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 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

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

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

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

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

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.

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

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

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 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 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

We study the joint distribution of the set of all marginals of a random Wishart matrix acting on a tensor product Hilbert space. We compute the limiting free mixed cumulants of the marginals, and we show that in the balanced asymptotical regime, the marginals are asymptotically free. We connect the matrix integrals relevant to the study of operators on tensor product spaces with the corresponding classes

This paper develops an outlier detection procedure for multinomial data when the number of categories tends to infinity. Most of the outlier detection methods are based on the assumption that the observations follow multivariate normal distribution, while in many modern applications, the observations either are measured on a discrete scale or naturally have some categorical structures. For such multinomial

We consider large-dimensional Hermitian or symmetric random matrices of the form W=M+𝜗V, where M is a Wigner matrix and V is a real diagonal matrix whose entries are independent of M. For a large class of diagonal matrices V, we prove that the fluctuations of linear spectral statistics of W for C2 test function can be decomposed into that of M and of V, and that each of those weakly converges to a

In this paper, we propose some new tests for high-dimensional covariance matrices that are applicable to generally distributed populations with finite fourth moments. The proposed test statistics are the maximum of the likelihood ratio test statistic and the statistic based on the Frobenius norm. The advantage of the new tests is the good performance in terms of power for both the traditional case

In this paper, we decompose the volatility of a diffusion process into systematic and idiosyncratic components, which are not identified with observations discretely sampled from univariate process. Using large dimensional high-frequency data and assuming a factor structure, we obtain consistent estimates of the Laplace transforms of the systematic and idiosyncratic volatility processes. Based on the

We prove a central limit theorem for the difference of linear eigenvalue statistics of a sample covariance matrix W̃ and its minor W. We find that the fluctuation of this difference is much smaller than those of the individual linear statistics, as a consequence of the strong correlation between the eigenvalues of W̃ and W. Our result identifies the fluctuation of the spatial derivative of the approximate