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

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

on the bootstrap for moran s i test for spatial dependence this paper is concerned with the use of the bootstrap for statistics in spatial econometric models with a focus on the test statistic for moran s i test for spatial dependence we show that for many statistics in spatial econometric models the bootstrap can be studied based on linear quadratic lq forms of disturbances by proving the uniform convergence of the cumulative distribution function for lq forms to that of a normal distribution we show that the bootstrap is generally consistent for test statistics that can be approximated by lq forms including moran s i possible asymptotic refinements of the bootstrap are most commonly studied using edgeworth expansions for spatial econometric models we may establish asymptotic refinements of the bootstrap based on asymptotic expansions of lq forms when the disturbances are normal we prove the existence of the usual edgeworth expansions for lq forms when the disturbances are not normal we establish an asymptotic expansion of lq forms based on martingales these results are applied to show the second order correctness of the bootstrap for moran s i test

fluctuations of the product of random matrices and generalized lyapunov exponent i present a general framework allowing to carry out explicit calculation of the moment generating function of random matrix products varpi n m nm n 1 ldots m 1 where m i s are i i d following tutubalin theor probab appl 10 1 15 27 1965 the calculation of the generating function is reduced to finding the largest eigenvalue of a certain transfer operator associated with a family of representations of the group the formalism is illustrated by considering products of random matrices from the group mathrm sl 2 mathbb r where explicit calculations are possible for concreteness i study in detail transfer matrix products for the one dimensional schrodinger equation where the random potential is a levy noise derivative of a levy process in this case i obtain a general formula for the variance of ln varpi n and for the variance of ln psi x where psi x is the wavefunction in terms of a single integral involving the fourier transform of the invariant density of the matrix product finally i discuss the continuum limit of random matrix products matrices close to the identity in particular i investigate a simple case where the spectral problem providing the generalized lyapunov exponent can be solved exactly

on the well posedness of the full compressible navier stokes system in critical besov spaces abstract we are concerned with the cauchy problem of the full compressible navier stokes equations satisfied by viscous and heat conducting fluids in r n we focus on the so called critical besov regularity framework in this setting it is natural to consider initial densities 0 velocity fields u 0 and temperatures 0 with a 0 0 1 b p 1 n p u 0 b p 1 n p 1 and 0 b p 1 n p 2 after recasting the whole system in lagrangian coordinates and working with the total energy along the flow rather than with the temperature we discover that the system may be solved by means of banach fixed point theorem in a critical functional framework whenever the space dimension is n 2 and 1 p 2 n back to eulerian coordinates this allows to improve the range of p s for which the system is locally well posed compared to 7

products of many large random matrices and gradients in deep neural networks we study products of random matrices in the regime where the number of terms and the size of the matrices simultaneously tend to infinity our main theorem is that the logarithm of the ell 2 norm of such a product applied to any fixed vector is asymptotically gaussian the fluctuations we find can be thought of as a finite temperature correction to the limit in which first the size and then the number of matrices tend to infinity depending on the scaling limit considered the mean and variance of the limiting gaussian depend only on either the first two or the first four moments of the measure from which matrix entries are drawn we also obtain explicit error bounds on the moments of the norm and the kolmogorov smirnov distance to a gaussian finally we apply our result to obtain precise information about the stability of gradients in randomly initialized deep neural networks with relu activations this provides a quantitative measure of the extent to which the exploding and vanishing gradient problem occurs in a fully connected neural network with relu activations and a given architecture

the triangle law for lyapunov exponents of large random matrices for products a t a t 1 a 1 of i i d n n random matrices with i i d entries a triangle law governs then distribution of lyapunov exponents much like wigner s quarter circle law governs the singular values ofa 1 our proof requires finite fourth moments and a bounded density the result was previously derived only in the gaussian case

the precise asymptotic behavior of parameter estimators in ornstein uhlenbeck process abstract in the present paper we study the asymptotic behavior for estimator of the drift parameter in an ornstein uhlenbeck process the l r convergence rate and the precise asymptotics in the law of iterated logarithm and in the law of logarithm for the estimator are obtained moreover we also get the complete moment convergence of this estimator the main method of this paper is the deviation inequality for the quadratic functional

new conditions on global stability of cohen grossberg neural networks in this letter we discuss the dynamics of the cohen grossberg neural networks we provide a new and relaxed set of sufficient conditions for the cohen grossberg networks to be absolutely stable and exponentially stable globally we also provide an estimate of the rate of convergence

transfer matrices non hermitian hamiltonians and resolvents some spectral identities i consider the n step transfer matrix t for a general block hamiltonian with eigenvalue equation where and are matrices and provide its explicit representation in terms of blocks of the resolvent of the hamiltonian matrix for the system of length n with boundary conditions i then introduce the related hamiltonian for the case and and provide an exact relation between the trace of its resolvent and together with an identity of thouless type connecting tr with the hamiltonian eigenvalues for the results are then extended to by showing that it is itself a transfer matrix besides being of mathematical interest the identities should be useful for an analytical approach in the study of spectral properties of a physically relevant class of transfer matrices

transfer matrices and tridiagonal block hamiltonians with periodic and scattering boundary conditions general hamiltonian matrices with tridiagonal block structure and the associated transfer matrices are investigated in the cases of periodic and scattering boundary conditions they arise from tight binding models with finite range hopping in one or more dimensions of space in the presence of a aharonov bohm flux or in multichannel scattering an identity relating the characteristic equation of the periodic hamiltonian with that of the transfer matrix is found allowing a detailed analysis of the bands a velocity matrix is defined with properties relevant for the band structure or for the channel structure in the scattering problem

calculating hausdorff dimension of julia sets and kleinian limit sets we present a new algorithm for efficiently computing the hausdorff dimension of sets x invariant under conformal expanding dynamical systems by locating the periodic points of period up to n we construct approximations s n which converge to dim x super exponentially fast in n this method can be used to give rigorous estimates for important examples including hyperbolic julia sets and limit sets of schottky and quasifuchsian groups

compressible navier stokes system large solutions and incompressible limit abstract here we prove the existence of global in time regular solutions to the two dimensional compressible navier stokes equations supplemented with arbitrary large initial velocity v 0 and almost constant density 0 for large volume bulk viscosity the result is generalized to the higher dimensional case under the additional assumption that the strong solution of the classical incompressible navier stokes equations supplemented with the divergence free projection of v 0 is global the systems are examined in r d with d 2 in the critical b 2 1 s besov spaces framework

exact rates of convergence in some martingale central limit theorems abstract renz 14 ouchti 13 el machkouri and ouchti 3 and mourrat 12 have established some tight bounds on the rate of convergence in the central limit theorem for martingales in the present paper a modification of the methods developed by bolthausen 1 and grama and haeusler 7 is applied for obtaining exact rates of convergence in the central limit theorem for martingales with differences having conditional moments of order 2 0 our results generalise and strengthen the bounds mentioned above

computable examples of the maximal lyapunov exponent some new examples are given of sequences of matrix valued random variables for which it is possible to compute the maximal lyapunov exponent the examples are constructed by using a sequence of stopping times to group the original sequence into commuting blocks if the original sequence is the outcome of independent bernoulli trials with success probability p then the maximal lyapunov exponent may be expressed in terms of power series in p with explicit formulae for the coefficients the convexity of the maximal lyapunov exponent as a function of p is discussed as is an application to branching processes in a random environment

explicit invariant measures for products of random matrices we construct explicit invariant measures for a family of infinite products of random independent identically distributed elements of sl 2 c the matrices in the product are such that one entry is gamma distributed along a ray in the complex plane when the ray is the positive real axis the products are those associated with a continued fraction studied by letac seshadri z wahr verw geb 62 1983 485 489 who showed that the distribution of the continued fraction is a generalised inverse gaussian we extend this result by finding the distribution for an arbitrary ray in the complex right half plane and thus compute the corresponding lyapunov exponent explicitly when the ray lies on the imaginary axis the matrices in the infinite product coincide with the transfer matrices associated with a one dimensional discrete schrodinger operator with a random gamma distributed potential hence the explicit knowledge of the lyapunov exponent may be used to estimate the exponential rate of localisation of the eigenstates

malliavin calculus and asymptotic expansion for martingales we present an asymptotic expansion of the distribution of a random variable which admits a stochastic expansion around a continuous martingale the emphasis is put on the use of the malliavin calculus the uniform nondegeneracy of the malliavin covariance under certain truncation plays an essential role as the cramer condition did in the case of independent observations applications to statistics are presented

on a lp estimate for the linearized compressible navier stokes equations with the dirichlet boundary conditions abstract in this paper a special l p estimate for the linearized compressible navier stokes in the lagrangian coordinates for the dirichlet boundary conditions is obtained the constant in the estimate does not depend on the length of time interval 0 t the result is essential to obtain an existence for regular solutions for the nonlinear problem with the lowest class of regularity in l p spaces

harmless delays in cohen grossberg neural networks without assuming monotonicity and differentiability of the activation functions and any symmetry of interconnections we establish some sufficient conditions for the globally asymptotic stability of a unique equilibrium for the cohen grossberg neural network with multiple delays lyapunov functionals and functions combined with the razumikhin technique are employed the criteria are all independent of the magnitudes of the delays and thus the delays under these conditions are harmless 2002 elsevier science b v all rights reserved msc 34d23 34k20 92b20

rates of convergence in the martingale central limit theorem we obtain a nonuniform estimate of the rate of convergence in the martingale central limit theorem for convergence to mixtures of normal distributions the uniform rates of convergence obtained by several other authors are special cases of our nonuniform estimate we also obtain a rate of convergence in b m brown s central limit theorem assuming only brown s elementary conditions this result is a martingale analogue of feller s generalization of the berry esseen theorem

kantorovich distance in the martingale clt and quantitative homogenization of parabolic equations with random coefficients the article begins with a quantitative version of the martingale central limit theorem in terms of the kantorovich distance this result is then used in the study of the homogenization of discrete parabolic equations with random i i d coefficients for smooth initial condition the rescaled solution of such an equation once averaged over the randomness is shown to converge polynomially fast to the solution of the homogenized equation with an explicit exponent depending only on the dimension polynomial rate of homogenization for the averaged heat kernel with an explicit exponent is then derived similar results for elliptic equations are also presented

the distribution of lyapunov exponents exact results for random matrices simple exact expressions are derived for all the lyapunov exponents of certainn dimensional stochastic linear dynamical systems in the case of the product of independent random matrices each of which has independent gaussian entries with mean zero and variance 1 n the exponents have an exponential distribution asn in the case of the time ordered product integral of exp n 1 2dw where the entries of then n matrixw t are independent standard wiener processes the exponents are equally spaced for fixedn and thus have a uniform distribution as n

the cauchy problem for the compressible navier stokes equations in the l p framework the global in time existence of regular solutions to the compressible navier stokes equations in the whole space is obtained the solutions are close to nontrivial equilibrium solutions moreover the result is sharp in the lp framework the velocity of the fluid belongs to wr2 1 and the initial density to w1r with r 3

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

on the largest lyapunov exponent for products of gaussian matrices the paper provides a new integral formula for the largest lyapunov exponent of gaussian matrices which is valid in the real complex and quaternion valued cases this formula is applied to derive asymptotic expressions for the largest lyapunov exponent when the size of the matrix is large and compare the lyapunov exponents in models with a spike and no spikes

on the rate of convergence in the central limit theorem for martingales with discrete and continuous time heyde and brown 1970 established a bound on the rate of convergence in the central limit theorem for discrete time martingales having finite moments of order 2 28 with 0 0 moreover an example is constructed demonstrating that this bound is asymptotically exact for all 8 0 the result for discrete time martingales is then used to derive the corresponding bound on the rate of convergence in the central limit theorem for locally square integrable martingales with continuous time

lyapunov exponents of free operators abstract lyapunov exponents of a dynamical system are a useful tool to gauge the stability and complexity of the system this paper offers a definition of lyapunov exponents for a sequence of free linear operators the definition is based on the concept of the extended fuglede kadison determinant we establish the existence of lyapunov exponents derive formulas for their calculation and show that lyapunov exponents of free variables are additive with respect to operator product we illustrate these results using an example of free operators whose singular values are distributed by the marchenko pastur law and relate this example to c m newman s triangle law for the distribution of lyapunov exponents of large random matrices with independent gaussian entries as an interesting by product of our results we derive a relation between the extended fuglede kadison determinant and voiculescu s s transform

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

martingales and character ratios some general connections between martingales and character ratios of finite groups are developed as an application we sharpen the convergence rate in a central limit theorem for the character ratio of a random representation of the symmetric group on transpositions a generalization of these results is given for jack measure on partitions we also give a probabilistic proof of a result of burnside and brauer on the decomposition of tensor products

a global existence result for the compressible navier stokes equations in the critical l p framework the present paper is dedicated to the global well posedness issue for the barotropic compressible navier stokes system in the whole space mathbb r d with d 2 we aim at extending the work by danchin inventiones mathematicae 141 3 579 614 2000 to a critical framework which is not related to the energy space for small perturbations of a stable equilibrium state in the sense of suitable lp type besov norms we establish the global existence as a consequence like for incompressible flows one may exhibit a class of large highly oscillating initial velocity fields for which global existence and uniqueness holds true in passing we obtain new estimates for the linearized and the paralinearized systems which may be of interest for future works on compressible flows

boundedness and stability for cohen grossberg neural network with time varying delays abstract in this paper a model is considered to describe the dynamics of cohen grossberg neural network with variable coefficients and time varying delays uniformly ultimate boundedness and uniform boundedness are studied for the model by utilizing the hardy inequality combining with the halanay inequality and the lyapunov functional method some new sufficient conditions are derived for the model to be globally exponentially stable the activation functions are not assumed to be differentiable or strictly increasing moreover no assumption on the symmetry of the connection matrices is necessary these criteria are important in signal processing and the design of networks

on moderate deviations for martingales let x n x t n f t n 0 t 1 be square integrable martingales with the quadratic characteristics x n n 1 2 we prove that the large deviations relation p x 1 n r 1 r 1 holds true for r growing to infinity with some rate depending on l 2 n z 0 t 1 x t n 2 28 and n 2 n e x n 1 1 1 where 0 and l 2 n 0 n 2 n 0 as n the exact bound for the remainder is also obtained

existence and global asymptotic stability of positive periodic solution of delayed cohen grossberg neural networks abstract in this paper a class of periodic cohen grossberg neural networks with discrete and distributed time varying delays is considered by an extension of the lyapunov krasovskii functional method a novel criterion for the existence and uniqueness and global asymptotic stability of positive periodic solution is derived in terms of m matrix without any restriction on uniform positiveness of the amplification functions comparison and illustrative examples are given to illustrate the effectiveness of the obtained results

asymptotic behavior of periodic cohen grossberg neural networks with delays without assuming the positivity of the amplification functions we prove some m matrix criteria for the r n global asymptotic stability of periodic cohen grossberg neural networks with delays by an extension of the lyapunov method we are able to include neural systems with multiple nonnegative periodic solutions and nonexponential convergence rate in our model and also include the lotka volterra system an important prototype of competitive neural networks as a special case the stability criteria for autonomous systems then follow as a corollary two numerical examples are provided to show that the limiting equilibrium or periodic solution need not be positive

multiple decorrelation and rate of convergence in multidimensional limit theorems for the prokhorov metric the motivation of this work is the study of the error term et x in the averaging method for differential equations perturbed by a dynamical system results of convergence in distribution for frac e t varepsilon x cdot sqrt varepsilon varepsilon 0 have been established in khas minskii theory probab appl 11 1966 211 228 kifer ergodic theory dynamical systems 15 1995 1143 1172 and pene esaim probab statist 6 2002 33 88 we are interested here in the question of the rate of convergence in distribution of the family of random variables frac e t varepsilon x cdot sqrt varepsilon varepsilon 0 when goes to 0 t 0 and x in bf r d being fixed we will make an assumption of multiple decorrelation property satisfied in several situations we start by establishing a simpler result the rate of convergence in the central limit theorem for regular multidimensional functions in this context we prove a result of convergence in distribution with rate of convergence in o n 1 2 for all 0 for the prokhorov metric this result can be seen as an extension of the main result of pene comm math phys 225 2002 91 119 to the case of d dimensional functions in a second time we use the same method to establish a result of convergence in distribution for frac e t varepsilon x cdot sqrt varepsilon varepsilon 0 with rate of convergence in o 1 2 for the prokhorov metric we close this paper with a discussion in the appendix about the behavior of the quantity vert sup 0 le t le t 0 vert e t varepsilon x cdot vert infty vert l p under less stringent hypotheses

on the limit of the largest eigenvalue of the large dimensional sample covariance matrix in this paper the authors show that the largest eigenvalue of the sample covariance matrix tends to a limit under certain conditions when both the number of variables and the sample size tend to infinity the above result is proved under the mild restriction that the fourth moment of the elements of the sample sums of squares and cross products sp matrix exist

global existence of strong solution for viscous shallow water system with large initial data on the irrotational part abstract we are interested in studying the cauchy problem for the viscous shallow water system in dimension n 2 we show the existence of global strong solutions with large initial data on the irrotational part of the velocity for the scaling of the equations more precisely our smallness assumption on the initial data is supercritical for the scaling of the equations it allows us to give a first kind of answer to the problem of the existence of global strong solution with large initial energy data in dimension n 2 to do this we introduce the notion of quasi solutions which consists in solving the pressureless viscous shallow water system we can obtain such solutions at least for irrotational data which are subject to regularizing effects both on the velocity and on the density this smoothing effect is purely nonlinear and is crucial in order to build solution of the viscous shallow water system as perturbations of the quasi solutions indeed the pressure term can be considered as a remainder term which becomes small in high frequencies for the scaling of the equations to finish we prove the existence of global strong solution with large initial data when n 2 provided that the mach number is sufficiently large

an upper bound for the largest lyapunov exponent of a markovian product of nonnegative matrices we derive an upper bound for the largest lyapunov exponent of a markovian product of nonnegative matrices using markovian type counting arguments the bound is expressed as the maximum of a nonlinear concave function over a finite dimensional convex polytope of probability distributions

initial boundary value problems for the equations of motion of compressible viscous and heat conductive fluids the equations of motion of compressible viscous and heat conductive fluids are investigated for initial boundary value problems on the half space and on the exterior domain of any bounded region the global solution in time is proved to exist uniquely and approach the stationary state ast provided the prescribed initial data and the external force are sufficiently small

lyapunov exponents for branching processes in a random environment the effect of information we consider multitype branching processes evolving in a markovian random environment to determine whether or not the branching process becomes extinct almost surely is akin to computing the maximal lyapunov exponent of a sequence of random matrices which is a notoriously difficult problem we define markov chains associated to the branching process and we construct bounds for the lyapunov exponent the bounds are obtained by adding or by removing information to add information results in a lower bound to remove information results in an upper bound and we show that adding less information improves the lower bound we give a few illustrative examples and we observe that the upper bound is generally more accurate than the lower bounds

maximal lyapunov exponents for random matrix products in this article we study the lyapunov exponent for random matrix products of positive matrices and express them in terms of associated complex functions this leads to new explicit formulae for the lyapunov exponents and to an efficient method for their computation

global existence in critical spaces for compressible navier stokes equations we investigate global strong solutions for isentropic compressible fluids with initial data close to a stable equilibrium we obtain the existence and uniqueness of a solution in a functional setting invariant by the scaling of the associated equations more precisely the initial velocity has the same critical regularity index as for the incompressible homogeneous navier stokes equations and one more derivative is needed for the density we point out a smoothing effect on the velocity and a l 1 decay on the difference between the density and the constant reference state the proof lies on uniform estimates for a mixed hyperbolic parabolic linear system with a convection term

asymptotics of finite system lyapunov exponents for some random matrix ensembles for products pn of n random matrices of size there is a natural notion of finite n lyapunov exponents in the case of standard gaussian random matrices with real complex or real quaternion elements and extended to the general variance case for methods known for the computation of are used to compute the large n form of the variances of the exponents analogous calculations are performed in the case that the matrices making up pn are products of sub blocks of random unitary matrices with haar measure furthermore we make some remarks relating to the coincidence of the lyapunov exponents and the stability exponents relating to the eigenvalues of pn

analyticity of solutions to the barotropic compressible navier stokes equations abstract in this paper we establish analyticity of solutions to the barotropic compressible navier stokes equations describing the motion of the density and the velocity field u in r 3 we assume that 0 is a small perturbation of 1 and 1 1 0 u 0 are analytic in besov spaces with analyticity radius 0 we show that the corresponding solutions are analytic globally in time when 1 1 0 u 0 are sufficiently small to do this we introduce the exponential operator e t d acting on 1 1 u where d is the differential operator whose fourier symbol is given by 1 1 2 3 and t is chosen to satisfy t globally in time

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