central limit theorem for linear eigenvalue statistics of random matrices with independent entries we consider n x n real symmetric and hermitian wigner random matrices n 1 2 w with independent modulo symmetry condition entries and the null sample covariance matrices n 1 x x with independent entries of m x n matrix x assuming first that the 4th cumulant excess 4 of entries of w and x is zero and that their 4th moments satisfy a lindeberg type condition we prove that linear statistics of eigenvalues of the above matrices satisfy the central limit theorem clt as n m m n c 0 oo with the same variance as for gaussian matrices if the test functions of statistics are smooth enough essentially of the class c 5 this is done by using a simple interpolation trick from the known results for the gaussian matrices and the integration by parts presented in the form of certain differentiation formulas then by using a more elaborated version of the techniques we prove the clt in the case of nonzero excess of entries again for essentially 5 test function here the variance of statistics contains an additional term proportional to 4 the proofs of all limit theorems follow essentially the same scheme

gaussian limit for determinantal random point fields author s soshnikov alexander abstract we prove that under fairly general conditions properly rescaled determinantal random point field converges to a generalized gaussian random process

correlation functions cluster functions and spacing distributions for random matrices the usual formulas for the correlation functions in orthogonal and symplectic matrix models express them as quaternion determinants from this representation one can deduce formulas for spacing probabilities in terms of fredholm determinants of matrix valued kernels the derivations of the various formulas are somewhat involved in this article we present a direct approach which leads immediately to scalar kernels for the unitary ensembles and matrix kernels for the orthogonal and symplectic ensembles and the representations of the correlation functions cluster functions and spacing distributions in terms of them

adjusted estimates and wald statistics for the ar 1 model with constant abstract adjusted profile likelihood of cox and reid 1993 journal of the royal statistical society b 55 467 471 for the autoregressive coefficient of the ar 1 model with constant yields an estimator which can be more accurate than the maximum likelihood estimator in small samples whether the process is stationary or a random walk and in the latter case also asymptotically a related estimator is asymptotically distributed compactly and symmetrically around zero under a random walk but is not consistent in general the adjusted wald statistic can follow the asymptotic standard normal distribution much better than the non adjusted under stationarity or a drifted random walk

universality of level correlation function of sparse random matrices the statistical properties of sparse random matrices ensembles are investigated by means of a supersymmetric approach with the use of a functional generalization of the hubbard stratonovich transformation when used to calculate the density of states the method is shown to be absolutely equivalent to the replica trick the model turns out to bear a close resemblance to the anderson model on the bethe lattice it possesses a delocalization transition that occurs with an increase in the mean connectivity parameter in the delocalized phase the level level correlation function proves to have a universal dyson form with the full density of states replaced by the contribution from the infinite cluster

clt for linear spectral statistics of large dimensional sample covariance matrices let bn 1 n t1 2nxnx nt1 2n where xn xij is n n with i i d complex standardized entries having finite fourth moment and t1 2n is a hermitian square root of the nonnegative definite hermitian matrix tn the limiting behavior as n with n n approaching a positive constant of functionals of the eigenvalues of bn where each is given equal weight is studied due to the limiting behavior of the empirical spectral distribution of bn it is known that these linear spectral statistics converges a s to a nonrandom quantity this paper shows their rate of convergence to be 1 n by proving after proper scaling that they form a tight sequence moreover if exppx211 0 and expp x11 4 2 or if x11 and tn are real and exppx411 3 they are shown to have gaussian limits

level spacings distribution for large random matrices gaussian fluctuations we study the level spacings distribution for eigenvalues of large n x n matrices from the classical compact groups in the scaling limit when the mean distance between nearest eigenvalues equals 1 defining by 77n s the number of nearest neighbors spacings greater than s 0 smaller than s 0 we prove functional limit theorem for the process 77n s e n s n1 2 giving weak convergence of this distribution to some gaussian random process on 0 x the limiting gaussian random process is universal for all classical compact groups it is holder continuous with any exponent less than 1 2 similar results can be obtained for the n level spacings distribution

on the influence of sampling and observation times on estimation of the bandwidth parameter of a gauss markov process the statistical problem of estimating the bandwidth parameter of a gauss markov gm process from a realization of fixed and finite duration t at selectable sampling interval spl delta is addressed in this paper as the observation time t is fixed and finite the variance of estimated autocorrelation and continuous time parameter does not vanish as spl delta approaches 0 this necessitates a second order taylor expansion in deriving the parameter estimator bias and variance the second order taylor expansion produces better bias and variance results than a first order one does the distribution of the estimator is also discussed according to the gradient change of the variance a key result is that three sample size regions which are termed finite large and very large corresponding to substantial gradual and very slight decrease in the variance of the parameter estimator respectively are quantified in terms of analysis bw the three regions are 23 35 35 55 and 55 spl infin db the characterization of the tradeoff between the variance decrease and sampling rate results in a practical guideline for choosing sampling rate these results are applied to the prediction problems of a time invariant gm process to show their value

a generalization of wigner s law we present a generalization of wigner s semicircle law we consider a sequence of probability distributions p 1 p 2 dots with mean value zero and take an n n real symmetric matrix with entries independently chosen from p n and analyze the distribution of eigenvalues if we normalize this distribution by its dispersion we show that as n for certain p n the distribution weakly converges to a universal distribution the result is a formula for the moments of the universal distribution in terms of the rate of growth of the k th moment of p n as a function of n and describe what this means in terms of the support of the distribution as a corollary when p n does not depend on n we obtain wigner s law if all moments of a distribution are finite the distribution of eigenvalues is a semicircle

bias in the estimation of the mean reversion parameter in continuous time models abstract it is well known that for continuous time models with a linear drift standard estimation methods yield biased estimators for the mean reversion parameter both in finite discrete samples and in large in fill samples in this paper we obtain two expressions to approximate the bias of the least squares maximum likelihood estimator of the mean reversion parameter in the ornstein uhlenbeck process with a known long run mean when discretely sampled data are available the first expression mimics the bias formula of marriott and pope 1954 for the discrete time model simulations show that this expression does not work satisfactorily when the speed of mean reversion is slow slow mean reversion corresponds to the near unit root situation and is empirically realistic for financial time series an improvement is made in the second expression where a nonlinear correction term is included into the bias formula it is shown that the nonlinear term is important in the near unit root situation simulations indicate that the second expression captures the magnitude the curvature and the non monotonicity of the actual bias better than the first expression

finite n fluctuation formulas for random matrices for the gaussian and laguerre random matrix ensembles the probability density function p d f for the linear statistic jn 1 xj x is computed exactly and shown to satisfy a central limit theorem asn for the circular random matrix ensemble the p d f s for the statistics jn 1 j and jn 1 log 2 sin j 2 are calculated exactly by using a constant term identity from the theory of the selberg integral and are also shown to satisfy a central limit theorem asn

theory of levy matrices we investigate the statistical properties of the spectrum of large symmetrical matrices with each element mathit h mathit i mathit j chosen according to a broad distribution ensuremath rho h decaying for large h as mathit h mathrm ensuremath 1 mathrm ensuremath mathrm ensuremath mu for ensuremath mu g2 mathit h 2 is finite and the well known gaussian orthogonal ensemble goe results are recovered when ensuremath mu 2 the semicircular law is replaced by a density which extends over the whole energy axis furthermore while all states are extended in the case of goe matrices we show numerically and analytically that two mobility edges appear separating extended from localized states with an intermediate mixed phase in between the unusual nature of these localized states is discussed

the circular unitary ensemble and the riemann zeta function the microscopic landscape and a new approach to ratios we show in this paper that after proper scalings the characteristic polynomial of a random unitary matrix converges to a random analytic function whose zeros which are on the real line form a determinantal point process with sine kernel our scaling is performed at the so called microscopic level that is we consider the characteristic polynomial at points whose distance to 1 has order 1 n we prove that the rescaled characteristic polynomial does not even have a moment of order one hence making the classical techniques of random matrix theory difficult to apply in order to deal with this issue we couple all the dimensions n on a single probability space in such a way that almost sure convergence occurs when n goes to infinity the strong convergence results in this setup provide us with a new approach to ratios we are able to solve open problems about the limiting distribution of ratios of characteristic polynomials evaluated at points of the form exp 2 i pi alpha n and related objects such as the logarithmic derivative we also explicitly describe the dependence relation for the logarithm of the characteristic polynomial evaluated at several points on the microscopic scale on the number theory side inspired by the work by keating and snaith we conjecture some new limit theorems for the value distribution of the riemann zeta function on the critical line at the level of stochastic processes

second order asymptotics for matrix models we study several matrix models and show that when the potential is convex and a small perturbation of the gaussian potential the first order correction to the free energy can be expressed as a generating function for the enumeration of maps of genus one in order to do that we prove a central limit theorem for traces of words of the weakly interacting random matrices defined by these matrix models and show that the variance is a generating function for the number of planar maps with two vertices with prescribed colored edges

the fixed trace β hermite ensemble of random matrices and the low temperature distribution of the determinant of an n n β hermite matrix the hermite ensemble he of tridiagonal n n random matrices of dumitriu and edelman 2002 j math phys 43 5830 is a continuum of ensembles in which the reciprocal of the temperature in the 2d electrostatic interpretation of the eigenvalue characteristics can take any value the eigenvalue distributions coincide with those of the classical gaussian ensembles goe gue gse for 1 2 4 a fixed trace hermite ensemble fthe is defined from the he and is used to extend the spherical ensembles of classical symmetries to spherical ensembles at low temperature when for a fixed value of n the asymptotic distributions of reduced determinants dn of random n n h and fth matrices are shown to be standard gaussians accordingly the fluctuations of the potential at the origin ln dn have a generalized gumbel distribution at low temperature for large n and large a ln n variance results from the strongly correlated fluctuations of eigenvalues around their equilibrium positions

central limit theorem for linear spectral statistics of large dimensional quaternion sample covariance matrices abstract laguerre wishart ensembles where known as dyson s threefold way are important in random matrix theory these classical matrices models have been considered by many authors in this paper we investigate the central limit theorem for linear spectral statistics of large dimensional quaternion sample covariance matrices which include the wishart quaternion ensembles as a special case combining with existing results a general central limit theorem where are parameters in the expressions of limit mean and variance is also given

central limit theorem for fluctuations of linear eigenvalue statistics of large random graphs diluted regime we study the linear eigenvalue statistics of large random graphs in the regimes when the mean number of edges for each vertex tends to infinity we prove that for a rather wide class of test functions the fluctuations of linear eigenvalue statistics converges in distribution to a gaussian random variable with zero mean and variance which coincides with non gaussian part of the wigner ensemble variance

noninformative priors and frequentist risks of bayesian estimators of vector autoregressive models in this study we examine posterior properties and frequentist risks of bayesian estimators based on several noninformative priors in vector autoregressive var models we prove existence of the posterior distributions and posterior moments under a general class of priors using a variety of priors in this class we conduct numerical simulations of posteriors we find that in most examples bayesian estimators with a shrinkage prior on the var coefficients and the reference prior of yang and berger ann statist 22 1994 1195 on the var covariance matrix dominate mle bayesian estimators with the diffuse prior and bayesian estimators with the prior used in rats we also examine the informative minnesota prior and find that its performance depends on the nature of the data sample and on the tightness of the minnesota prior a tightly set minnesota prior is better when the data generating processes are similar to random walks but the shrinkage prior or constant prior can be better otherwise

gaussian fluctuations for linear spectral statistics of wigner beta ensembles abstract as an important topic in mathematical physics and statistics random matrices theory has found uses in many aspects of modern physics and multivariate analysis this paper is to investigate the gaussian fluctuations for linear spectral statistics lss of wigner beta ensembles we establish a central limit theorem clt for lss of wigner quaternion matrices

deviations from the circular law consider ginibre s ensemble of n n non hermitian random matrices in which all entries are independent complex gaussians of mean zero and variance open image in new window as n the normalized counting measure of the eigenvalues converges to the uniform measure on the unit disk in the complex plane in this note we describe fluctuations about this circular law first we obtain finite n formulas for the covariance of certain linear statistics of the eigenvalues asymptotics of these objects coupled with a theorem of costin and lebowitz then result in central limit theorems for a variety of these statistics

fluctuations of eigenvalues and second order poincare inequalities linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems the proofs of such results are usually rather difficult involving hard computations specific to the model in question in this article we attempt to formulate a unified technique for deriving such results via relatively soft arguments in the process we introduce a notion of second order poincare inequalities just as ordinary poincare inequalities give variance bounds second order poincare inequalities give central limit theorems the proof of the main result employs stein s method of normal approximation a number of examples are worked out some of which are new one of the new results is a clt for the spectrum of gaussian toeplitz matrices

regression based methods for using control variates in monte carlo experiments abstract methods based on linear regression provide an easy way to use the information in control variates to improve the efficiency with which certain features of the distributions of estimators and test statistics are estimated in monte carlo experiments we propose a new technique that allows these methods to be used when the quantities of interest are quantiles we also propose new ways to obtain approximately optimal control variates in many cases of interest these methods seem to work well in practice and can greatly reduce the number of replications required to obtain a given level of accuracy

bias assessment and reduction in linear error correction models abstract the small sample bias of the ordinary least squares coefficient estimator for dynamic regression models with innovation errors and lagged dependent and strongly exogenous explanatory variables is approximated through both small disturbance and large sample asymptotics results for the standard armax p 0 k model are obtained and also for such models under linear parameter constraints and variable transformations these approximations are then used to construct corrected estimators for the parameters of interest in higher order dynamic models including the empirically highly relevant linear error correction model by simulating two empirical cases the corrected estimators obtained via large sample asymptotics are shown to have more attractive location and efficiency properties than ordinary least squares

gaussian fluctuation for the number of particles in airy bessel sine and other determinantal random point fields we prove the central limit theorem clt for the number of eigenvalues near the spectrum edge for certain hermitian ensembles of random matrices to derive our results we use a general theorem essentially due to costin and lebowitz concerning the gaussian fluctuation of the number of particles in random point fields with determinantal correlation functions as another corollary of the costin lebowitz theorem we prove the clt for the empirical distribution function of the eigenvalues of random matrices from classical compact groups

spectrum of large random reversible markov chains heavy tailed weights on the complete graph we consider the random reversible markov kernel k obtained by assigning i i d non negative weights to the edges of the complete graph over n vertices and normalizing by the corresponding row sum the weights are assumed to be in the domain of attraction of an alpha stable law with alpha in 0 2 when 1 alpha 2 we show that for a suitable regularly varying sequence kappa n of index 1 1 alpha the limiting spectral distribution mu alpha of kappa n k coincides with the one of the random symmetric matrix of the un normalized weights levy matrix with i i d entries in contrast when 0 alpha 1 we show that the empirical spectral distribution of k converges without rescaling to a non trivial law wmu alpha supported on 1 1 whose moments are the return probabilities of the random walk on the poisson weighted infinite tree pwit introduced by aldous the limiting spectral distributions are given by the expected value of the random spectral measure at the root of suitable self adjoint operators defined on the pwit this characterization is used together with recursive relations on the tree to derive some properties of mu alpha and wmu alpha we also study the limiting behavior of the invariant probability measure of k

the onsager formula the fisher hartwig conjecture and their influence on research into toeplitz operators this is not primarily a paper about applications of mathematics to statistical physics but rather a report on how a particular problem of statistical physics has resulted in an extensive mathematical theory the problem alluded to is the computation of the spontaneous magnetizationm o t of the two dimensional ising model with nearest neighbor interactions whose solution for temperaturest below the curie pointt c was given by the famous formula of lars onsager in 1948 the theory grown out of this formula is the edifice of toeplitz determinants matrices and operators

second order freeness and fluctuations of random matrices i gaussian and wishart matrices and cyclic fock spaces abstract we extend the relation between random matrices and free probability theory from the level of expectations to the level of fluctuations we introduce the concept of second order freeness and interpret the global fluctuations of gaussian and wishart random matrices by a general limit theorem for second order freeness by introducing cyclic fock space we also give an operator algebraic model for the fluctuations of our random matrices in terms of the usual creation annihilation and preservation operators we show that orthogonal families of gaussian and wishart random matrices are asymptotically free of second order

toeplitz and wiener hopf determinants with piecewise continuous symbols abstract the asymptotics for determinants of toeplitz and wiener hopf operators with piecewise continuous symbols are obtained in this paper if w is the wiener hopf operator defined on l2 0 with piecewise continuous symbol having a finite number of discontinuities at r then under appropriate conditions it is shown that det w g r2k where g exp log 0 r 1 2 log r r and k is a completely determined constant an analogous result is obtained for toeplitz operators the main point of the paper is to obtain a result in the wiener hopf case since the toeplitz case had been treated earlier in the toeplitz case it was discovered that one could obtain asymptotics fairly easily for symbols with several singularities if for each singularity one could find a single example of a symbol with a singularity of that kind whose associated asymptotics were known fortunately in the toeplitz case such asymptotics were known the difficulty in the wiener hopf case is that there was not a single singular case where the determinant was explicitly known this problem was overcome by using the fact that wiener hopf determinants when discretized become toeplitz determinants whose entries depend on the size of the matrix no theorem on toeplitz matrices can be applied directly but these theorems are modified to obtain the desired results

distribution functions for random variables for ensembles of positive hermitian matrices distribution functions for random variables that depend on a parameter are computed asymptotically for ensembles of positive hermitian matrices the inverse fourier transform of the distribution is shown to be a fredholm determinant of a certain operator that is an analogue of a wiener hopf operator the asymptotic formula shows that up to the terms of order o 1 the distributions are gaussian

on fluctuations of eigenvalues of random band matrices we consider the fluctuations of linear eigenvalue statistics of random band n times n matrices whose entries have the form mathcal m ij b 1 2 u 1 2 i j b tilde w ij with i i d tilde w ij possessing the 4 varepsilon th moment where the function u has a finite support c c so that m has only 2c b 1 nonzero diagonals the parameter b called the bandwidth is assumed to grow with n in a way such that b n rightarrow 0 without any additional assumptions on the growth of b we prove clt for linear eigenvalue statistics for a rather wide class of test functions thus we improve and generalize the results of the previous papers jana et al arxiv 1412 2445 li et al random matrices 2 04 2013 where clt was proven under the assumption n b n 1 2 moreover we develop a method which allows to prove automatically the clt for linear eigenvalue statistics of the smooth test functions for almost all classical models of random matrix theory deformed wigner and sample covariance matrices sparse matrices diluted random matrices matrices with heavy tales etc

a mirror image invariance for m estimators much research into the performance of econometric estimators utilizes monte carlo experiments either as a primary tool or as a check on the accuracy of first or higher order asymptotic approximations it is important to design efficient experiments and to extract all the information they provide but it is also important to avoid constructing experiments which give an overly optimistic view of the performance of an estimator or test this note gives a theorem concerning choice of covariate distribution which is relevant to both these considerations it is shown that when a subset of covariates x has a symmetric conditional distribution given the remaining covariates z a mirror image invariance applies to m estimators huber 1977 of the coefficients that measure the impact of x on the conditional distribution of the response variates namely that reversal of the signs of the coefficients causes the marginal2 sampling distributions of their m estimators to be reflected around the origin while the marginal sampling distributions of estimators of other parameters are unaffected it follows that when covariate distributions are symmetric results for one half of the parameter space can be deduced from results for the other half for example if bias b 3 and mse b p are respectively the bias and mean squared error assumed for the moment to exist of an estimator 3 when the true value of 3 is b then bias b 13 bias b 3 and mse b 3 mse b p it follows that bias o 13 0 and more generally the result implies that sampling distributions are symmetric when coefficients associated with symmetrically distributed covariates are zero this typically removes the n 1 2 term from the edgeworth expansion of the estimator and associated t statistics therefore choosing a symmetric covariate distribution in a monte carlo experiment may lead to a false view of the accuracy of first order approximations to distributions of certain estimators and test statistics the theorem highlights the potential sensitivity of the results of monte carlo experiments to the choice of covariate distribution the mirror image result and the associated symmetry result are unusual because they do not require strong assumptions about the distribution of the response variate in contrast to the symmetry results described by andrews 1986 and the mirror image invariance given by cryer nankervis and savin 1989 apart from the symmetry of the conditional distribution of a subset of covariates the main requirements are that the coefficients of interest 3 satisfy two related single index restrictions namely that they

strong szegő asymptotics and zeros of the zeta function assuming the riemann hypothesis we prove the weak convergence of linear statistics of the zeros of the riemann function to a gaussian field with covariance structure corresponding to the h1 2 norm of the test functions for this purpose we obtain an approximate form of the explicit formula relying on selberg s smoothed expression for and the helffer sjostrand functional calculus our main result is an analogue of the strong szeg theorem known for toeplitz operators and random matrix theory 2014 wiley periodicals inc

bulk universality for generalized wigner matrices with few moments in this paper we consider n times n real generalized wigner matrices whose entries are only assumed to have finite 2 varepsilon th moment for some fixed but arbitrarily small varepsilon 0 we show that the stieltjes transforms m n z of these matrices satisfy a weak local semicircle law on the nearly smallest possible scale when eta mathfrak i z is almost of order n 1 as a consequence we establish bulk universality for local spectral statistics of these matrices at fixed energy levels both in terms of eigenvalue gap distributions and correlation functions meaning that these statistics converge to those of the gaussian orthogonal ensemble in the large n limit

local law and tracy widom limit for sparse random matrices we consider spectral properties and the edge universality of sparse random matrices the class of random matrices that includes the adjacency matrices of the erd s renyi graph model g n p we prove a local law for the eigenvalue density up to the spectral edges under a suitable condition on the sparsity we also prove that the rescaled extremal eigenvalues exhibit goe tracy widom fluctuations if a deterministic shift of the spectral edge due to the sparsity is included for the adjacency matrix of the erd s renyi graph this establishes the tracy widom fluctuations of the second largest eigenvalue when p is much larger than n 2 3 with a deterministic shift of order np 1

a clt for a band matrix model a law of large numbers and a central limit theorem are derived for linear statistics of random symmetric matrices whose on or above diagonal entries are independent but neither necessarily identically distributed nor necessarily all of the same variance the derivation is based on systematic combinatorial enumeration study of generating functions and concentration inequalities of the poincare type special cases treated with an explicit evaluation of limiting variances are generalized wigner and wishart matrices

levy khintchine random matrices and the poisson weighted infinite skeleton tree we study a class of hermitian random matrices which includes wigner matrices heavy tailed random matrices and sparse random matrices such as adjacency matrices of erd os renyi random graphs with pn 1 our n n random matrices have real entries which are i i d up to symmetry the distribution of entries depends on n and we require sums of rows to converge in distribution it is then well known that the limit distribution must be infinitely divisible we show that a limiting empirical spectral distribution lsd exists and via local weak convergence of associated graphs the lsd corresponds to the spectral measure associated to the root of a graph which is formed by connecting infinitely many poisson weighted infinite trees using a backbone structure of special edges called cords to infinity one example covered by the results are matrices with i i d entries having infinite second moments but normalized to be in the gaussian domain of attraction in this case the limiting graph is n rooted at 1 so the lsd is the semi circle law the results also extend to self adjoint complex matrices and wishart matrices

limiting laws of linear eigenvalue statistics for hermitian matrix models we study the variance and the laplace transform of the probability law of linear eigenvalue statistics of unitary invariant matrix models of n n hermitian matrices as n assuming that the test function of statistics is smooth enough and using the asymptotic formulas by deift et al commun pure appl math 52 1325 1425 1999 for orthogonal polynomials with varying weights we show first that if the support of the density of states of the model consists of q 2 intervals then in the global regime the variance of statistics is a quasiperiodic function of n as n generically in the potential determining the model we show next that the exponent of the laplace transform of the probability law is not in general 1 2 variance as it should be if the central limit theorem would be valid and we find the asymptotic form of the laplace transform of the probability law in certain cases

stable non gaussian random processes stochastic models with infinite variance stable random variables on the real line multivariate stable distributions stable stochastic integrals dependence structures of multivariate stable distributions non linear regression complex stable stochastic integrals and harmonizable processes self similar processes chentsov random fields introduction to sample path properties boundedness continuity and oscillations measurability integrability and absolute continuity boundedness and continuity via metric entropy integral representation historical notes and extensions

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

new distribution theory for the estimation of structural break point in mean abstract based on the girsanov theorem this paper obtains the exact distribution of the maximum likelihood estimator of structural break point in a continuous time model the exact distribution is asymmetric and tri modal indicating that the estimator is biased these two properties are also found in the finite sample distribution of the least squares ls estimator of structural break point in the discrete time model suggesting the classical long span asymptotic theory is inadequate the paper then builds a continuous time approximation to the discrete time model and develops an in fill asymptotic theory for the ls estimator the in fill asymptotic distribution is asymmetric and tri modal and delivers good approximations to the finite sample distribution to reduce the bias in the estimation of both the continuous time and the discrete time models a simulation based method based on the indirect estimation ie approach is proposed monte carlo studies show that ie achieves substantial bias reductions

approximate bias correction in econometrics abstract this paper discusses methods for reducing the bias of consistent estimators that are biased in finite samples these methods are available whenever the bias function which relates the bias of the parameter estimates to the values of the parameters can be estimated by computer simulation or by some other method if so bias can be reduced by one full order in the sample size and in some cases that may not be unrealistic virtually eliminated unfortunately reducing bias may increase the variance or even the mean squared error of an estimator whether it does so depends on the shape of the bias function the results of the paper are illustrated by applying them to two problems estimating the autoregressive parameter in an ar 1 model with a constant term and estimating a logic model

fractional brownian motion with hurst index h 0 and the gaussian unitary ensemble the goal of this paper is to establish a relation between characteristic polynomials of n n gue random matrices h as n and gaussian processes with logarithmic correlations we introduce a regularized version of fractional brownian motion with zero hurst index which is a gaussian process with stationary increments and logarithmic increment structure then we prove that this process appears as a limit of d n z log det h zi on mesoscopic scales as n by employing a fourier integral representation we use this to prove a continuous analogue of a result by diaconis and shahshahani j appl probab 31a 1994 49 62 on the macroscopic scale d n x gives rise to yet another type of gaussian process with logarithmic correlations we give an explicit construction of the latter in terms of a chebyshev fourier random series

stein s method and characters of compact lie groups stein s method is used to study the trace of a random element from a compact lie group or symmetric space central limit theorems are proved using very little information character values on a single element and the decomposition of the square of the trace into irreducible components this is illustrated for lie groups of classical type and dyson s circular ensembles the approach in this paper will be useful for the study of higher dimensional characters where normal approximations need not hold

central limit theorem for fluctuations of linear eigenvalue statistics of large random graphs we consider the adjacency matrix a of a large random graph and study fluctuations of the function fn z u 1 n k 1nexp ugkk z with g z z ia 1 we prove that the moments of fluctuations normalized by n 1 2 in the limit n satisfy the wick relations for the gaussian random variables this allows us to prove central limit theorem for tr g z and then extend the result on the linear eigenvalue statistics tr a of any function r r which increases together with its first two derivatives at infinity not faster than an exponential

the central limit theorem for local linear statistics in classical compact groups and related combinatorial identities author s soshnikov alexander abstract we discuss clt for the global and local linear statistics of random matrices from classical compact groups the main part of our proofs are certain combinatorial identities much in the spirit of works by kac and spohn

the finite sample effects of var dimensions on ols bias ols variance and minimum mse estimators vector autoregressions vars are important tools in time series analysis however relatively little is known about the finite sample behaviour of parameter estimators we address this issue by investigating ordinary least squares ols estimators given a data generating process that is a purely nonstationary first order var specifically we use monte carlo simulation and numerical optimisation to derive response surfaces for ols bias and variance in terms of var dimensions given correct specification and several types of over parameterisation of the model we include a constant and a constant and trend and introduce excess lags we then examine the correction factors that are required for the least squares estimator to attain the minimum mean squared error mse our results improve and extend one of the main finite sample multivariate analytical bias results of abadir hadri and tzavalis abadir k m hadri k tzavalis e 1999 the influence of var dimensions on estimator biases econometrica 67 163 181 generalise the univariate variance and mse findings of abadir abadir k m 1995 unbiased estimation as a solution to testing for random walks economics letters 47 263 268 to the multivariate setting and complement various asymptotic studies

the strong ekeland variational principle abstract in this paper we consider the strong ekeland variational principle due to georgiev p g georgiev the strong ekeland variational principle the strong drop theorem and applications j math anal appl 131 1988 1 21 we discuss it for functions defined on banach spaces and on compact metric spaces we also prove the distance version of it

the limiting distributions of large heavy wigner and arbitrary random matrices abstract a heavy wigner matrix x n is defined similarly to a classical wigner one it is hermitian with independent sub diagonal entries the diagonal entries and the non diagonal entries are identically distributed nevertheless the moments of the entries of n x n tend to infinity with n as for matrices with truncated heavy tailed entries or adjacency matrices of sparse erdos renyi graphs consider a family x n of independent heavy wigner matrices and an independent family y n of arbitrary random matrices with a bound condition and converging in distribution in the sense of free probability we characterize the possible limiting joint distributions of x n y n giving explicit formulas for joint moments we find that they depend on more than the distribution of y n and that in general x n and y n are not asymptotically free we use the traffic distributions and the associated notion of independence 21 to encode the information on y n and describe the limiting distribution of x n y n we develop this approach for related models and give recurrence relations for the limiting distribution of heavy wigner and independent diagonal matrices

prepivoting to reduce level error of confidence sets summary approximate confidence sets for a parameter 0 may be obtained by referring a function of 0 and of the sample to an estimated quantile of that function s sampling distribution we call this function the root of the confidence set either asymptotic theory or bootstrap methods can be used to estimate the desired quantile when the root is not a pivot in the sense of classical statistics the actual level of the approximate confidence set may differ substantially from the intended level prepivoting is the transformation of a confidence set root by its estimated bootstrap cumulative distribution function prepivoting can be iterated bootstrap confidence sets generated from a root prepivoted one or more times have smaller error in level than do confidence sets based on the original root the first prepivoting is nearly equivalent to studentizing when that operation is appropriate further iterations of prepivoting make higher order corrections automatically

spectrum of non hermitian heavy tailed random matrices let x jk j k 1 be i i d complex random variables such that x jk is in the domain of attraction of an alpha stable law with 0 alpha 2 our main result is a heavy tailed counterpart of girko s circular law namely under some additional smoothness assumptions on the law of x jk we prove that there exists a deterministic sequence a n n 1 alpha and a probability measure mu alpha on c depending only on alpha such that with probability one the empirical distribution of the eigenvalues of the rescaled matrix a n 1 x jk 1 j k n converges weakly to mu alpha as n tends to infinity our approach combines aldous steele s objective method with girko s hermitization using logarithmic potentials the underlying limiting object is defined on a bipartized version of aldous poisson weighted infinite tree recursive relations on the tree provide some properties of mu alpha in contrast with the hermitian case we find that mu alpha is not heavy tailed

on barnes beta distributions and applications to the maximum distribution of the 2d gaussian free field a new family of barnes beta distributions on 0 infty is introduced and its infinite divisibility moment determinacy scaling and factorization properties are established the morris integral probability distribution is constructed from barnes beta distributions of types 1 0 and 2 2 and its moment determinacy and involution invariance properties are established for application the maximum distributions of the 2d gaussian free field on the unit interval and circle with a non random logarithmic potential are conjecturally related to the critical selberg and morris integral probability distributions respectively and expressed in terms of sums of barnes beta distributions of types 1 0 and 2 2

generalized distance and existence theorems in complete metric spaces abstract in this paper we first introduce the concept of distance on a metric space which is a generalized concept of both w distance and tataru s distance we also improve the generalizations of the banach contraction principle caristi s fixed point theorem ekeland s variational principle and the nonconvex minimization theorem according to takahashi further we discuss the relation between w distance and tataru s distance

on the moments of traces of matrices of classical groups we consider random matrices belonging to the groups u n o n so n and sp n and distributed according to the corresponding unit haar measure we prove that the moments of traces of powers of the matrices coincide with the moments of certain gaussian random variables if the order of moments is low enough corresponding formulas proved partly before by various methods are obtained here in the framework of a unique method reminiscent of the method of correlation equations of statistical mechanics the equations are derived by using a version of the integration by parts

characterizations of reflexivity and compactness via the strong ekeland variational principle it is well known that the famous ekeland variational principle characterizes the metric completeness of underlying spaces in this paper we prove that some versions of the strong ekeland variational principle characterize the reflexivity and the compactness of underlying spaces

mock gaussian behaviour for linear statistics of classical compact groups we consider the scaling limit of linear statistics for eigenphases of a matrix taken from one of the classical compact groups we compute their moments and find that the first few moments are gaussian whereas the limiting distribution is not the precise number of gaussian moments depends upon the particular statistic considered

functional limit theorems for random regular graphs consider d uniformly random permutation matrices on n labels consider the sum of these matrices along with their transposes the total can be interpreted as the adjacency matrix of a random regular graph of degree 2d on n vertices we consider limit theorems for various combinatorial and analytical properties of this graph or the matrix as n grows to infinity either when d is kept fixed or grows slowly with n in a suitable weak convergence framework we prove that the finite but growing in length sequences of the number of short cycles and of cyclically non backtracking walks converge to distributional limits we estimate the total variation distance from the limit using stein s method as an application of these results we derive limits of linear functionals of the eigenvalues of the adjacency matrix a key step in this latter derivation is an extension of the kahn szemeredi argument for estimating the second largest eigenvalue for all values of d and n

the spectrum of heavy tailed random matrices let x n be an n n random symmetric matrix with independent equidistributed entries if the law p of the entries has a finite second moment it was shown by wigner 14 that the empirical distribution of the eigenvalues of x n once renormalized by sqrt n converges almost surely and in expectation to the so called semicircular distribution as n goes to infinity in this paper we study the same question when p is in the domain of attraction of an stable law we prove that if we renormalize the eigenvalues by a constant a n of order n frac 1 alpha the corresponding spectral distribution converges in expectation towards a law mu alpha which only depends on we characterize mu alpha and study some of its properties it is a heavy tailed probability measure which is absolutely continuous with respect to lebesgue measure except possibly on a compact set of capacity zero

estimating functions in indirect inference summary there are models for which the evaluation of the likelihood is infeasible in practice for these models the metropolis hastings acceptance probability cannot be easily computed this is the case for instance when only departure times from a g g 1 queue are observed and inference on the arrival and service distributions are required indirect inference is a method to estimate a parameter in models whose likelihood function does not have an analytical closed form but from which random samples can be drawn for fixed values of first an auxiliary model is chosen whose parameter can be directly estimated next the parameters in the auxiliary model are estimated for the original data leading to an estimate the parameter is also estimated by using several sampled data sets simulated from the original model for different values of the original parameter finally the parameter which leads to the best match to is chosen as the indirect inference estimate we analyse which properties an auxiliary model should have to give satisfactory indirect inference we look at the situation where the data are summarized in a vector statistic t and the auxiliary model is chosen so that inference on is drawn from t only under appropriate assumptions the asymptotic covariance matrix of the indirect estimators is proportional to the asymptotic covariance matrix of t and componentwise inversely proportional to the square of the derivative with respect to of the expected value of t we discuss how these results can be used in selecting good estimating functions we apply our findings to the queuing problem

central limit theorems for linear statistics of heavy tailed random matrices we show central limit theorems clt for the stieltjes transforms or more general analytic functions of symmetric matrices with independent heavy tailed entries including entries in the domain of attraction of alpha stable laws and entries with moments exploding with the dimension as in the adjacency matrices of erdos renyi graphs for the second model we also prove a central limit theorem of the moments of its empirical eigenvalues distribution the limit laws are gaussian but unlike to the case of standard wigner matrices the normalization is the one of the classical clt for independent random variables

random point fields associated with certain fredholm determinants i fermion poisson and boson point processes we introduce certain classes of random point fields including fermion and boson point processes which are associated with fredholm determinants of certain integral operators and study some of their basic properties limit theorems correlation functions palm measures etc also we propose a conjecture on an analogue of the determinant and permanent

localization and delocalization of eigenvectors for heavy tailed random matrices consider an n times n hermitian random matrix with above the diagonal independent entries with alpha stable symmetric distribution and 0 2 we give vanishing bounds on the l p norm of the eigenvectors normalized to have unit l 2 norm on the contrary when 0 alpha 2 3 we prove that these eigenvectors are localized

asymptotic properties of large random matrices with independent entries we study the normalized trace gn z n 1 tr h zi 1 of the resolvent of n n real symmetric matrices h 1 jk wjk n j k 1n assuming that their entries are independent but not necessarily identically distributed random variables we develop a rigorous method of asymptotic analysis of moments of gn z for iz 0 where 0 is determined by the second moment of wjk by using this method we find the asymptotic form of the expectation e gn z and of the connected correlator e gn z1 gn z2 e gn z1 e gn z2 we also prove that the centralized trace ngn z e ngn z has the gaussian distribution in the limit n based on these results we present heuristic arguments supporting the universality property of the local eigenvalue statistics for this class of random matrix ensembles

eigenvalue distribution of large weighted random graphs we study eigenvalue distribution of the adjacency matrix a n p of weighted random graphs n p we assume that the graphs have n vertices and the average number of edges attached to one vertex is p to each edge of the graph eij we assign a weight given by a random variable aij with zero mathematical expectation and all moments finite in the first part of the paper we consider the moments of normalized eigenvalue counting function n p of a n p assuming all moments of a finite we obtain recurrent relations that determine the moments of the limiting measure p limn n p the method developed is applied to the laplace operator closely related with a n p using the recurrent relations we analyze the form of p for the both of random matrix families in the second part of the paper we consider the resolvents g a z of a n p and of n p and study the functions fn a z u 1 n k 1n exp ugkk a z in the limit n we derive closed equations that uniquely determine the limiting functio

boundedly connected sets and the distance to the intersection of two sets abstract we prove that if x d is a metric space c is a closed subset of x and x x then the distance of x to r s agrees with the maximum of the distances of x to r and s for every closed subsets r s c such that c r s if and only if c is x boundedly connected

the strong ekeland variational principle the strong drop theorem and applications on donne une nouvelle demonstration du theoreme de la goutte generalisee en utilisant des formes renforcees du principe variationnel d ekeland et du theoreme de la goutte et donne de nouveaux resultats et de nouvelles demonstrations de resultats connus sur les proprietes generiques des problemes de minimisation convexes et non convexes