direct sums of tensor products and non commutative independence abstract we point out the connection between the three known fundamental notions of non commutative independence in quantum probability and the three basic choices for forming an associative direct sum of alternating tensor products of two vector spaces

positive definite functions on coxeter groups with applications to operator spaces and noncommutative probability a new class of positive definite functions related to colour length function on arbitrary coxeter group is introduced extensions of positive definite functions called the riesz coxeter product from the riesz product on the rademacher abelian coxeter group to arbitrary coxeter group is obtained applications to harmonic analysis operator spaces and noncommutative probability are presented characterization of radial and colour radial functions on dihedral groups and infinite permutation group are shown

remarks on the r and δ convolutions in this paper we study the properties of the r deformation introduced in b1 we observe that the associated convolution coming from the conditionally free convolution is associative only for r 1 and r 0 we give the realization of some r gaussian random variables and obtain haagerup pisier buchholz type inequalities we also study another convolution defined with the use of the r deformation through a moment cumulant formula ky1 and show that it is associative and in general not positive

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

relations between cumulants in noncommutative probability abstract we express classical free boolean and monotone cumulants in terms of each other using combinatorics of heaps pyramids tutte polynomials and permutations we completely determine the coefficients of these formulas with the exception of the formula for classical cumulants in terms of monotone cumulants whose coefficients are only partially computed

de finetti s coherence and statistical inference on donne la definition d une inference coherente en conformite avec une theorie des probabilites conditionnelles derivee du principe de coherence de de finetti et on effectue une comparaison critique d une telle definition avec celle proposee par heath lane et sudderth

a generalization of the bernstein polynomials abstract we give an interesting generalization of the bernstein polynomials we find sufficient and necessary conditions for uniform convergence by the new polynomials and we generalize the bernstein theorem

a noncommutative de finetti theorem for boolean independence abstract we introduce a family of quantum semigroups and its natural coactions on noncommutative polynomials we define three invariance conditions for the joint distribution of sequences of selfadjoint noncommutative random variables associated with these coactions for one of the invariance conditions we show that the joint distribution of an infinite sequence of noncommutative random variables satisfy it is equivalent to the fact that the sequence of the random variables is 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 also discuss the other two invariance conditions which lead to some trivial results

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

the norm estimates of the q bernstein operators for varying q 1 the aim of this paper is to present norm estimates in c 0 1 for the q bernstein basic polynomials and the q bernstein operators b n q in the case q 1 while for 01 the norm b n q increases rather rapidly as q in this study it is proved that b n q c nq n n 1 2 q withc n 2n 1 1n n 1 moreover it is shown that b n q 2q n n 1 2ne as n q the results of the paper are illustrated by numerical examples

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

a note on approximation properties of q durrmeyer operators in this paper the approximation properties of q durrmeyer operators d n q f x for f c 0 1 are discussed the exact class of continuous functions satisfying approximation process lim n d n q f x f x is determined the results of the paper provide an elaboration of the previously known ones on operators d n q

the quantum group suq 2 and a q analogue of the boson operators a new realisation of the quantum group suq 2 is constructed by means of a q analogue to the jordan schwinger mapping determining thereby both the complete representation structure and q analogues to the wigner and racah operators to achieve this realisation a new elementary object is defined a q analogue to the harmonic oscillator the uncertainty relation for position and momentum in a q harmonic oscillator is quite unusual

properties of convergence for ω q bernstein polynomials in this paper we discuss properties of the q bernstein polynomials bn q f x introduced by s lewanowicz and p wo ny in s lewanowicz p wo ny generalized bernstein polynomials bit 44 1 2004 63 78 where f c 0 1 q 0 1 q 1 q n 1 when 0 we recover the q bernstein polynomials introduced by g m phillips bernstein polynomials based on the q integers ann numer math 4 1997 511 518 when q 1 we recover the classical bernstein polynomials we compute the second moment of bn q t2 x and demonstrate that if f is convex and q 0 1 or 1 then bn q f x are monotonically decreasing in n for all x 0 1 we prove that for 0 1 qn 0 1 the sequence bn qn f n 1 converges to f uniformly on 0 1 for each f c 0 1 if and only if limn qn 1 for fixed q 0 1 we prove that the sequence bn q f converges for each f c 0 1 and obtain the estimates for the rate of convergence of bn q f by the modulus of continuity of f and the estimates are sharp in the sense of order for lipschitz continuous functions

a q deformed poisson distribution based on orthogonal polynomials we introduce certain q deformations of the binomial distribution and the poisson distribution by virtue of q deformed sequences of orthogonal polynomials they correspond to the case of free independence for q 0 and are reduced to the usual commutative case when q 1 furthermore we determine the probability measure for a q deformed poisson distribution by using the formulae for the al salam chihara polynomials of askey and ismail

cumulants in noncommutative probability theory iv noncrossing cumulants de finetti s theorem and lp inequalities abstract de finetti s theorem states that any exchangeable sequence of classical random variables is conditionally i i d with respect to some algebra in this paper we prove a free noncommutative analog of this theorem namely we show that any noncrossing exchangeability system with a faithful state which satisfies a so called weak singleton condition can be embedded into an free product with amalgamation over a certain subalgebra such that the interchangeable algebras remain interchangeable with respect to the operator valued expectation vanishing of crossing cumulants can be verified by checking a certain weak freeness condition and the weak singleton condition is satisfied e g when the state is tracial the proof follows the classical proof of de finetti s theorem the main technical tool being a noncommutative l p inequality for i i d sums of centered noncommutative random variables in noncrossing exchangeability systems

approximation properties of complex q szasz mirakjan operators in compact disks this paper deals with approximating properties of the newly defined q generalization of the szasz mirakjan operators in the case q 1 quantitative estimates of the convergence the voronovskaja s theorem and saturation of convergence for complex q szasz mirakjan operators attached to analytic functions in compact disks are given in particular it is proved that for functions analytic in email protected c z 2q the rate of approximation by the q szasz mirakjan operators q 1 is of order q n versus 1 n for the classical szasz mirakjan operators

noncommutative levy processes for generalized particularly anyon statistics let t mathbb r d let a function qt 2 to mathbb c satisfy q s t overline q t s and q s t 1 a generalized statistics is described by creation operators partial t dagger and annihilation operators t t in t which satisfy the q commutation relations partial s partial dagger t q s t partial dagger t partial s delta s t partial s partial t q t s partial t partial s partial dagger s partial dagger t q t s partial dagger t partial dagger s from the point of view of physics the most important case of a generalized statistics is the anyon statistics for which q s t is equal to q if s t here q in mathbb c q 1 we start the paper with a detailed discussion of a q fock space and operators partial t dagger partial t t in t in it which satisfy the q commutation relations next we consider a noncommutative stochastic process white noise omega t partial t dagger partial t lambda partial t dagger partial t t in t here lambda in mathbb r is a fixed parameter the case 0 corresponds to a q analog of brownian motion while 0 corresponds to a centered q poisson process we study q hermite q charlier respectively polynomials of infinitely many noncommutatative variables omega t t in t the main aim of the paper is to explain the notion of independence for a generalized statistics and to derive corresponding levy processes to this end we recursively define q cumulants of a field xi t t in t this allows us to define a q levy process as a field xi t t in t whose values at different points of t are q independent and which possesses a stationarity of increments in a certain sense we present an explicit construction of a q levy process and derive a nualart schoutens type chaotic decomposition for such a process

q bernstein polynomials of the cauchy kernel due to the fact that in the case q 1 q bernstein polynomials are not positive linear operators on c 0 1 the study of their approximation properties is essentially more difficult than that for 0 1 is still open in this paper the q bernstein polynomials bn q fa z of the cauchy kernel fa 1 z a a c 0 1 are found explicitly and their properties are investigated in particular it is proved that if q 1 then polynomials bn q fa z converge to fa uniformly on any compact set k z z a the sequence bn q fa z is not even uniformly bounded

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

saturation of convergence for q bernstein polynomials in the case q 1 abstract in the note we discuss voronovskaya type theorem and saturation of convergence for q bernstein polynomials for a function analytic in the disc u r z z r r q for arbitrary fixed q 1 we give explicit formulas of voronovskaya type for the q bernstein polynomials for q 1 we show that the rate of convergence for the q bernstein polynomials is o q n q 1 for infinite number of points having an accumulation point on u r q if and only if f is linear

on the integrability of zero range chipping models with factorized steady states the conditions of the integrability of general zero range chipping models with factorized steady states which were proposed in evans et al 2004 j phys a math gen 37 l275 are examined we find a three parametric family of hopping probabilities for the models solvable by the bethe ansatz which includes most of known integrable stochastic particle models as limiting cases the solution is based on the quantum binomial formula for two elements of an associative algebra obeying generic homogeneous quadratic relations which is proved as a byproduct we use the bethe ansatz to solve an eigenproblem for the transition matrix of the markov process on its basis we conjecture an integral formula for the green function of the evolution operator for the model on an infinite lattice and derive the bethe equations for the spectrum of the model on a ring

generalised brownian motion and second quantisation a new approach to the generalised brownian motion introduced by m bouzejko and r speicher is described based on symmetry rather than deformation the symmetrisation principle is provided by joyal s notions of tensorial and combinatorial species any such species v gives rise to an endofunctor fv of the category of hilbert spaces with contractions a generalised brownian motion is an algebra of creation and annihilation operators acting on fv h for arbitrary hilbert spaces h and having a prescription for the calculation of vacuum expectations in terms of a function t on pair partitions the positivity is encoded by a semigroup of broken pair partitions whose representation space with respect to t is v the existence of the second quantisation as functor t from hilbert spaces to noncommutative probability spaces is proved to be equivalent to the multiplicative property of the function t for a certain one parameter interpolation between the fermionic and the free brownian motion it is shown that the field algebras k are type ii1 factors when k is infinite dimensional

approximation in compact sets by q stancu faber polynomials q 1 we prove that the q stancu faber polynomials q 1 attached to a compact set email protected c and to an analytic function on g can represent good alternatives to the partial sums of the faber series attached to the same function and subset both giving the geometric progression order of approximation q n

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

bounding parameters in a linear regression model with a mismeasured regressor using additional information this paper discusses a linear regression model with a mismeasured regressor in which the measurement error is correlated with both the latent variable and the regression error we use a linear structure to capture the correlation between the measurement error and the latent variable this paper shows that the variance of the latent variable is very useful for revealing information on the parameters which otherwise cannot be obtained with such a nonclassical measurement error the main result is that the finite bounds on the parameters can be found using the variance of the latent variable regardless of how severely the measurement error and the regression error are correlated if the mismeasured regressor contains enough information on the latent one this paper also discusses the special but interesting case of the latent variable being dichotomous in this case the mean of the latent variable may even reveal information on the correlation between the measurement error and the regression error all the bounds developed in the paper are tight

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

noncommutative analogs of probabilistic notions and results abstract considering a random variable as a multiplication operator by a measurable function a natural generalization consists in allowing noncommuting and unbounded operators defined on a common invariant domain with cyclic vector by multiplication and addition these operators generate a algebra which in turn can be considered as a representation of an abstract tensor algebra moments are replaced by m a 1 a n a 1 a n in analogy to the classical case the notions of cumulants addition of independent random variables and infinite divisibility are introduced as well as gaussianness as a generalization of normal random variables previous results are briefly reviewed including a characterization of infinite divisibility among the new results are noncommutative analogs of versions of the central limit theorem and of cramer s theorem all results have direct applications to representations of lie algebras to quantum field theory and to statistical mechanics

convergence properties and iterations for q stancu polynomials in compact disks this paper deals with approximating properties and convergence results of the iterates for complex q stancu polynomials attached to analytic functions on compact disks in the case q 1 some estimates on the rate of convergence are given

on the analyticity of functions approximated by their q bernstein polynomials when q 1 abstract since in the case q 1 the q bernstein polynomials bn q are not positive linear operators on c 0 1 the investigation of their convergence properties for q 1 turns out to be much harder than the one for 0 1 create a serious obstacle for the numerical experiments with the q bernstein polynomials despite the intensive research conducted in the area lately the class of functions which are uniformly approximated by their q bernstein polynomials on 0 1 is yet to be described in this paper we prove that if f 0 1 c is analytic at 0 and can be uniformly approximated by its q bernstein polynomials q 1 on 0 1 then f admits an analytic continuation from 0 1 into z z

cumulants in noncommutative probability theory i noncommutative exchangeability systems cumulants linearize convolution of measures we use a formula of good to define noncommutative cumulants in a very general setting it turns out that the essential property needed is exchangeability of random variables roughly speaking the formula says that cumulants are moments of a certain discrete fourier transform of a random variable this provides a simple unified method to understand the known examples of cumulants like classical free and various q cumulants

operator khintchine inequality in non commutative probability abstract for the operator t sum n k 1 a k otimes r k where a k belongs to the schatten class s 2n and where r k are non commutative random variables with mixed moments satisfying a specific condition we prove the following khintchine inequality vert t vert 2n leq d 2n left left vert left sum n k001 a ka k right frac 1 2 right vert s 2n quad quad left vert left sum n k 1 a k a k right frac 1 2 right vert s 2n right we find the optimal constants d 2n in the case when r k are the q gaussian and circular random variables moreover we show that the moments of any probability symmetric measure appear as the optimal constants for some random variables

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

second order freeness and fluctuations of random matrices ii unitary random matrices abstract we extend the relation between random matrices and free probability theory from the level of expectations to the level of fluctuations we show how the concept of second order freeness which was introduced in part i allows one to understand global fluctuations of haar distributed unitary random matrices in particular independence between the unitary ensemble and another ensemble goes in the large n limit over into asymptotic second order freeness two important consequences of our general theory are i we obtain a natural generalization of a theorem of diaconis and shahshahani to the case of several independent unitary matrices ii we can show that global fluctuations in unitarily invariant multi matrix models are not universal

the norm estimates for the q bernstein operator in the case q 1 the q bernstein basis with 0 1 the behavior of the q bernstein basic polynomials on 0 1 combines the fast increase in magnitude with sign oscillations this seriously complicates the study of q bernstein polynomials in the case of q 1 the aim of this paper is to present norm estimates in c 0 1 for the q bernstein basic polynomials and the q bernstein operator b n q in the case q 1 while for 0 1 the norm b n q increases rather rapidly as n oo we prove here that b n q c q q n n 1 2 n n oo with c q 2 q 2 q 2 e such a fast growth of norms provides an explanation for the unpredictable behavior of q bernstein polynomials q 1 with respect to convergence

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

an obstruction for q deformation of the convolution product we consider two independent q gaussian random variables xo and x and a function y chosen in such a way that y xo and xo have the same distribution for q e 0 1 we find that at least the fourth moments of x 4 xi and y x o xi are different we conclude that no deformed convolution product parallelling the known cases q 0 and q 1 can exist

proper posteriors from improper priors for an unidentified errors in variables model the problem considered is inference in a simple errors in variables model where consistent estimation is impossible without introducing additional exact prior information the probabilistic prior information required for bayesian analysis is found to be surprisingly light despirte the model s lack of identification a proper posterior is guaranteed for any bounded prior density including those representing improper priors this result is illustrated with the improper uniform prior which implies marginal posterior densities obtainable by integrating the likelihood function

generalized szasz mirakjan type operators via q calculus and approximation properties abstract the aim of this paper is to construct q analogue of generalized szasz mirakjan type operators whose construction depend on a real valued function we prove that the new operators provide better weighted uniform approximation over 0 in terms of weighted moduli of smoothness we obtain degrees of approximation associated with the function also a voronovskaya type result is obtained finally we give some graphical examples for these operators and show that the new operators are more flexible in view of rate of convergence to the function f which depends on the selection of un q and vn q

analytical properties of the lupas q transform abstract the lupas q transform emerges in the study of the limit q lupas operator the latter comes out naturally as a limit for a sequence of the lupas q analogues of the bernstein operator given q 0 1 f c 0 1 the q lupas transform of f is defined by q f z 1 z q k 0 f 1 q k q k k 1 2 q q k z k the transform is closely related to both the q deformed poisson probability distribution which is used widely in the q boson operator calculus and to valiron s method of summation for divergent series in general q f is a meromorphic function whose poles are contained in the set j q q j j 0 in this paper we study the connection between the behaviour of f on 0 1 and the decay of q f as z

on the lupas q transform the lupasq transform emerges in the study of the limit q lupas operator the latter comes out naturally as a limit for a sequence of the lupas q analogues of the bernstein operator lately it has been studied by several authors from different perspectives in mathematical analysis and approximation theory this operator is closely related to the q deformed poisson probability distribution which is used widely in the q boson operator calculus given email protected 0 1 email protected c 0 1 the q lupas transform of f is defined by l qf z 1 z q email protected email protected k 0 f 1 q k q k k 1 2 q q kz k in this paper we study some analytic properties of l qf z in particular we examine the conditions under which l qf can either be an entire function or a rational one

de finetti theorem on the car algebra the symmetric states on a quasi local c algebra on the infinite set of indices j are those invariant under the action of the group of the permutations moving only a finite but arbitrary number of elements of j the celebrated de finetti theorem describes the structure of the symmetric states i e exchangeable probability measures in classical probability in the present paper we extend the de finetti theorem to the case of the car algebra that is for physical systems describing fermions namely after showing that a symmetric state is automatically even under the natural action of the parity automorphism we prove that the compact convex set of such states is a choquet simplex whose extremal i e ergodic w r t the action of the group of permutations previously described are precisely the product states in the sense of araki moriya in order to do that we also prove some ergodic properties naturally enjoyed by the symmetric states which have a self containing interest

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 deformed binomial distribution and its asymptotic behaviour the q analogue of the binomial distribution is defined by virtue of the q binomial theorem which takes the euler distribution as its limiting form and is new to the literature

a new completeness relation in the q deformed two mode fock space in two mode fock space a new completeness relation for the q analogue of the single complex variable state is proved by virtue of some q deformed combinatorial identities which are also derived here for the first time

the saturation of convergence on the interval 0 1 for the q bernstein polynomials in the case q 1 abstract in the note we consider saturation of convergence on the interval 0 1 for the q bernstein polynomials of a continuous function f for arbitrary fixed q 1 we show that the rate of uniform convergence on 0 1 is o q n if and only if f is linear the result is sharp in the following sense it ceases to be true if we replace o by o

bayesian moment based inference in a regression model with misclassification error we present a bayesian analysis of a regression model with a binary covariate that may have classification measurement error prior research demonstrates that the regression coefficient is only partially identified we take a bayesian approach which adds assumptions in the form of priors on the unknown misclassification probabilities the approach is intermediate between the frequentist bounds of previous literature and strong assumptions which achieve point identification and thus preferable in many settings we present two simple algorithms to sample from the posterior distribution when the likelihood function is not fully parametric but only satisfies a set of moment restrictions we focus on how varying amounts of information contained in a prior distribution on the misclassification probabilities change the posterior of the parameters of interest while the priors add information to the model they do not necessarily tighten the identified set however the information is sufficient to tighten bayesian inferences we also consider the case where the mismeasured binary regressor is endogenous we illustrate the use of our bayesian approach in a simulated data set and an empirical application investigating the association between narcotic pain reliever use and earnings

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

on the q oscillator and the quantum algebra suq 1 1 the authors discuss some problems of the q deformed harmonic oscillator different q bosonisations of the quantum suq 1 1 algebra are given and the corresponding infinite dimensional representations of discrete series are analysed

approximation by q durrmeyer type polynomials in compact disks in the case q 1 abstract recently agarwal and gupta 2012 1 studied some approximation properties of the complex q durrmeyer type operators in the case 0 q 1 in this paper this study is extended to the case q 1 more precisely approximation properties of the newly defined generalization of this operators in the case q 1 are studied quantitative estimates of the convergence the voronovskaja type theorem and saturation of convergence for complex q durrmeyer type polynomials attached to analytic functions in compact disks are given in particular it is proved that for functions analytic in z c z r r q the rate of approximation by the q durrmeyer type polynomials q 1 is of order q n versus 1 n for the classical q 1 durrmeyer type polynomials explicit formulas of voronovskaya type for the q durrmeyer type operators for q 1 are also given this paper represents an answer to the open problem initiated by gal 2013 6

stancu type generalization of modified schurer operators based on q integers in the present paper we introduce the stancu type generalization s n p a b q x of the modified schurer operators based on q integers first we prove the basic convergence of the operators s n p a b q x and then obtain the voronovskaja type theorem next we discuss local direct results for these operators finally we study the statistical convergence of s n p a b q x and also obtain some convergence properties of the limit q modified schurer stancu operators

non crossing cumulants of type b we establish connections between the lattices of non crossing partitions of type b introduced by v reiner and the framework of the free probability theory of d voiculescu lattices of non crossing partitions of type a up to now have played an important role in the combinatorics of free probability primarily via the non crossing cumulants of r speicher here we introduce the concept of non crossing cumulant of type b the inspiration for its definition is found by looking at an operation of restricted convolution of multiplicative functions studied in parallel for functions on symmetric groups in type a and on hyperoctahedral groups in type b the non crossing cumulants of type b live in an appropriate framework of non commutative probability space of type b and are closely related to a type b analogue for the r transform of voiculescu which is the free probabilistic counterpart of the fourier transform by starting from a condition of vanishing of mixed cumulants of type b we obtain an analogue of type b for the concept of free independence for random variables in a non commutative probability space

free boolean independence for pairs of algebras abstract 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 addition we provide a moments condition for free boolean independence a central limit law for free boolean independence is studied

on the q bernstein polynomials of rational functions with real poles abstract the paper aims to investigate the convergence of the q bernstein polynomials b n q f x attached to rational functions in the case q 1 the problem reduces to that for the partial fractions x j j n the already available results deal with cases where either the pole is simple or q m m n 0 consequently the present work is focused on the polynomials b n q f x for the functions of the form f x x q m j with j 2 for such functions it is proved that the interval of convergence of b n q f x depends not only on the location but also on the multiplicity of the pole a phenomenon which has not been considered previously

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

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

dunkl analogue of szasz operators abstract the main purpose of this paper is to construct linear positive operators generated by dunkl generalization of exponential function we present qualitative result with the help of universal korovkin type theorem and weighted korovkin type theorem and also get quantitative estimates in terms of classical modulus of continuity the class of lipschitz functions peetre s k functional second order modulus of continuity and weighted modulus of continuity by dunkl analogue of szasz operators

r cyclic families of matrices in free probability we introduce the concept of r cyclic family of matrices with entries in a noncommutative probability space the definition consists in asking that only the cyclic noncrossing cumulants of the entries of the matrices are allowed to be nonzero let a1 as be an r cyclic family of d d matrices over a noncommutative probability space a we prove a convolution type formula for the explicit computation of the joint distribution of a1 as considered in md a with the natural state in terms of the joint distribution considered in the original space a of the entries of the s matrices several important situations of families of matrices with tractable joint distributions arise by application of this formula moreover let a1 as be a family of d d matrices over a noncommutative probability space a let d md a denote the algebra of scalar diagonal matrices and let c be the subalgebra of md a generated by a1 as d we prove that the r cyclicity of a1 as is equivalent to a property of c namely that c is free from md c with amalgamation over d

general de finetti type theorems in noncommutative probability we prove general de finetti type theorems for classical and free independence the de finetti type theorems work for all non easy quantum groups which generalize a recent work of banica curran and speicher we determine maximal distributional symmetries which means the corresponding de finetti type theorem fails if a sequence of random variables satisfy more symmetry relations other than the maximal one in addition we define boolean quantum semigroups in analogous to the easy quantum groups by universal conditions on matrix coordinate generators and an orthogonal projection then we show a general de finetti type theorem for boolean independence