universal microscopic correlation functions for products of independent ginibre matrices we consider the product of n complex non hermitian independent random matrices each of size n n with independent identically distributed gaussian entries ginibre matrices the joint probability distribution of the complex eigenvalues of the product matrix is found to be given by a determinantal point process as in the case of a single ginibre matrix but with a more complicated weight given by a meijer g function depending on n using the method of orthogonal polynomials we compute all eigenvalue density correlation functions exactly for finite n and fixed n they are given by the determinant of the corresponding kernel which we construct explicitly in the large n limit at fixed n we first determine the microscopic correlation functions in the bulk and at the edge of the spectrum after unfolding they are identical to that of the ginibre ensemble with n 1 and thus universal in contrast the microscopic correlations we find at the origin differ for each n 1 and generalize the known bessel law in the complex plane for n 2 to a new hypergeometric kernel 0fn 1

products of random matrices we derive analytic expressions for infinite products of random 2 x 2 matrices the determinant of the target matrix is log normally distributed whereas the remainder is a surprisingly complicated function of a parameter characterizing the norm of the matrix and a parameter characterizing its skewness the distribution may have importance as an uncommitted prior in statistical image analysis

raney distributions and random matrix theory recent works have shown that the family of probability distributions with moments given by the fuss catalan numbers permit a simple parameterized form for their density we extend this result to the raney distribution which by definition has its moments given by a generalization of the fuss catalan numbers such computations begin with an algebraic equation satisfied by the stieltjes transform which we show can be derived from the linear differential equation satisfied by the characteristic polynomial of random matrix realizations of the raney distribution for the fuss catalan distribution an equilibrium problem characterizing the density is identified the stieltjes transform for the limiting spectral density of the singular values squared of the matrix product formed from q inverse standard gaussian matrices and s standard gaussian matrices is shown to satisfy a variant of the algebraic equation relating to the raney distribution supported on 0 infty we show that it too permits a simple functional form upon the introduction of an appropriate choice of parameterization as an application the leading asymptotic form of the density as the endpoints of the support are approached is computed and is shown to have some universal features

probability of all eigenvalues real for products of standard gaussian matrices with xi independent n n standard gaussian random matrices the probability that all eigenvalues are real for the matrix product pm xmxm 1 x1 is expressed in terms of an n 2 n 2 n even and n 1 2 n 1 2 n odd determinant the entries of the determinant are certain meijer g functions in the case m 2 high precision computation indicates that the entries are rational multiples of 2 with the denominator a power of 2 and that to leading order in n decays as we are able to show that for general m and large n with an explicit bm an analytic demonstration that as m is given

robust probabilistic multivariate calibration model in this article we propose a robust probabilistic multivariate calibration rpmc model in an attempt to identify linear relationships between two sets of observed variables contaminated with outliers instead of the gaussian assumptions that predominate in classical statistical models rpmc is closely related with the multivariate student t distribution over noises and latent variables thus rpmc diminishes the effect of outlying data points by regulating the thickness of the distribution tails rpmc is essentially a robustified version of the supervised probabilistic principal component analysis sppca that has emerged recently we show that rpmc encompasses probabilistic principal component analysis and sppca as limiting cases we also derive an efficient em algorithm for parameter estimation in rpmc based on a probabilistic description of latent variables we present a procedure for the detection of outliers the experimental results from both simulated examples and real life data sets demonstrate t

singular value correlation functions for products of wishart random matrices we consider the product of m quadratic random matrices with complex elements and no further symmetry where all matrix elements of each factor have a gaussian distribution this generalizes the classical wishart laguerre gaussian unitary ensemble with m 1 in this paper we first compute the joint probability distribution for the singular values of the product matrix when the matrix size n and the number m are fixed but arbitrary this leads to a determinantal point process which can be realized in two different ways first it can be written as a one matrix singular value model with a nonstandard jacobian or second for m 2 as a two matrix singular value model with a set of auxiliary singular values and a weight proportional to the meijer g function for both formulations we determine all singular value correlation functions in terms of the kernels of biorthogonal polynomials which we explicitly construct they are given in terms of the hypergeometric and meijer g functions generalizing the laguerre polynomials for m 1 our investigation was motivated from applications in telecommunication of multi layered scattering multiple input and multiple output channels we present the ergodic mutual information for finite n for such a channel model with m 1 layers of scatterers as an example

products of independent quaternion ginibre matrices and their correlation functions we discuss the product of independent induced quaternion 4 ginibre matrices and the eigenvalue correlations of this product matrix the joint probability density function for the eigenvalues of the product matrix is shown to be identical to that of a single ginibre matrix but with a more complicated weight function we find the skew orthogonal polynomials corresponding to the weight function of the product matrix and use the method of skew orthogonal polynomials to compute the eigenvalue correlation functions for product matrices of finite size the radial behavior of the density of states is studied in the limit of large matrices and the macroscopic density is discussed the microscopic limit at the origin at the edge s and in the bulk is also discussed for the radial behavior of the density of states

products of rectangular random matrices singular values and progressive scattering we discuss the product of m rectangular random matrices with independent gaussian entries which have several applications including wireless telecommunication and econophysics for complex matrices an explicit expression for the joint probability density function is obtained using the harish chandra itzykson zuber integration formula explicit expressions for all correlation functions and moments for finite matrix sizes are obtained using a two matrix model and the method of biorthogonal polynomials this generalizes the classical result for the so called wishart laguerre gaussian unitary ensemble or chiral unitary ensemble at m 1 and previous results for the product of square matrices the correlation functions are given by a determinantal point process where the kernel can be expressed in terms of meijer g functions we compare the results with numerical simulations and known results for the macroscopic level density in the limit of large matrices the location of the end points of support for the latter are analyzed in detail for general m finally we consider the so called ergodic mutual information which gives an upper bound for the spectral efficiency of a mimo communication channel with multifold scattering

lyapunov exponents for products of rectangular real complex and quaternionic ginibre matrices we study the joint density of eigenvalues for products of independent rectangular real complex and quaternionic ginibre matrices in the limit where the number of matrices tends to infinity it is shown that the joint probability density function for the eigenvalues forms a permanental point process for all three classes the moduli of the eigenvalues become uncorrelated and log normal distributed while the distribution for the phases of the eigenvalues depends on whether real complex or quaternionic ginibre matrices are considered in the derivation for a product of real matrices we explicitly use the fact that all eigenvalues become real when the number of matrices tends to infinity finally we compare our results with known results for the lyapunov exponents as well as numerical simulations

random calibration with many measurements an application of stein estimation in the problem considered a vector of many imprecise measurements e g spectroscopic is used to linearly predict a quantity whose precise measurement is difficult or expensive the regression vector is estimated from a calibration experiment having both types of measurements for a random sample most previous approaches to this problem adjust for approximate multicollinearity which often results from the correlations among the imprecise measurements by inverting an approximation e g factor analytic to their covariance matrix in the approach here it is argued that the regression vector should lie in a lower dimensional suhspace determined by the principal components of the covariance matrix it is then estimated by applying a stein contraction of the least squares estimator to the principal components regression estimator examples using real data are presented in which the proposed estimator substantially improves on the ordinary least squares estimation

universal microscopic correlation functions for products of truncated unitary matrices we investigate the spectral properties of the product of m complex non hermitian random matrices that are obtained by removing l rows and columns of larger unitary random matrices uniformly distributed on the group u n l such matrices are called truncated unitary matrices or random contractions we first derive the joint probability distribution for the complex eigenvalues of the product matrix for fixed n l and m given by a standard determinantal point process in the complex plane the weight however is non standard and can be expressed in terms of the meijer g function the explicit knowledge of all eigenvalue correlation functions and the corresponding kernel allows us to take various large n and l limits at fixed m at strong non unitarity with l n finite the eigenvalues condense on a domain inside the unit circle at the edge and in the bulk we find the same universal microscopic kernel as for a single complex non hermitian matrix from the ginibre ensemble at the origin we find the same new universality classes labeled by m as for the product of m matrices from the ginibre ensemble keeping a fixed size of truncation l when n goes to infinity leads to weak non unitarity with most eigenvalues on the unit circle as for unitary matrices here we find a new microscopic edge kernel that generalizes the known results for m 1 we briefly comment on the case when each product matrix results from a truncation of different size lj

multivariate calibration with more variables than observations multivariate calibration involves the use of an estimated linear relationship between a q variate response vector y and a p dimensional explanatory vector x to estimate or predict future unknown x from further observed y in some applications for example when y represents automatic intensity measurements at q different wavelengths on n chemical standard samples the feasible sample size n may be restricted whereas the dimension q can be chosen quite large in this article the singular cases appearing when n q the traditional solutions to the estimation and prediction problems that is the generalized least squares estimator and the estimated best linear predictor are both unique whereas for smaller n q n 1 dimensional hyperplanes of solutions are obtained the same in both problems the properties of the predictor are also empirically studied in an example with p 1 q 6 and varying n

prediction diagnostics and updating in multivariate calibration summary in multivariate calibration the relationship between a q variate response vector y and p explanatory variables x are estimated from training data in order to predict an unknown x denoted by e from further observed responses when q p both the profile likelihood for e and bayesian inference for e depend on a prediction inconsistency diagnostic which highlights those response vectors used for prediction which are internally inconsistent in the prediction of e when several further response vectors together display systematic anomalies one is led to questioning the estimated model the information in prediction data about changes in parameters is investigated under various assumptions the results indicate systematic anomalies in prediction data may be detected in a variety of ways but corrected only under strong assumptions so that recalibration might be needed

products of independent elliptic random matrices for fixed m 1 we study the product of m independent n times n elliptic random matrices as n tends to infinity our main result shows that the empirical spectral distribution of the product converges with probability 1 to the m th power of the circular law regardless of the joint distribution of the mirror entries in each matrix this leads to a new kind of universality phenomenon the limit law for the product of independent random matrices is independent of the limit laws for the individual matrices themselves our result also generalizes earlier results of gotze tikhomirov on the asymptotic spectrum of products of independent random matrices available at http arxiv org abs 1012 2710 and o rourke soshnikov j probab 16 81 2219 2245 2011 concerning the product of independent iid random matrices

conditional value at risk in stochastic programs with mixed integer recourse in classical two stage stochastic programming the expected value of the total costs is minimized recently mean risk models studied in mathematical finance for several decades have attracted attention in stochastic programming we consider conditional value at risk as risk measure in the framework of two stage stochastic integer programming the paper addresses structure stability and algorithms for this class of models in particular we study continuity properties of the objective function both with respect to the first stage decisions and the integrating probability measure further we present an explicit mixed integer linear programming formulation of the problem when the probability distribution is discrete and finite finally a solution algorithm based on lagrangean relaxation of nonanticipativity is proposed

a note on the limiting mean distribution of singular values for products of two wishart random matrices the product of m complex random gaussian matrices of size n has recently been studied by akemann kieburg and wei they showed that for fixed m and n the joint probability distribution for the squared singular values of the product matrix forms a determinantal point process with a correlation kernel determined by certain biorthogonal polynomials that can be explicitly constructed we find that in the case m 2 the relevant biorthogonal polynomials are actually special cases of multiple orthogonal polynomials associated with macdonald functions modified bessel functions of the second kind which was first introduced by van assche and yakubovich with known results on asymptotic zero distribution of these polynomials and general theory on multiple orthogonal polynomial ensembles it is then easy to obtain an explicit expression for the distribution of squared singular values for the product of two complex random gaussian matrices in the limit of large matrix dimensions

universal distribution of lyapunov exponents for products of ginibre matrices starting from exact analytical results on singular values and complex eigenvalues of products of independent gaussian complex random n n matrices also called the ginibre ensemble we rederive the lyapunov exponents for an infinite product we show that for a large number t of product matrices the distribution of each lyapunov exponent is normal and we compute its t dependent variance as well as corrections in a large t expansion originally lyapunov exponents are defined for the singular values of the product matrix that represents a linear time evolution surprisingly a similar construction for the moduli of the complex eigenvalues yields the very same exponents and normal distributions to leading order we discuss a general mechanism for 2 2 matrices showing why the singular values and the radii of complex eigenvalues collapse onto the same value in the large t limit we thereby rederive newman s triangular law which has a simple interpretation as the radial density of complex eigenvalues in the circular law and study the commutativity of the two limits and on the global scale and on the local scale we show as a mathematical byproduct that a particular asymptotic expansion of a meijer g function with large index leads to a gaussian

singular values of products of ginibre random matrices multiple orthogonal polynomials and hard edge scaling limits akemann ipsen and kieburg recently showed that the squared singular values of products of m rectangular random matrices with independent complex gaussian entries are distributed according to a determinantal point process with a correlation kernel that can be expressed in terms of meijer g functions we show that this point process can be interpreted as a multiple orthogonal polynomial ensemble we give integral representations for the relevant multiple orthogonal polynomials and a new double contour integral for the correlation kernel which allows us to find its scaling limits at the origin hard edge the limiting kernels generalize the classical bessel kernels for m 2 they coincide with the scaling limits found by bertola gekhtman and szmigielski in the cauchy laguerre two matrix model which indicates that these kernels represent a new universality class in random matrix theory

isotropic brownian motions over complex fields as a solvable model for may wigner stability analysis we consider matrix valued stochastic processes known as isotropic brownian motions and show that these can be solved exactly over complex fields while these processes appear in a variety of questions in mathematical physics our main motivation is their relation to a may wigner like stability analysis for which we obtain a stability phase diagram the exact results establish the full joint probability distribution of the finite time lyapunov exponents and may be used as a starting point for a more detailed analysis of the stability instability phase transition our derivations rest on an explicit formulation of a fokker planck equation for the lyapunov exponents this formulation happens to coincide with an exactly solvable class of models of the calgero sutherland type originally encountered for a model of phase coherent transport the exact solution over complex fields describes a determinantal point process of biorthogonal type similar to recent results for products of random matrices and is also closely related to hermitian matrix models with an external source

from stochastic dominance to mean risk models semideviations as risk measures abstract two methods are frequently used for modeling the choice among uncertain outcomes stochastic dominance and mean risk approaches the former is based on an axiomatic model of risk averse preferences but does not provide a convenient computational recipe the latter quantifies the problem in a lucid form of two criteria with possible trade off analysis but cannot model all risk averse preferences in particular if variance is used as a measure of risk the resulting mean variance markowitz model is in general not consistent with stochastic dominance rules this paper shows that the standard semideviation square root of the semivariance as the risk measure makes the mean risk model consistent with the second degree stochastic dominance provided that the trade off coefficient is bounded by a certain constant similar results are obtained for the absolute semideviation and for the absolute and standard deviations in the case of symmetric or bounded distributions in the analysis we use a new tool the outcome risk o r diagram which appears to be particularly useful for comparing uncertain outcomes

spectral statistics for the difference of two wishart matrices in this work we consider the weighted difference of two independent complex wishart matrices and derive the joint probability density function of the corresponding eigenvalues in a finite dimension scenario using two distinct approaches the first derivation involves the use of unitary group integral while the second one relies on applying the derivative principle the latter relates the joint probability density of eigenvalues of a matrix drawn from a unitarily invariant ensemble to the joint probability density of its diagonal elements exact closed form expressions for an arbitrary order correlation function are also obtained and spectral densities are contrasted with monte carlo simulation results analytical results for moments as well as probabilities quantifying positivity aspects of the spectrum are also derived additionally we provide a large dimension asymptotic result for the spectral density using the stieltjes transform approach for algebraic random matrices finally we point out the relationship of these results with the corresponding results for difference of two random density matrices and obtain some explicit and closed form expressions for the spectral density and absolute mean

asymptotic analysis of rayleigh product channels a free probability approach the rayleigh product channel model is useful in capturing the performance degradation due to rank deficiency of mimo channels in this paper such a performance degradation is investigated via the distribution of mutual information assuming the block fading channels and the uniform power transmission scheme using techniques of free probability theory the asymptotic variance of mutual information is derived when the dimensions of the channel matrices approach infinity in this asymptotic regime the mutual information is rigorously proven to be gaussian distributed using the obtained results a fundamental tradeoff between multiplexing gain and diversity gain of rayleigh product channels under the uniform power transmission can be characterized by the closed form expression at any finite signal to noise ratio numerical results are provided to compare the outage performance between the rayleigh product channels and the conventional rayleigh mimo channels

differential equations for singular values of products of ginibre random matrices it was proved by akemann et al 2013 phys rev e 88 052118 that squared singular values of products of m complex ginibre random matrices form a determinantal point process whose correlation kernel is expressible in terms of meijer s g functions kuijlaars and zhang arxiv 1308 1003 recently showed that at the edge of the spectrum this correlation kernel has a remarkable scaling limit which can be understood as a generalization of the classical bessel kernel of random matrix theory in this paper we investigate the fredholm determinant of the operator with the kernel where j is a disjoint union of intervals and is the characteristic function of the set j this fredholm determinant is equal to the probability that j contains no particles of the limiting determinantal point process defined by the gap probability we derive the hamiltonian differential associated with the corresponding fredholm determinant and relate them with the monodromy preserving deformation equations of the jimbo miwa mori ueno and sato theory in the special case we give a formula for the gap probability in terms of a solution of a system of nonlinear ordinary differential equations

large n limit for the product of two coupled random matrices for a pair of coupled rectangular random matrices we consider the squared singular values of their product which form a determinantal point process we show that the limiting mean distribution of these squared singular values is described by the second component of the solution to a vector equilibrium problem this vector equilibrium problem is defined for three measures with an upper constraint on the first measure and an external field on the second measure we carry out the steepest descent analysis for a 4 times 4 matrix valued riemann hilbert problem which characterizes the correlation kernel and is related to mixed type multiple orthogonal polynomials associated with the modified bessel functions a careful study of the vector equilibrium problem combined with this asymptotic analysis ultimately leads to the aforementioned convergence result for the limiting mean distribution an explicit form of the associated spectral curve as well as local sine meijer g and airy universality results for the squared singular values considered

limit theorems for biorthogonal ensembles and related combinatorial identities abstract we study the fluctuations of certain biorthogonal ensembles for which the underlying family p q satisfies a finite term recurrence relation of the form x p x j p x for polynomial linear statistics of such ensembles we reformulate the cumulant method introduced in 53 in terms of counting certain lattice paths on the adjacency graph of the recurrence matrix j in the spirit of 12 we show that the asymptotic fluctuations are described by the right limits of the matrix j moreover whenever the right limit is a laurent matrix we show that the clt is equivalent to soshnikov s main combinatorial lemma we discuss several applications to unitary invariant hermitian random matrices in particular we provide a general central limit theorem clt and a law of large numbers in the one cut regime we also prove a clt for the square singular values of the product of independent complex rectangular ginibre matrices as well as for the laguerre and jacobi biorthogonal ensembles introduced in 7 and we explain how to recover the equilibrium measure from the asymptotics of the recurrence coefficients finally we discuss the connection with the strong szeg limit theorem where this combinatorial method originates

universality of local spectral statistics of products of random matrices we derive exact analytical expressions for correlation functions of singular values of the product of m ginibre matrices of size n in the double scaling limit m n ensuremath rightarrow ensuremath infty the singular value statistics is described by a determinantal point process with a kernel that interpolates between gaussian unitary ensemble statistic and dirac delta picket fence statistic in the thermodynamic limit n ensuremath rightarrow ensuremath infty the interpolation parameter is given by the limiting quotient a n m one of our goals is to find an explicit form of the kernel at the hard edge in the bulk and at the soft edge for any a we find that in addition to the standard scaling regimes there is a transitional regime which interpolates between the hard edge and the bulk we conjecture that these results are universal and that they apply to a broad class of products of random matrices from the gaussian basin of attraction including correlated matrices we corroborate this conjecture by numerical simulations additionally we show that the local spectral statistics of the considered random matrix products is identical with the local statistics of dyson brownian motion with the initial condition given by equidistant positions with the crucial difference that this equivalence holds only locally finally we have identified a mesoscopic spectral scale at the soft edge which is crucial for the unfolding of the spectrum

restricted least squares estimation of the spectra and concentration of two unknown constituents available in mixtures the problem considered is the identification of two unknown chemical compounds and the estimation of their proportions in a set of unknown mixtures of the two compounds given data that are vectors of measurements on their mixtures it is assumed that the expected value of a mixture vector is an unknown convex linear combination of two unknown component vectors and least squares estimation is used to obtain a set of possible solutions of the mixing proportions and the component vectors obtaining a unique solution requires additional constraints or information the solution set is interpreted geometrically and examples involving amino acids and light absorbance data are given

errors in variables estimation in multivariate calibration a set of q responses y y 1 y 2 y q t is related to a set of p explanatory variables x x 1 x 2 x p t through the classical linear regression model y t a t x t b e t first the unknown parameters a and b are estimated using a calibrafion set the statistical problem that is considered here is that of estimating the vector x o that underlies a new observed vector of responses y o using the parameter estimates obtained from the first procedure these two procedures are commonly referred to as calibration and prediction or inverse prediction and sometimes jointly referred to as calibration the prediction procedure can be viewed as parameter estimation in errors in variables regression the maximum likelihood estimator assuming normally distributed measurement errors is proposed for the prediction procedure unlike the classical estimator used in the prediction procedure the proposed estimator is consistent with respect to the number of response variables the performan

matrices coupled in a chain i eigenvalue correlations the general correlation function for the eigenvalues of p complex hermitian matrices coupled in a chain is given as a single determinant for this we use a slight generalization of a theorem of dyson

the cauchy two matrix model we introduce a new class of two multi matrix models of positive hermitian matrices coupled in a chain the coupling is related to the cauchy kernel and differs from the exponential coupling more commonly used in similar models the correlation functions are expressed entirely in terms of certain biorthogonal polynomials and solutions of appropriate riemann hilbert problems thus paving the way to a steepest descent analysis and universality results the interpretation of the formal expansion of the partition function in terms of multicolored ribbon graphs is provided and a connection to the o 1 model a steepest descent analysis of the partition function reveals that the model is related to a trigonal curve three sheeted covering of the plane much in the same way as the hermitian matrix model is related to a hyperelliptic curve

integrable structure of products of finite complex ginibre random matrices abstract we consider the squared singular values of the product of m standard complex gaussian matrices since the squared singular values form a determinantal point process with a particular meijer g function kernel the gap probabilities are given by a fredholm determinant based on this kernel it was shown by strahov 2014 that a hard edge scaling limit of the gap probabilities is described by hamiltonian differential equations which can be formulated as an isomonodromic deformation system similar to the theory of the kyoto school we generalize this result to the case of finite matrices by first finding a representation of the finite kernel in integrable form as a result we obtain the hamiltonian structure for finite size matrices and formulate it in terms of a m 1 m 1 matrix schlesinger system the case m 1 reproduces the tracy and widom theory which results in the painleve v equation for the 0 s gap probability some integrals of motion for m 2 are identified and a coupled system of differential equations in two unknowns is presented which uniquely determines the corresponding 0 s gap probability

multivariate calibration when the error covariance matrix is structured this article concerns multivariate calibration in linear models the error covariance matrix is assumed to have linear factor structure a new approach to calibration incorporating distributional properties of the observations of the calibration set is proposed the proposed predictor is based on estimating the parameters in the best linear predictor in the assumed model the predictor is tested on two data sets meat data and fish data

generating random density matrices we study various methods to generate ensembles of random density matrices of a fixed size n obtained by partial trace of pure states on composite systems structured ensembles of random pure states invariant with respect to local unitary transformations are introduced to analyze statistical properties of quantum entanglement in bi partite systems we analyze the distribution of schmidt coefficients of random pure states such a distribution is derived in the case of a superposition of k random maximally entangled states for another ensemble obtained by performing selective measurements in a maximally entangled basis on a multi partite system we show that this distribution is given by the fuss catalan law and find the average entanglement entropy a more general class of structured ensembles proposed containing also the case of bures forms an extension of the standard ensemble of structureless random pure states described asymptotically as n by the marchenko pastur distribution

local central limit theorem for determinantal point processes we prove a local central limit theorem lclt for the number of points n j in a region j in mathbb r d specified by a determinantal point process with an hermitian kernel the only assumption is that the variance of n j tends to infinity as j rightarrow infty this extends a previous result giving a weaker central limit theorem for these systems our result relies on the fact that the lee yang zeros of the generating function for e k j the probabilities of there being exactly k points in j all lie on the negative real z axis in particular the result applies to the scaled bulk eigenvalue distribution for the gaussian unitary ensemble gue and that of the ginibre ensemble for the gue we can also treat the properly scaled edge eigenvalue distribution using identities between gap probabilities the lclt can be extended to bulk eigenvalues of the gaussian symplectic ensemble a lclt is also established for the probability density function of the k th largest eigenvalue at the soft edge and of the spacing between k th neighbors in the bulk

weak commutation relations and eigenvalue statistics for products of rectangular random matrices we study the joint probability density of the eigenvalues of a product of rectangular real complex or quaternion random matrices in a unified way the random matrices are distributed according to arbitrary probability densities whose only restriction is the invariance under left and right multiplication by orthogonal unitary or unitary symplectic matrices respectively we show that a product of rectangular matrices is statistically equivalent to a product of square matrices hereby we prove a weak commutation relation of the random matrices at finite matrix sizes which previously has been discussed for infinite matrix size moreover we derive the joint probability densities of the eigenvalues to illustrate our results we apply them to a product of random matrices drawn from ginibre ensembles and jacobi ensembles as well as a mixed version thereof for these weights we show that the product of complex random matrices yields a determinantal point process while the real and quaternion matrix ensembles correspond to pfaffian point processes our results are visualized by numerical simulations furthermore we present an application to a transport on a closed disordered chain coupled to a particle bath

double points in nonlinear calibration summary in nonlinear calibration and regression a double point is defined with respect to estimation and to confidence sets loosely speaking if a future observation is a double point then multiple point estimates or disconnected confidence regions will occur we investigate the extent of the set of double points a diagnostic the probability of their occurrence is computed

continuum regression cross validated sequentially constructed prediction embracing ordinary least squares partial least squares and principal components regression read before the royal statistical society at a meeting organized by the research section on wednesday october 25th 1989 professor d v hinkley in the chair summary the paper addresses the evergreen problem of construction of regressors for use in least squares multiple regression in the context of a general sequential procedure for doing this it is shown that with a particular objective criterion for the construction the procedures of ordinary least squares and principal components regression occupy the opposite ends of a continuous spectrum with partial least squares lying in between there are two adjustable parameters controlling the procedure alpha in the continuum 0 11 and omega the number of regressors finally accepted these control parameters are chosen by crossvalidation the method is illustrated by a range of examples of its application

induced ginibre ensemble of random matrices and quantum operations a generalization of the ginibre ensemble of non hermitian random square matrices is introduced the corresponding probability measure is induced by the ensemble of rectangular gaussian matrices via a quadratization procedure we derive the joint probability density of eigenvalues for such an induced ginibre ensemble and study various spectral correlation functions for complex and real matrices and analyse universal behaviour in the limit of large dimensions in this limit the eigenvalues of the induced ginibre ensemble cover uniformly a ring in the complex plane the real induced ginibre ensemble is shown to be useful to describe the statistical properties of evolution operators associated with random quantum operations for which the dimensions of the input state and the output state do differ

the supersymmetry method for chiral random matrix theory with arbitrary rotation invariant weights in the past few years the supersymmetry method has been generalized to real symmetric hermitian and hermitian self dual random matrices drawn from ensembles invariant under the orthogonal unitary and unitary symplectic groups respectively we extend this supersymmetry approach to chiral random matrix theory invariant under the three chiral unitary groups in a unifying way thereby we generalize a projection formula providing a direct link and hence a short cut between the probability density in ordinary space and that in superspace we emphasize that this point was one of the main problems and shortcomings of the supersymmetry method since only implicit dualities between ordinary space and superspace were known before to provide examples we apply this approach to the calculation of the supersymmetric analogue of a lorentzian cauchy ensemble and an ensemble with a quartic potential moreover we consider the partially quenched partition function of the three chiral gaussian ensembles corresponding to four dimensional continuum quantum chromodynamics we identify a natural splitting of the chiral lagrangian in its lowest order into a part for the physical mesons and a part associated with source terms generating the observables e g the level density of the dirac operator

eigenvalue statistics for product complex wishart matrices the eigenvalue statistics for complex wishart matrices where is equal to the product of r complex gaussian matrices and the inverse of s complex gaussian matrices are considered in the case r s the exact form of the global density is computed the averaged characteristic polynomial for the corresponding generalized eigenvalue problem is calculated in terms of a particular generalized hypergeometric function for finite n the eigenvalue probability density function is computed and is shown to be an example of a biorthogonal ensemble a double contour integral form of the corresponding correlation kernel is derived which allows the hard edge scaled limit to be computed the limiting kernel is given in terms of certain meijer g functions and is identical to that found in the recent work of kuijlaars and zhang in the case s 0 properties of the kernel and corresponding correlation functions are discussed

product of ginibre matrices fuss catalan and raney distributions squared singular values of a product of s square random ginibre matrices are asymptotically characterized by probability distributions p s x such that their moments are equal to the fuss catalan numbers of order s we find a representation of the fuss catalan distributions p s x in terms of a combination of s hypergeometric functions of the type s f s 1 the explicit formula derived here is exact for an arbitrary positive integer s and for s 1 it reduces to the marchenko pastur distribution using similar techniques involving the mellin transform and the meijer g function we find exact expressions for the raney probability distributions the moments of which are given by a two parameter generalization of the fuss catalan numbers these distributions can also be considered as a two parameter generalization of the wigner semicircle law

dropping the independence singular values for products of two coupled random matrices we study the singular values of the product of two coupled rectangular random matrices as a determinantal point process each of the two factors is given by a parameter dependent linear combination of two independent complex gaussian random matrices which is equivalent to a coupling of the two factors via an itzykson zuber term we prove that the squared singular values of such a product form a biorthogonal ensemble and establish its exact solvability the parameter dependence allows us to interpolate between the singular value statistics of the laguerre ensemble and that of the product of two independent complex ginibre ensembles which are both known we give exact formulae for the correlation kernel in terms of a complex double contour integral suitable for the subsequent asymptotic analysis in particular we derive a christoffel darboux type formula for the correlation kernel based on a five term recurrence relation for our biorthogonal functions it enables us to find its scaling limit at the origin representing a hard edge the resulting limiting kernel coincides with the universal meijer g kernel found by several authors in different ensembles we show that the central limit theorem holds for the linear statistics of the singular values and give the limiting variance explicitly

exact evaluations of some meijer g functions and probability of all eigenvalues real for the product of two gaussian matrices we provide a proof to a recent conjecture by forrester 2014 j phys a math theor 47 065202 regarding the algebraic and arithmetic structure of meijer g functions which appear in the expression for probability of all eigenvalues real for the product of two real gaussian matrices in the process we come across several interesting identities involving meijer g functions

universality conjecture and results for a model of several coupled positive definite matrices the paper contains two main parts in the first part we analyze the general case of p geq 2 matrices coupled in a chain subject to cauchy interaction similarly to the itzykson zuber interaction model the eigenvalues of the cauchy chain form a multi level determinantal point process we first compute all correlations functions in terms of cauchy biorthogonal polynomials and locate them as specific entries of a p 1 times p 1 matrix valued solution of a riemann hilbert problem in the second part we fix the external potentials as classical laguerre weights we then derive strong asymptotics for the cauchy biorthogonal polynomials when the support of the equilibrium measures contains the origin as a result we obtain a new family of universality classes for multi level random determinantal point fields which include the bessel nu universality for 1 level and the meijer g universality for 2 level our analysis uses the deift zhou nonlinear steepest descent method and the explicit construction of a p 1 times p 1 origin parametrix in terms of meijer g functions the solution of the full riemann hilbert problem is derived rigorously only for p 3 but the general framework of the proof can be extended to the cauchy chain of arbitrary length p

spectral density of a wishart model for nonsymmetric correlation matrices the wishart model for real symmetric correlation matrices is defined as mathsf w mathsf aa t where matrix mathsf a is usually a rectangular gaussian random matrix and mathsf a t is the transpose of mathsf a analogously for nonsymmetric correlation matrices a model may be defined for two statistically equivalent but different matrices mathsf a and mathsf b as mathsf ab t the corresponding wishart model thus is defined as mathbf c mathsf ab t mathsf ba t we study the spectral density of mathbf c for the case when mathsf a and mathsf b are not statistically independent the ensemble average of such nonsymmetric matrices therefore does not simply vanishes to a null matrix in this paper we derive a pastur self consistent equation which describes spectral density of mathbf c at large matrix dimension we complement our analytic results with numerics

random graph states maximal flow and fuss catalan distributions for any graph consisting of k vertices and m edges we construct an ensemble of random pure quantum states which describe a system composed of 2m subsystems each edge of the graph represents a bipartite maximally entangled state each vertex represents a random unitary matrix generated according to the haar measure which describes the coupling between subsystems dividing all subsystems into two parts one may study entanglement with respect to this partition a general technique to derive an expression for the average entanglement entropy of random pure states associated with a given graph is presented our technique relies on weingarten calculus and flow problems we analyze the statistical properties of spectra of such random density matrices and show for which cases they are described by the free poissonian marchenko pastur distribution we derive a discrete family of generalized fuss catalan distributions and explicitly construct graphs which lead to ensembles of random states characterized by these novel distributions of eigenvalues

spectral density of generalized wishart matrices and free multiplicative convolution we investigate the level density for several ensembles of positive random matrices of a wishart like structure w xx where x stands for a non hermitian random matrix in particular making use of the cauchy transform we study the free multiplicative powers of the marchenko pastur mp distribution mp s which for an integer s yield fuss catalan distributions corresponding to a product of s independent square random matrices x x 1 x s new formulas for the level densities are derived for s 3 and s 1 3 moreover the level density corresponding to the generalized bures distribution given by the free convolution of arcsine and mp distributions is obtained we also explain the reason of such a curious convolution the technique proposed here allows for the derivation of the level densities for several other cases

eigenvalues and singular values of products of rectangular gaussian random matrices we derive exact analytic expressions for the distributions of eigenvalues and singular values for the product of an arbitrary number of independent rectangular gaussian random matrices in the limit of large matrix dimensions we show that they both have power law behavior at zero and determine the corresponding powers we also propose a heuristic form of finite size corrections to these expressions which very well approximates the distributions for matrices of finite dimensions

spectrum of the product of independent random gaussian matrices we show that the eigenvalue density of a product x x1x2 xm of m independent nxn gaussian random matrices in the limit n infinity is rotationally symmetric in the complex plane and is given by a simple expression rho z z 1 mpisigma 2 m z 2 2 m for z sigma the parameter sigma corresponds to the radius of the circular support and is related to the amplitude of the gaussian fluctuations this form of the eigenvalue density is highly universal it is identical for products of gaussian hermitian non hermitian and real or complex random matrices it does not change even if the matrices in the product are taken from different gaussian ensembles we present a self contained derivation of this result using a planar diagrammatic technique additionally we conjecture that this distribution also holds for any matrices whose elements are independent centered random variables with a finite variance or even more generally for matrices which fulfill pastur lindeberg s condition we provide a numerical evidence supporting this conjecture

eigenvector statistics of the product of ginibre matrices we develop a method to calculate left right eigenvector correlations of the product of m independent n n complex ginibre matrices for illustration we present explicit analytical results for the vector overlap for a couple of examples for small m and n we conjecture that the integrated overlap between left and right eigenvectors is given by the formula o 1 m 2 n 1 and support this conjecture by analytical and numerical calculations we derive an analytical expression for the limiting correlation density as n for the product of ginibre matrices as well as for the product of elliptic matrices in the latter case we find that the correlation function is independent of the eccentricities of the elliptic laws