The Pearcey process is a universal point process in random matrix theory and depends on a parameter ρ∈ℝ. Let N(x) be the random variable that counts the number of points in this process that fall in the interval [−x,x]. In this note, we establish the following global rigidity upper bound: lims→∞ℙsupx>sN(x)−334πx43−3ρ2πx23logx≤423π+𝜖=1, where 𝜖>0 is arbitrary. We also obtain a similar upper bound

In this paper, the exact distribution of the largest eigenvalue of a singular random matrix for multivariate analysis of variance (MANOVA) is discussed. The key to developing the distribution theory of eigenvalues of a singular random matrix is to use heterogeneous hypergeometric functions with two matrix arguments. In this study, we define the singular beta F-matrix and extend the distributions of

For linear spectral statistics of Wigner matrix, we establish a CLT when the entries are independent, standardised with however inhomogeneous fourth moments. Formulas for the expectation and the variance of the Gaussian limiting distribution are given. An application to the normalized stochastic block model is further proposed.

We prove the Marchenko–Pastur law for the eigenvalues of p×p sample covariance matrices in two new situations where the data does not have independent coordinates. In the first scenario — the block-independent model — the p coordinates of the data are partitioned into blocks in such a way that the entries in different blocks are independent, but the entries from the same block may be dependent. In

In this paper, we study the recurrence coefficients of a deformed Hermite polynomials orthogonal with respect to the weight w(x;t,α):=e−x2|x−t|α(A+B⋅𝜃(x−t)),x∈(−∞,∞), where α>−1,A≥0,A+B≥0 and t∈ℝ. It is an extension of Chen and Feigin [J. Phys. A., Math. Gen. 39 (2006) 12381–12393]. By using the ladder operator technique, we show that the recurrence coefficients satisfy a particular Painlevé IV equation

We study the limiting distribution of a pair counting statistics of the form ∑1≤i≠j≤Nf(LN(𝜃i−𝜃j)) for the circular β-ensemble (CβE) of random matrices for sufficiently smooth test function f and LN=O(N). For β=2 and LN=N our results are inspired by a classical result of Montgomery on pair correlation of zeros of Riemann zeta function.

Subordination is the basis of the analytic approach to free additive and multiplicative convolution. We extend this approach to a more general setting and prove that the conditional expectation 𝔼φ(z−X−f(X)Yf∗(X))−1|X for free random variables X,Y and a Borel function f is a resolvent again. This result allows the explicit calculation of the distribution of noncommutative polynomials of the form X+f(X)Yf∗(X)

Patterned random matrices such as the reverse circulant, the symmetric circulant, the Toeplitz and the Hankel matrices and their almost sure limiting spectral distribution (LSD), have attracted much attention. Under the assumption that the entries are taken from an i.i.d. sequence with finite variance, the LSD are tied together by a common thread — the 2kth moment of the limit equals a weighted sum

In this paper, we give bounds on the variance of the number of points of the Circular and the Gaussian β Ensemble in arcs of the unit circle or intervals of the real line. These bounds are logarithmic with respect to the renormalized length of these sets, which is expected to be optimal up to a multiplicative constant depending only on β.

In our previous paper [R. Feng, G. Tian and D. Wei, Spectrum of SYK model, Peking Math. J.2 (2019) 41–70], we derived the almost sure convergence of the global density of eigenvalues of random matrices of the SYK model. In this paper, we will prove the central limit theorem for the linear statistics of eigenvalues of the SYK model and compute its variance.

We consider the three finite free convolutions for polynomials studied in a recent paper by Marcus, Spielman and Srivastava. Each can be described either by direct explicit formulae or in terms of operations on randomly rotated matrices. We present an alternate approach to the equivalence between these descriptions, based on combinatorial Weingarten methods for integration over the unitary and orthogonal

We provide a generalization and new proofs of the formulas of Collins et al. for the spectrum of polynomials in cyclic monotone elements. This is applied to Random Matrices with discrete spectrum.

We discuss the process convergence of the time dependent fluctuations of linear eigenvalue statistics of random circulant matrices with independent Brownian motion entries, as the dimension of the matrix tends to ∞. Our derivation is based on the trace formula of circulant matrix, method of moments and some combinatorial techniques.

Let Nn be an n×n complex random matrix, each of whose entries is an independent copy of a centered complex random variable z with finite nonzero variance σ2. The strong circular law, proved by Tao and Vu, states that almost surely, as n→∞, the empirical spectral distribution of Nn/(σn) converges to the uniform distribution on the unit disc in ℂ. A crucial ingredient in the proof of Tao and Vu, which

The aim of this paper is to investigate the Kolmogorov distance of the Circular Law to the empirical spectral distribution of non-Hermitian random matrices with independent entries. The optimal rate of convergence is determined by the Ginibre ensemble and is given by n−1/2. A smoothing inequality for complex measures that quantitatively relates the uniform Kolmogorov-like distance to the concentration

It is known from [G. Filipuk and W. Van Assche, Discrete orthogonal polynomials with hypergeometric weights and Painlevé VI, Symmetry Integr. Geom. Methods Appl.14 (2018), Article ID: 088, 19 pp.] that the recurrence coefficients of discrete orthogonal polynomials on the nonnegative integers with hypergeometric weights satisfy a system of nonlinear difference equations. There is also a connection to

Determinantal point processes are point processes whose correlation functions are given by determinants of matrices. The entries of these matrices are given by one fixed function of two variables, which is called the kernel of the point process. It is well known that there are different kernels that induce the same correlation functions. We classify all the possible transformations of a kernel that

We describe the non-backtracking spectrum of a stochastic block model with connection probabilities pin,pout=ω(logn)/n. In this regime we answer a question posed in [L. Dall’Amico, R. Couillet and N. Tremblay, Revisiting the Bethe–Hessian: Improved community detection in sparse heterogeneous graphs, in Advances in Neural Information Processing Systems (2019), pp. 4039–4049] regarding the existence

We investigate traces of powers of random matrices whose distributions are invariant under rotations (with respect to the Hilbert–Schmidt inner product) within a real-linear subspace of the space of n×n matrices. The matrices, we consider may be real or complex, and Hermitian, antihermitian, or general. We use Stein’s method to prove multivariate central limit theorems, with convergence rates, for

We analyze the limiting behavior of the eigenvalue and singular value distribution for random convolution operators on large (not necessarily Abelian) groups, extending the results by Meckes for the Abelian case. We show that for regular sequences of groups, the limiting distribution of eigenvalues (respectively singular values) is a mixture of eigenvalue (respectively singular value) distributions

We focus on regression problems in which the predictors are naturally in the form of matrices. Reduced rank regression and related regularized method have been adapted to matrix regression. However, linear methods are restrictive in their expressive power. In this work, we consider a class of semiparametric additive models based on series estimation of nonlinear functions which interestingly induces

The asymptotic behavior of the distribution of the squared singular values of the sample autocovariance matrix between the past and the future of a high-dimensional complex Gaussian uncorrelated sequence is studied. Using Gaussian tools, it is established that the distribution behaves as a deterministic probability measure whose support 𝒮 is characterized. It is also established that the squared singular

The spectral density for random matrix β ensembles can be written in terms of the average of the absolute value of the characteristic polynomial raised to the power of β, which for even β is a polynomial of degree β(N−1). In the cases of the classical Gaussian, Laguerre, and Jacobi weights, we show that this polynomial, and moreover, the spectral density itself, can be characterized as the solution

We give an upper bound on the total variation distance between the linear eigenvalue statistic, properly scaled and centered, of a random matrix with a variance profile and the standard Gaussian random variable. The second-order Poincaré inequality-type result introduced in [S. Chatterjee, Fluctuations of eigenvalues and second order poincaré inequalities, Prob. Theory Rel. Fields143(1) (2009) 1–40

In order to further improve the efficiency of parameter estimation and reduce the number of estimated parameters, we adopt dimension reduction ideas of partial envelope model proposed by [Su and Cook, Partial envelopes for efficient estimation in multivariate linear regression, Biometrika98 (2011) 133–146.] to center on some predictors of special interest. Based on the research results of [Cook et

In this paper, we derive the Tracy–Widom law for the largest eigenvalue of sample covariance matrix generated by the vector autoregressive moving average model when the dimension is comparable to the sample size. This result is applied to make inference on the vector autoregressive moving average model. Simulations are conducted to demonstrate the finite sample performance of our inference.

In this text, based on elementary computations, we provide a perturbative expansion of the coordinates of the eigenvectors of a Hermitian matrix of large size perturbed by a random matrix with small operator norm whose entries in the eigenvector basis of the first one are independent, centered, with a variance profile. This is done through a perturbative expansion of spectral measures associated to

We present a definition for second-order freeness in the quaternionic case. We demonstrate that this definition on a second-order probability space is asymptotically satisfied by independent symplectically invariant quaternionic matrices. This definition is different from the natural definition for complex and real second-order probability spaces, those motivated by the asymptotic behavior of unitarily

We consider properties of the ground state density for the d-dimensional Fermi gas in an harmonic trap. Previous work has shown that the d-dimensional Fourier transform has a very simple functional form. It is shown that this fact can be used to deduce that the density itself satisfies a third-order linear differential equation, previously known in the literature but from other considerations. It is

In this note, we give a combinatorial and noncomputational proof of the asymptotics of the integer moments of the moments of the characteristic polynomials of Haar distributed unitary matrices as the size of the matrix goes to infinity. This is achieved by relating these quantities to a lattice point count problem. Our main result is a new explicit expression for the leading order coefficient in the

We use free probability to compute the limiting spectral properties of the harmonic mean of n i.i.d. Wishart random matrices Wi whose limiting aspect ratio is γ∈(0,1) when 𝔼[Wi]=I. We demonstrate an interesting phenomenon where the harmonic mean H of the n Wishart matrices is closer in operator norm to 𝔼[Wi] than the arithmetic mean A for small n, after which the arithmetic mean is closer. We also

The Bessel point process is a rigid point process on the positive real line and its conditional measure on a bounded interval [0,R] is almost surely an orthogonal polynomial ensemble. In this paper, we show that if R tends to infinity, one almost surely recovers the Bessel point process. In fact, we show this convergence for a deterministic class of probability measures, to which the conditional measure

In this paper, we study the convergent limits and rates of the eigenvalues and eigenvectors for spiked sample covariance matrices whose spectrum can have multiple bulk components. Our model is an extension of Johnstone’s spiked covariance matrix model. Based on our results, we can extend many statistical applications based on Johnstone’s spiked covariance matrix model.

We consider the empirical eigenvalue distribution of an m×m principal submatrix of an n×n random unitary matrix distributed according to Haar measure. For n and m large with mn=α, the empirical spectral measure is well approximated by a deterministic measure μα supported on the unit disc. In earlier work, we showed that for fixed n and m, the bounded-Lipschitz distance dBL between the empirical spectral

We establish a few properties of eigenvalues and eigenvectors of the quaternionic Ginibre ensemble (QGE), analogous to what is known in the complex Ginibre case (see [7, 11, 14]). We first recover a version of Kostlan’s theorem that was already at the heart of an argument by Rider [1], namely, that the set of the squared radii of the eigenvalues is distributed as a set of independent gamma variables

We continue to explore the connections between large deviations for objects coming from random matrix theory and sum rules. This connection was established in [Sum rules via large deviations, J. Funct. Anal. 270(2) (2016) 509–559] for spectral measures of classical ensembles (Gauss–Hermite, Laguerre, Jacobi) and it was extended to spectral matrix measures of the Hermite and Laguerre ensemble in [Sum

Koloğlu, Kopp and Miller compute the limiting spectral distribution of a certain class of real random matrix ensembles, known as k-block circulant ensembles, and discover that it is exactly equal to the eigenvalue distribution of an k×k Gaussian unitary ensemble. We give a simpler proof that under very general conditions which subsume the cases studied by Koloğlu–Kopp–Miller, real-symmetric ensembles

We present some new results on the joint distribution of an arbitrary subset of the ordered eigenvalues of complex Wishart, double Wishart, and Gaussian hermitian random matrices of finite dimensions, using a tensor pseudo-determinant operator. Specifically, we derive compact expressions for the joint probability distribution function of the eigenvalues and the expectation of functions of the eigenvalues

The aim of this paper is to give fine asymptotics for random variables with moments of Gamma type. Among the examples, we consider random determinants of Laguerre and Jacobi beta ensembles with varying dimensions (the number of observed variables and the number of measurements vary and may be different). In addition to the Dyson threefold way of classical random matrix models (GOE, GUE, GSE), we study

This paper considers inhomogeneous Erdős–Rényi random graphs 𝔾N on N vertices in the non-sparse non-dense regime. The edge between the pair of vertices {i,j} is retained with probability 𝜀Nf(iN,jN), 1≤i≠j≤N, independently of other edges, where f:[0,1]×[0,1]→[0,∞) is a continuous function such that f(x,y)=f(y,x) for all x,y∈[0,1]. We study the empirical distribution of both the adjacency matrix AN

We present new expressions for the densities and distributions of the largest eigenvalue and the trace of a Beta–Wishart matrix. The series expansions for these expressions involve fewer terms than previously known results. For the trace, we also present a new algorithm that is linear in the size of the matrix and the degree of truncation, which is optimal.

Roy’s largest root is a common test statistic in multivariate analysis, statistical signal processing and related fields. According to Anderson [An Introduction to Multivariate Statistical Analysis, 3rd edn. (Wiley, New York, 2003)], it is numerically known that compared with the other three tests of linear hypotheses, Roy’s largest root test has the highest power under rank-one alternatives. Therefore

We consider products of independent n×n non-Hermitian random matrices X(1),…,X(m). Assume that their entries, Xjk(q),1≤j,k≤n,q=1,…,m, are independent identically distributed random variables with zero mean, unit variance. Götze and Tikhomirov [On the asymptotic spectrum of products of independent random matrices, preprint (2010), arXiv:1012.2710] and O’Rourke and Sochnikov [Products of independent

Recently, subclasses of polynomial ensembles for additive and multiplicative matrix convolutions were identified which were called Pólya ensembles (or polynomial ensembles of derivative type). Those ensembles are closed under the respective convolutions and, thus, build a semi-group when adding by hand a unit element. They even have a semi-group action on the polynomial ensembles. Moreover, in several

We consider a panel data partially linear single-index models (PDPLSIM) with errors correlated in space and time. A serially correlated error structure is adopted for the correlation in time. We propose using a semiparametric minimum average variance estimation (SMAVE) to obtain estimators for both the parameters and unknown link function. We not only establish an asymptotically normal distribution

In this paper, we innovatively propose an extremely flexible semi-parametric regression model called Multi-response Trans-Elliptical Regression (MTER) Model, which can capture the heavy-tail characteristic and tail dependence of both responses and covariates. We investigate the feature screening procedure for the MTER model, in which Kendall’ tau-based canonical correlation estimators are proposed

In this paper, we consider the spectral properties of quaternion sample covariance matrices. Let Bn=1nTn1/2XnXn∗Tn1/2, where Tn1/2 is the square root of a p×p quaternion Hermitian non-negative definite matrix Tn and Xn is a p×n matrix consisting of i.i.d. standard quaternion entries. Under the framework of random matrix theory, i.e. p/n→c∈(0,∞) as n→∞, we prove that if the fourth moment of the entries

In this paper, we highlight the role played by orthogonal and symplectic Harish-Chandra integrals in the study of real-valued matrix product ensembles. By making use of these integrals and the matrix-valued Fourier-Laplace transform, we find the explicit eigenvalue distributions for particular Hermitian anti-symmetric matrices and Hermitian anti-self dual matrices, involving both sums and products

Change-point detection is an integral component of statistical modeling and estimation. For high-dimensional data, classical methods based on the Mahalanobis distance are typically inapplicable. We propose a novel testing statistic by combining a modified Euclidean distance and an extreme statistic, and its null distribution is asymptotically normal. The new method naturally strikes a balance between

In this paper, we study the Hankel determinant associated with the degenerate Laguerre unitary ensemble (dLUE). This problem originates from the largest or smallest eigenvalue distribution of the dLUE. We derive the ladder operators and its compatibility condition with respect to a general perturbed weight function. By applying the ladder operators to our problem, we obtain two auxiliary quantities

This paper considers empirical likelihood inference for a high-dimensional partially functional linear model. An empirical log-likelihood ratio statistic is constructed for the regression coefficients of non-functional predictors and proved to be asymptotically normally distributed under some regularity conditions. Moreover, maximum empirical likelihood estimators of the regression coefficients of

We examine the probability that at least two eigenvalues of a Hermitian matrix-valued Gaussian process, collide. In particular, we determine sharp conditions under which such probability is zero. As an application, we show that the eigenvalues of a real symmetric matrix-valued fractional Brownian motion of Hurst parameter H, collide when H<1/2 and do not collide when H>12, while those of a complex

We continue the study of joint statistics of eigenvectors and eigenvalues initiated in the seminal papers of Chalker and Mehlig. The principal object of our investigation is the expectation of the matrix of overlaps between the left and the right eigenvectors for the complex N×N Ginibre ensemble, conditional on an arbitrary number k=1,2,… of complex eigenvalues. These objects provide the simplest generalization

We explore the boundaries of sine kernel universality for the eigenvalues of Gaussian perturbations of large deterministic Hermitian matrices. Equivalently, we study for deterministic initial data the time after which Dyson’s Brownian motion exhibits sine kernel correlations. We explicitly describe this time span in terms of the limiting density and rigidity of the initial points. Our main focus lies

We study the joint probability generating function for k occupancy numbers on disjoint intervals in the Bessel point process. This generating function can be expressed as a Fredholm determinant. We obtain an expression for it in terms of a system of coupled Painlevé V equations, which are derived from a Lax pair of a Riemann–Hilbert problem. This generalizes a result of Tracy and Widom [C. A. Tracy

We consider a Wigner-type ensemble, i.e. large hermitian N×N random matrices H=H∗ with centered independent entries and with a general matrix of variances Sxy=𝔼|Hxy|2. The norm of H is asymptotically given by the maximum of the support of the self-consistent density of states. We establish a bound on this maximum in terms of norms of powers of S that substantially improves the earlier bound 2∥S∥∞1/2

In this paper, we study whether two networks arising from two stochastic block models have the same connection structures by comparing their adjacency matrices. We conduct Monte Carlo simulations study to examine the finite sample performance of the proposed method. A real data example is used to illustrate the proposed methodology.

We study largest singular values of large random matrices, each with mean of a fixed rank K. Our main result is a limit theorem as the number of rows and columns approach infinity, while their ratio approaches a positive constant. It provides a decomposition of the largest K singular values into the deterministic rate of growth, random centered fluctuations given as explicit linear combinations of

We study the fluctuations associated to the a.s. convergence of the outliers established by Belinschi–Bercovici–Capitaine of an Hermitian polynomial in a complex Wigner matrix and a spiked deterministic real diagonal matrix. Thus, we extend the nonuniversality phenomenon established by Capitaine–Donati-Martin–Féral for additive deformations of complex Wigner matrices, to any Hermitian polynomial. The

We study the densities of limiting distributions of squared singular values of high-dimensional matrix products composed of independent complex Gaussian (complex Ginibre) and truncated unitary matrices which are taken from Haar distributed unitary matrices with appropriate dimensional growth. In the general case, we develop a new approach to obtain complex integral representations for densities of