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

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

high dimensional correlation matrices the central limit theorem and its applications summary statistical inferences for sample correlation matrices are important in high dimensional data analysis motivated by this the paper establishes a new central limit theorem for a linear spectral statistic of high dimensional sample correlation matrices for the case where the dimension p and the sample size n are comparable this result is of independent interest in large dimensional random matrix theory we also further investigate the sample correlation matrices of a high dimensional vector whose elements have a special correlated structure and the corresponding central limit theorem is developed meanwhile we apply the linear spectral statistic to an independence test for p random variables and then an equivalence test for p factor loadings and n factors in a factor model the finite sample performance of the test proposed shows its applicability and effectiveness in practice an empirical application to test the independence of household incomes from various cities in china is also conducted

testing independence among a large number of high dimensional random vectors capturing dependence among a large number of high dimensional random vectors is a very important and challenging problem by arranging n random vectors of length p in the form of a matrix we develop a linear spectral statistic of the constructed matrix to test whether the n random vectors are independent or not specifically the proposed statistic can also be applied to n random vectors each of whose elements can be written as either a linear stationary process or a linear combination of independent random variables the asymptotic distribution of the proposed test statistic is established for the case of as n to avoid estimating the spectrum of each random vector a modified test statistic which is based on splitting the original n vectors into two equal parts and eliminating the term that contains the inner structure of each random vector or time series is constructed the facts that the limiting distribution is normal and there is no need to know the inner structure of each investigated rando

a note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices recently johansson and johnstone proved that the distribution of the properly rescaled largest principal component of the complex real wishart matrix x x xtx converges to the tracy widom law as n p the dimensions of x tend to in some ratio n p 0 we extend these results in two directions first of all we prove that the joint distribution of the first second third etc eigenvalues of a wishart matrix converges after a proper rescaling to the tracy widom distribution second of all we explain how the combinatorial machinery developed for wigner random matrices in refs 27 38 and 39 allows to extend the results by johansson and johnstone to the case of x with non gaussian entries provided n p o p1 3 we also prove that max n1 2 p1 2 2 o p1 2 log p a e for general 0

pairs of analytical eigenfunctions for the x2 λ x2 1 gx2 interaction a general method is discussed for deriving pairs of exact analytical eigenfunctions of the x2 x2 1 gx2 interaction recently discussed by blecher and leach and commented on by gallas it is shown that those pairs always consist of an odd and an even parity solution in each case and g are related by the conditions 0 and g as a result of their investigations the authors conjecture the possible existence of an infinite number of solution pairs having and g connected by several different relations

a quantum quasi harmonic nonlinear oscillator with an isotonic term the properties of a nonlinear oscillator with an additional term kg x2 characterizing the isotonic oscillator are studied the nonlinearity affects to both the kinetic term and the potential and combines two nonlinearities associated to two parameters and kg in such a way that for 0 all the characteristics of the standard isotonic system are recovered the first part is devoted to the classical system and the second part to the quantum system this is a problem of quantization of a system with position dependent mass of the form m x 1 1 x2 with a dependent non polynomial rational potential and with an additional isotonic term the schrodinger equation is exactly solved and the kg dependent wave functions and bound state energies are explicitly obtained for both 0

hypothesis testing in time series via the empirical characteristic function a generalized spectral density approach abstract the standardized spectral density completely describes serial dependence of a gaussian process for non gaussian processes however it may become an inappropriate analytic tool because it misses the nonlinear processes with zero autocorrelation by generalizing the concept of the standardized spectral density i propose a new spectral tool suitable for both linear and nonlinear time series analysis the generalized spectral density is indexed by frequency and a pair of auxiliary parameters it is well defined for both continuous and discrete random variables and requires no moment condition introduction of the auxiliary parameters renders the spectrum able to capture all pairwise dependencies including those with zero autocorrelation the standardized spectral density can be derived by properly differentiating the generalized spectral density with respect to the auxiliary parameters at the origin the consistency of a class of parzen s kernel type estimators for the generalized spectral dens

the schrodinger equation for the x2 λx2 1 gx2 interaction certain exact odd and even wavefunctions for the potential x2 x2 1 gx2 x in are obtained the coefficients of the wavefunctions and expressions for the energies are obtained using the symbolic manipulation code reduce the chief use of these results is to provide a check for the effectiveness of numerical algorithms the results supplement those of flessas 1982 for the same potential on xe 0

eigenvalues from power series expansions an alternative approach an appropriate rational approximation to the eigenfunction of the schrodinger equation for anharmonic oscillators enables one to obtain the eigenvalue accurately as the limit of a sequence of roots of hankel determinants the convergence rate of this approach is greater than that for a well established method based on power series expansions weighted by a gaussian factor with an adjustable parameter the so called hill determinant method

lower bounds for quartic anharmonic and double well potentials rigorous and remarkably accurate lower bounds to the lower eigenvalue spectrum of the schrodinger equation with quartic anharmonic and symmetric double well potentials of the form v a b ax2 2 bx4 b 0 are presented this procedure exploits some exactly soluble model potentials and appears to be of quite general utility

on the distribution of the largest eigenvalue in principal components analysis let x 1 denote the square of the largest singular value of an n p matrix x all of whose entries are independent standard gaussian variates equivalently x 1 is the largest principal component variance of the covariance matrix x x or the largest eigenvalue of a p variate wishart distribution on n degrees of freedom with identity covariance consider the limit of large p and n with n p gamma ge 1 when centered by mu p sqrt n 1 sqrt p 2 and scaled by sigma p sqrt n 1 sqrt p 1 sqrt n 1 1 sqrt p 1 3 the distribution of x 1 approaches the tracey widom law of order 1 which is defined in terms of the painleve ii differential equation and can be numerically evaluated and tabulated in software simulations show the approximation to be informative for n and p as small as 5 the limit is derived via a corresponding result for complex wishart matrices using methods from random matrix theory the result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts

a comparison of tests of the independence of two covariance stationary time series abstract the approximate slopes of several tests of the independence of two covariance stationary time series are derived and compared it is shown that the approximate slopes of regression tests are at least as great as those based on the residuals of univariate arima models and that there are cases in which the former are arbitrarily great while the latter are arbitrarily small these analytical findings are supported by a monte carlo study that shows that in samples of size 100 and 250 the asymptotic distribution theory under the null hypothesis is adequate for all tests but under alternatives to the null hypothesis the rate of type ii error for the test based on arima model residuals is often more than double that of the regression tests

on the distribution of the length of the longest increasing subsequence of random permutations let sn be the group of permutations of 1 2 n if 7r e sn we say that 7 i1 7f ik is an increasing subsequence in 7r if il i2 ik and 7r ii 7r i2 7r ik let 1n r be the length of the longest increasing subsequence for example if n 5 and 7r is the permutation 5 1 3 2 4 in one line notation thus 7r 1 5 7r 2 1 then the longest increasing subsequences are 1 2 4 and 1 3 4 and n 3 equip sn with uniform distribution

asymptotic distribution theory for nonparametric entropy measures of serial dependence entropy is a classical statistical concept with appealing properties establishing asymptotic distribution theory for smoothed nonparametric entropy measures of dependence has so far proved challenging in this paper we develop an asymptotic theory for a class of kernel based smoothed nonparametric entropy measures of serial dependence in a time series context we use this theory to derive the limiting distribution of granger and lin s 1994 normalized entropy measure of serial dependence which was previously not available in the literature we also apply our theory to construct a new entropy based test for serial dependence providing an alternative to robinson s 1991 approach to obtain accurate inferences we propose and justify a consistent smoothed bootstrap procedure the naive bootstrap is not consistent for our test our test is useful in for example testing the random walk hypothesis evaluating density forecasts and identifying important lags of a time series it is asymptotically locally more powerful than robinson s 1991 test as is confirmed in our simulation an application to the daily s p 500 stock price index illustrates our approach

exact solutions for nonpolynomial potentials in n space dimensions using a factorization method and supersymmetry a supersymmetry inspired factorization method is used to obtain exact analytic solutions for one or a few quantum states of a broad class of nonpolynomial potentials in an arbitrary space dimension under suitable constraints on the potential parameters for the specific nonpolynomial oscillator potential v x x2 x2 1 gx2 previous analytic results including algebraic and integral type in one and three dimensions are easily obtainable from these compact expressions interesting aspects of the existence of pairs of solutions for certain combinations of and g are also discussed

a test of cross section dependence for a linear dynamic panel model with regressors abstract this paper proposes a new testing procedure for detecting error cross section dependence after estimating a linear dynamic panel data model with regressors using the generalised method of moments gmm the test is valid when the cross sectional dimension of the panel is large relative to the time series dimension importantly our approach allows one to examine whether any error cross section dependence remains after including time dummies or after transforming the data in terms of deviations from time specific averages which will be the case under heterogeneous error cross section dependence finite sample simulation based results suggest that our tests perform well particularly the version based on the blundell r bond s 1998 initial conditions and moment restrictions in dynamic panel data models journal of econometrics 87 115 143 system gmm estimator in addition it is shown that the system gmm estimator based only on partial instruments consisting of the regressors can be a reliable alternative to the standard gmm estimators under heterogeneous error cross section dependence the proposed tests are applied to employment equations using uk firm data and the results show little evidence of heterogeneous error cross section dependence

testing for independence between two covariance stationary time series summary a one sided asymptotically normal test for independence between two stationary time series is proposed by first prewhitening the two time series and then basing the test on the residual cross correlation function the test statistic is a properly standardised version of the sum of weighted squares of residual cross correlations with weights depending on a kernel function haugh s 1976 test can be viewed as a special case of our approach in the sense that it corresponds to the use of the truncated kernel many kernels deliver better power than haugh s test a simulation study shows that the new test has good power against short and long cross correlations

on the elementary schrodinger bound states and their multiplets the problem of the existence of elementary bound states is discussed a trivial observation that every elementary wave function i r is an exact bound state for an appropriate potential v r v i r r is shown to lead to a very transparent form of the quasiexact qe solvability condition v i v j for doublets and multiplets of the s in this sense the particular class of elementary ansatze i r r polynomial r2 exp r polynomial r2 also defines the particular class of qe solvable potentials they have an elementary nonpolynomial rational form possibly also with a strongly singular repulsive core at the origin the properties of these forces are discussed in detail

bootstrapping spectral statistics in high dimensions author s lopes me blandino a aue a abstract 2019 biometrika trust statistics derived from the eigenvalues of sample covariance matrices are called spectral statistics and they play a central role in multivariate testing although bootstrap methods are an established approach to approximating the laws of spectral statistics in low dimensional problems such methods are relatively unexplored in the high dimensional setting the aim of this article is to focus on linear spectral statistics as a class of prototypes for developing a new bootstrap in high dimensions a method we refer to as the spectral bootstrap in essence the proposed method originates from the parametric bootstrap and is motivated by the fact that in high dimensions it is difficult to obtain a nonparametric approximation to the full data generating distribution from a practical standpoint the method is easy to use and allows the user to circumvent the difficulties of complex asymptotic formulas for linear spectral statistics in addition to proving the consistency of the proposed method we present encouraging empirical results in a variety of settings lastly and perhaps most interestingly we show through simulations that the method can be applied successfully to statistics outside the class of linear spectral statistics such as the largest sample eigenvalue and others

the distribution and moments of the smallest eigenvalue of a random matrix of wishart type abstract given a random rectangular m n matrix with elements from a normal distribution what is the distribution of the smallest singular value to pose an equivalent question in the language of multivariate statistics what is the distribution of the smallest eigenvalue of a matrix from the central wishart distribution in the null case we give new results giving the distribution as a simple recursion this includes the more difficult case when n m is an even integer without resorting to zonal polynomials and hypergeometric functions of matrix arguments with the recursion one can obtain exact expressions for the density and the moments of the distribution in terms of functions usually no more complicated than polynomials exponentials and at worst ordinary hypergeometric functions we further elaborate on the special cases when n m 0 1 2 and 3 and give a numerical table of the expected values for 2 m 25 and 0 n m 25

finite difference calculations of eigenvalues for various potentials the energy levels of the schrodinger equation for the potentials v x sigma 2n 2 lambda 2nx2n v x z2x2 x2n and v x x2 lambda x2n 1 gx2 with 2n 4 6 8 18 20 have been calculated by using a finite difference method for various values of lambda and the quantum number n the obtained results are compared with previously available results

on the simultaneous eigenproblem for the x2 λ x2 1 gx2 1 interaction extension of gallas results an ansatz for the eigenfunctions of the x2 x2 1 gx2 1 interaction has been recently introduced by blecher and leach using this ansatz gallas has obtained five solution pairs for 2g 3g 2 gallas results are generalised and extended here

generalized spectral tests for serial dependence summary two tests for serial dependence are proposed using a generalized spectral theory in combination with the empirical distribution function the tests are generalizations of the cramer von mises and kolmogorov smirnov tests based on the standardized spectral distribution function they do not involve the choice of a lag order and they are consistent against all types of pairwise serial dependence including those with zero autocorrelation they also require no moment condition and are distribution free under serial independence a simulation study compares the finite sample performances of the new tests and some closely related tests the asymptotic distribution theory works well in finite samples the generalized cramer von mises test has good power against a variety of dependent alternatives and dominates the generalized kolmogorov smirnov test a local power analysis explains some important stylized facts on the power of the tests based on the empirical distribution function

the schrodinger equation for the f x g x interaction a natural extension of the omega 2x2 lambda x2 1 gx2 interaction is omega 2x2 f x g x where f and g are polynomials in x2 the method of obtaining quasi exact solutions of the schrodinger equation is outlined and some simple examples are given solutions which are valid for part of the real line are obtained when g x has real zeros

double exponential sinc collocation method for solving the energy eigenvalues of harmonic oscillators perturbed by a rational function we show that the double exponential sinc collocation method provides an efficient uniformly accurate solution to the one dimensional time independent schrodinger equation for a general class of rational potentials of the form v x p x q x the derived algorithm is based on the discretization of the hamiltonian of the schrodinger equation using sinc expansions this discretization results in a generalized eigenvalue problem the eigenvalues of which correspond to approximations of the energy values of the starting hamiltonian a systematic numerical study is conducted beginning with test potentials with known eigenvalues and moving to rational potentials of increasing degree

some observations on the nature of solutions for the interaction v x x2 λx2 1 gx2 the authors study the nature of exact solutions for the nonpolynomial interaction in particular it is shown that all the solutions are supersymmetric it is also shown that this problem admits a dynamical su 2 symmetry so that an arbitrarily large part of the spectrum but not the whole spectrum can be determined exactly

some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size this paper analyzes whether standard covariance matrix tests work when dimensionality is large and in particular larger than sample size in the latter case the singularity of the sample covariance matrix makes likelihood ratio tests degenerate but other tests based on quadratic forms of sample covariance matrix eigenvalues remain well defined we study the consistency property and limiting distribution of these tests as dimensionality and sample size go to infinity together with their ratio converging to a finite nonzero limit we find that the existing test for sphericity is robust against high dimensionality but not the test for equality of the covariance matrix to a given matrix for the latter test we develop a new correction to the existing test statistic that makes it robust against high dimensionality

two timescale analysis of the alopex algorithm for optimization alopex is a correlation based gradient free optimization technique useful in many learning problems however there are no analytical results on the asymptotic behavior of this algorithm this article presents a new version of alopex that can be analyzed using techniques of two timescale stochastic approximation method it is shown that the algorithm asymptotically behaves like a gradient descent method though it does not need or estimate any gradient information it is also shown through simulations that the algorithm is quite effective

study of a class of non polynomial oscillator potentials we develop a variational method to obtain accurate bounds for the eigenenergies of h v in arbitrary dimensions n 1 where v r is the nonpolynomial oscillator potential the variational bounds are compared with results previously obtained in the literature an infinite set of exact solutions is also obtained and used as a source of comparison eigenvalues

exact analytical eigenfunctions for the x2 λ x2 1 gx2 interaction the author discusses an ansatz for the eigenfunctions of the x2 x2 1 gx2 interaction introduced by blecher and leach 1987 the author shows that the ansatz provides pairs of solutions in five cases having and g connected by 6g2 4g and conjectures the possible existence of an infinite number of such solution pairs

corrections to lrt on large dimensional covariance matrix by rmt in this paper we give an explanation to the failure of two likelihood ratio procedures for testing about covariance matrices from gaussian populations when the dimension p is large compared to the sample size n next using recent central limit theorems for linear spectral statistics of sample covariance matrices and of random f matrices we propose necessary corrections for these lr tests to cope with high dimensional effects the asymptotic distributions of these corrected tests under the null are given simulations demonstrate that the corrected lr tests yield a realized size close to nominal level for both moderate p around 20 and high dimension while the traditional lr tests with 2 approximation fails another contribution from the paper is that for testing the equality between two covariance matrices the proposed correction applies equally for non gaussian populations yielding a valid pseudo likelihood ratio test

multivariate stochastic approximation using a simultaneous perturbation gradient approximation the problem of finding a root of the multivariate gradient equation that arises in function minimization is considered when only noisy measurements of the function are available a stochastic approximation sa algorithm for the general kiefer wolfowitz type is appropriate for estimating the root the paper presents an sa algorithm that is based on a simultaneous perturbation gradient approximation instead of the standard finite difference approximation of keifer wolfowitz type procedures theory and numerical experience indicate that the algorithm can be significantly more efficient than the standard algorithms in large dimensional problems

the quantum harmonic oscillator on the sphere and the hyperbolic plane κ dependent formalism polar coordinates and hypergeometric functions a nonlinear model representing the quantum harmonic oscillator on the sphere and the hyperbolic plane is solved in polar coordinates r by making use of a curvature dependent formalism the curvature is considered as a parameter and then the radial schrodinger equation becomes a dependent gauss hypergeometric equation the energy spectrum and the wave functions are exactly obtained in both the sphere s2 0 and the hyperbolic plane h2 0 a comparative study between the spherical and the hyperbolic quantum results is presented

assessing cross sectional correlation in panel data abstract consider drawing a sample of n experimental units where each unit is observed over t time periods are the draws independent a test statistic introduced by breusch and pagan 1980 is one measure used to quantify the amount of cross sectional correlation it is effective when t is large relative to n and as shown by breusch and pagan has desirable asymptotic in t properties in this paper we consider the case often encountered in the analysis of panel data where n is large relative to t here the breusch and pagan statistic does not enjoy the same desirable asymptotic in n properties in fact we show that the asymptotic distribution depends on the parent population even under the hypothesis of no cross sectional correlations thus we introduce a distribution free statistic that does not have this drawback asymptotic properties of the new statistic are established when the data are drawn from identical distributions and when the statistic is evaluated using residuals from a complex model both the breusch and pagan statistic and the new statistic are shown to be members of a more general family by considering this generalization finite sample properties of members of the family are easily established further it turns out that a statistic introduced by friedman 1937 is another special case of this family