameter p and scale parameter 2a 1 and such that x has a gig distribution with parameters p a and b the direct part for a b was obtained in matsumoto and yor the result is partially extended theorem 5 1 to the case where x and y are valued in the cone v of symmetric positive definite r r real matrices as follows under a hypothesis of smoothness of densities we prove that u x y 1 and v x 1 x y 1 are independent if and only if there exists p r 1 2 and a and b in v such that y is wishart distributed with shape parameter p and scale parameter 2a 1 and such that x has a matrix gig distribution with parameters p a and b the direct result is also extended to singular wishart distributions theorem 3 1

multiplicative functions on the lattice of non crossing partitions and free convolution the lattice of non crossing partitions was introduced by kreweras kre in 1972 and examined further by poupard pou and edelman edel ede2 in the last time there has been an increased interest in this lattice on one hand from the purely combinatorial point of view siu eds sire bss and on the other hand from a quantum probabilistic point of view gss spel spe2 spe3 we became interested in this lattice when we noticed that it is connected with some new form of convolution for probability measures the free convolution introduced by voiculescu voil voi2 voi3 this connection between the free convolution and the lattice of non crossing partitions is exactly analogous to the connection between the usual convolution and the lattice of all partitions we shall work out this connection in the last part of this work the intention of this article is to show that both the purely combinatorial and the probabilistic point of view can benefit from each other motivated by voiculescu s formula for the r series of free convolution we examine multiplicative functions on the lattice of non crossing partitions and prove our main theorem which is much in the spirit of voiculescu s formula we shall then use this main theorem for deriving some known results on non crossing partitions in a unified way this part of our work is purely combinatorial the free convolution serves only as a motivation for our theorem in the last section we shall switch to the probabilistie side and show how the lattice of non crossing partitions determines the structure of the free convolution voiculescu s formula will then follow as an easy corollary of our main theorem in this respect we shall get a purely combinatorial proof of this formula without any operator algebraic or functional analytic tools