We consider the probability that no points lie on g large intervals in the bulk of the Airy point process. We make a conjecture for all the terms in the asymptotics up to and including the oscillations of order 1, and we prove this conjecture for g=1.

We find the asymptotic spectral distribution of random Kummer matrix. Then we formulate and prove a free analogue of HV independence property, which is known for classical Kummer and Gamma random variables and for Kummer and Wishart matrices. We also prove a related characterization of free-Kummer and free-Poisson (Marchenko–Pastur) non-commutative random variables.

We characterize the limiting fluctuations of traces of several independent Wigner matrices and deterministic matrices under mild conditions. A CLT holds but in general the families are not asymptotically free of second-order and the limiting covariance depends the limiting ∗-distribution of the deterministic matrices and their transposes and Hadamard products.

Assuming a covariance structure with blocked compound symmetry, it was showed that unbiased estimators for the covariance matrices are optimal under normality. In this paper, we derive the asymptotic distribution of the correlation matrix using unbiased estimators and discuss its use in hypothesis testing. The accuracy of the result is investigated through numerical simulation and the method is applied

We establish a moderate deviation principle for linear eigenvalue statistics of β-ensembles in the one-cut regime with a real-analytic potential. The main ingredient is to obtain uniform estimates for the correlators of a family of perturbations of β-ensembles using the loop equations.

We prove a central limit theorem (CLT) for the product of a class of random singular matrices related to a random Hill’s equation studied by Adams–Bloch–Lagarias. The CLT features an explicit formula for the variance in terms of the distribution of the matrix entries and this allows for exact calculation in some examples. Our proof relies on a novel connection to the theory of m-dependent sequences

We consider the singular linear statistic of the Laguerre unitary ensemble (LUE) consisting of the sum of the reciprocal of the eigenvalues. It is observed that the exponential generating function for this statistic can be written as a Toeplitz determinant with entries given in terms of particular K Bessel functions. Earlier studies have identified the same determinant, but with the K Bessel functions

Complex Wishart matrices are a class of random matrices with numerous emerging applications. In particular, the statistical characterization of such class of random matrices is essential for solving problems in various fields, including statistics, finance, physics and engineering. This paper establishes a new way to solve such problems based on the statistical moments and correlation of the minors

Large-dimensional (LD) random matrix theory, RMT for short, which originates from the research field of quantum physics, has shown tremendous capability in providing deep insights into large-dimensional systems. With the fact that we have entered an unprecedented era full of massive amounts of data and large complex systems, RMT is expected to play more important roles in the analysis and design of

Additivity violation of minimum output entropy, which shows non-classical properties in quantum communication, had been proved in most cases for random quantum channels defined by Haar-distributed unitary matrices. In this paper, we investigate random completely positive maps made of Gaussian Unitary Ensembles and Ginibre Ensembles regarding this matter. Using semi-circular systems and circular systems

We investigate the orthogonal polynomials associated with a singularly perturbed Pollaczek–Jacobi type weight wPJ2(x,t;α,β)=xα(1−x)βe−tx(1−x), where t∈[0,∞), α>0, β>0 and 0

Optimal portfolio selection problems are determined by the (unknown) parameters of the data generating process. If an investor wants to realize the position suggested by the optimal portfolios, he/she needs to estimate the unknown parameters and to account for the parameter uncertainty in the decision process. Most often, the parameters of interest are the population mean vector and the population

In this paper, a ridgelized Hotelling’s T2 test is developed for a hypothesis on a large-dimensional mean vector under certain moment conditions. It generalizes the main result of Chen et al. [A regularized Hotelling’s t2 test for pathway analysis in proteomic studies, J. Am. Stat. Assoc.106(496) (2011) 1345–1360.] by relaxing their Gaussian assumption. This is achieved by establishing an exact four-moment

Consider two random vectors C11/2x∈ℝp and C21/2y∈Rq, where the entries of x and y are i.i.d. random variables with mean zero and variance one, and C1 and C2 are respectively, p×p and q×q deterministic population covariance matrices. With n independent samples of (C11/2x,C21/2y), we study the sample correlation between these two vectors using canonical correlation analysis. Under the high-dimensional

We study the high-dimensional asymptotic regimes of correlated Wishart matrices d−1𝒴𝒴T, where 𝒴 is a n×d Gaussian random matrix with correlated and non-stationary entries. We prove that under different normalizations, two distinct regimes emerge as both n and d grow to infinity. The first regime is the one of central convergence, where the law of the properly renormalized Wishart matrices becomes

With nonignorable dropouts and outliers, we propose robust statistical inference and variable selection methods for linear quantile regression models based on composite quantile regression and empirical likelihood (EL) that accommodate both the within-subject correlations and nonignorable dropouts. The purpose of our study is threefold. First, we apply the generalized method of moments to estimate

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)

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

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

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.

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.

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

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. Fields 143(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