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Asymptotic freeness of unitary matrices in tensor product spaces for invariant states Random Matrices Theory Appl. (IF 1.209) Pub Date : 2022-08-10 Benoît Collins, Pierre Yves Gaudreau Lamarre, Camille Male
In this paper, we pursue our study of asymptotic properties of families of random matrices that have a tensor structure. In [6], the first and second authors provided conditions under which tensor products of unitary random matrices are asymptotically free with respect to the normalized trace. Here, we extend this result by proving that asymptotic freeness of tensor products of Haar unitary matrices
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Characteristic polynomials of random truncations: Moments, duality and asymptotics Random Matrices Theory Appl. (IF 1.209) Pub Date : 2022-07-28 Alexander Serebryakov, Nick Simm, Guillaume Dubach
We study moments of characteristic polynomials of truncated Haar distributed matrices from the three classical compact groups O(N), U(N) and Sp(2N). For finite matrix size we calculate the moments in terms of hypergeometric functions of matrix argument and give explicit integral representations highlighting the duality between the moment and the matrix size as well as the duality between the orthogonal
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Limiting eigenvalue behavior of a class of large dimensional random matrices formed from a Hadamard product Random Matrices Theory Appl. (IF 1.209) Pub Date : 2022-07-28 Jack W. Silverstein
This paper investigates the strong limiting behavior of the eigenvalues of the class of matrices 1N(Dn∘Xn)(Dn∘Xn)∗, studied in [V. L. Girko, Theory of Stochastic Canonical Equations: Vol. 1 (Kluwer Academic Publishers, Dordrecht, 2001)]. Here, Xn=(xij) is an n×N random matrix consisting of independent complex standardized random variables, Dn=(dij), n×N, has nonnegative entries, and ∘ denotes Hadamard
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On the analytic structure of second-order non-commutative probability spaces and functions of bounded Fréchet variation Random Matrices Theory Appl. (IF 1.209) Pub Date : 2022-07-22 Mario Diaz, James A. Mingo
In this paper, we propose a new approach to the central limit theorem (CLT) based on functions of bounded Fréchet variation for the continuously differentiable linear statistics of random matrix ensembles which relies on a weaker form of a large deviation principle for the operator norm; a Poincaré-type inequality for the linear statistics; and the existence of a second-order limit distribution. This
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Universal scaling limits of the symplectic elliptic Ginibre ensemble Random Matrices Theory Appl. (IF 1.209) Pub Date : 2022-07-22 Sung-Soo Byun, Markus Ebke
We consider the eigenvalues of symplectic elliptic Ginibre matrices which are known to form a Pfaffian point process whose correlation kernel can be expressed in terms of the skew-orthogonal Hermite polynomials. We derive the scaling limits and the convergence rates of the correlation functions at the real bulk/edge of the spectrum, which in particular establishes the local universality at strong non-Hermiticity
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Three-dimensional Gaussian fluctuations of spectra of overlapping stochastic Wishart matrices Random Matrices Theory Appl. (IF 1.209) Pub Date : 2022-07-22 Jeffrey Kuan, Zhengye Zhou
In [I. Dumitriu and E. Paquette, Spectra of overlapping Wishart matrices and the gaussian free field, Random Matrices: Theory Appl.07(2) (2018) 1850003], the authors consider eigenvalues of overlapping Wishart matrices and prove that its fluctuations asymptotically convergence to the Gaussian free field. In this brief note, their result is extended to show that when the matrix entries undergo stochastic
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Necessary and sufficient conditions for convergence to the semicircle distribution Random Matrices Theory Appl. (IF 1.209) Pub Date : 2022-07-14 Calvin Wooyoung Chin
We consider random Hermitian matrices with independent upper triangular entries. Wigner’s semicircle law says that under certain additional assumptions, the empirical spectral distribution converges to the semicircle distribution. We characterize convergence to semicircle in terms of the variances of the entries, under natural assumptions such as the Lindeberg condition. The result extends to certain
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On random matrices arising in deep neural networks: General I.I.D. case Random Matrices Theory Appl. (IF 1.209) Pub Date : 2022-07-14 Leonid Pastur, Victor Slavin
We study the eigenvalue distribution of random matrices pertinent to the analysis of deep neural networks. The matrices resemble the product of the sample covariance matrices, however, an important difference is that the analog of the population covariance matrix is now a function of random data matrices (synaptic weight matrices in the deep neural network terminology). The problem has been treated
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Operator level limit of the circular Jacobi β-ensemble Random Matrices Theory Appl. (IF 1.209) Pub Date : 2022-07-12 Yun Li, Benedek Valkó
We prove an operator level limit for the circular Jacobi β-ensemble. As a result, we characterize the counting function of the limit point process via coupled systems of stochastic differential equations. We also show that the normalized characteristic polynomials converge to a random analytic function, which we characterize via the joint distribution of its Taylor coefficients at zero and as the solution
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Some characterization results on classical and free Poisson thinning Random Matrices Theory Appl. (IF 1.209) Pub Date : 2022-06-29 Soumendu Sundar Mukherjee
Poisson thinning is an elementary result in probability, which is of great importance in the theory of Poisson point processes. In this paper, we record a couple of characterization results on Poisson thinning. We also consider several free probability analogues of Poisson thinning, which we collectively dub as free Poisson thinning, and prove characterization results for them, similar to the classical
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The partition function of log-gases with multiple odd charges Random Matrices Theory Appl. (IF 1.209) Pub Date : 2022-05-18 Elisha D. Wolff, Jonathan M. Wells
We use techniques in the shuffle algebra to present a formula for the partition function of a one-dimensional log-gas comprised of particles of (possibly) different integer charges at certain inverse temperature β in terms of the Berezin integral of an associated non-homogeneous alternating tensor. This generalizes previously known results by removing the restriction on the number of species of odd
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Edge fluctuations for random normal matrix ensembles Random Matrices Theory Appl. (IF 1.209) Pub Date : 2022-05-16 David García-Zelada
A famous result going back to Eric Kostlan states that the moduli of the eigenvalues of random normal matrices with radial potential are independent yet non-identically distributed. This phenomenon is at the heart of the asymptotic analysis of the edge, and leads in particular to the Gumbel fluctuation of the spectral radius when the potential is quadratic. In the present work, we show that a wide
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Large deviations for spectral measures of some spiked matrices Random Matrices Theory Appl. (IF 1.209) Pub Date : 2022-05-04 Nathan Noiry, Alain Rouault
We prove large deviations principles for spectral measures of perturbed (or spiked) matrix models in the direction of an eigenvector of the perturbation. In each model under study, we provide two approaches, one of which relying on large deviations principle of unperturbed models derived in the previous work “Sum rules via large deviations” (Gamboa et al. [Sum rules via large deviations, J. Funct.
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Relating random matrix map enumeration to a universal symbol calculus for recurrence operators in terms of Bessel–Appell polynomials Random Matrices Theory Appl. (IF 1.209) Pub Date : 2022-04-29 Nicholas M. Ercolani, Patrick Waters
Maps are polygonal cellular networks on Riemann surfaces. This paper analyzes the construction of closed form general representations for the enumerative generating functions associated to maps of fixed but arbitrary genus. The method of construction developed here involves a novel asymptotic symbol calculus for difference operators based on the relation between spectral asymptotics for Hermitian random
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Weak convergence of a collection of random functions defined by the eigenvectors of large dimensional random matrices Random Matrices Theory Appl. (IF 1.209) Pub Date : 2022-04-23 Jack W. Silverstein
For each n, let Un be Haar distributed on the group of n × n unitary matrices. Let xn,1,…,xn,m denote orthogonal nonrandom unit vectors in ℂn and let un,k = (uk1,…,u kn)∗ = U n∗x n,k, k = 1,…,m. Define the following functions on [0, 1]: Xnk,k(t) = n∑ i=1[nt](|u ki|2 − 1 n), Xnk,k′(t) = 2n∑i=1[nt]ū kiu k′i, k < k′. Then it is proven that Xnk,k,ℜX nk,k′, ℑXnk,k′, considered as random processes in D[0
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Adaptive singular value shrinkage estimate for low rank tensor denoising Random Matrices Theory Appl. (IF 1.209) Pub Date : 2022-04-19 Zerui Tao, Zhouping Li
Recently, tensors are widely used to represent higher-order data with internal spatial or temporal relations, e.g. images, videos, hyperspectral images (HSIs). While the true signals are usually corrupted by noises, it is of interest to study tensor recovery problems. To this end, many models have been established based on tensor decompositions. Traditional tensor decomposition models, such as the
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Adaptive singular value shrinkage estimate for low rank tensor denoising Random Matrices Theory Appl. (IF 1.209) Pub Date : 2022-04-19 Zerui Tao, Zhouping Li
Recently, tensors are widely used to represent higher-order data with internal spatial or temporal relations, e.g. images, videos, hyperspectral images (HSIs). While the true signals are usually corrupted by noises, it is of interest to study tensor recovery problems. To this end, many models have been established based on tensor decompositions. Traditional tensor decomposition models, such as the
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A new decomposition for multivalued 3 × 3 matrices Random Matrices Theory Appl. (IF 1.209) Pub Date : 2022-04-12 Aymen Ammar, Aref Jeribi, Bilel Saadaoui
In this paper, a new concept for a 3 × 3 block relation matrix is studied in a Banach space. It is shown that, under certain condition, we can investigate the Frobenius–Schur decomposition of relation matrices. Furthermore, we present some conditions which should allow the multivalued 3 × 3 matrices linear operator to be closable.
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High–low temperature dualities for the classical β-ensembles Random Matrices Theory Appl. (IF 1.209) Pub Date : 2022-04-05 Peter J. Forrester
The loop equations for the β-ensembles are conventionally solved in terms of a 1/N expansion. We observe that it is also possible to fix N and expand in inverse powers of β. At leading order, for the one-point function W1(x) corresponding to the average of the linear statistic A =∑j=1N1/(x − λ j) and after specialising to the classical weights, this reclaims well known results of Stieltjes relating
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High–low temperature dualities for the classical β-ensembles Random Matrices Theory Appl. (IF 1.209) Pub Date : 2022-04-05 Peter J. Forrester
The loop equations for the β-ensembles are conventionally solved in terms of a 1/N expansion. We observe that it is also possible to fix N and expand in inverse powers of β. At leading order, for the one-point function W1(x) corresponding to the average of the linear statistic A=∑j=1N1/(x−λj) and after specialising to the classical weights, this reclaims well known results of Stieltjes relating the
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On the operator norm of a Hermitian random matrix with correlated entries Random Matrices Theory Appl. (IF 1.209) Pub Date : 2022-03-19 Jana Reker
We consider a correlated N × N Hermitian random matrix with a polynomially decaying metric correlation structure. By calculating the trace of the moments of the matrix and using the summable decay of the cumulants, we show that its operator norm is stochastically dominated by one.
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Generalized heterogeneous hypergeometric functions and the distribution of the largest eigenvalue of an elliptical Wishart matrix Random Matrices Theory Appl. (IF 1.209) Pub Date : 2022-03-17 Aya Shinozaki, Koki Shimizu, Hiroki Hashiguchi
In this paper, we derive the exact distributions of eigenvalues of a singular Wishart matrix under the elliptical model. We define the generalized heterogeneous hypergeometric functions with two matrix arguments and provide the convergence conditions of these functions. The joint density of eigenvalues and the distribution function of the largest eigenvalue for a singular elliptical Wishart matrix
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Random matrices with independent entries: Beyond non-crossing partitions Random Matrices Theory Appl. (IF 1.209) Pub Date : 2022-03-08 Arup Bose, Koushik Saha, Arusharka Sen, Priyanka Sen
The scaled standard Wigner matrix (symmetric with mean zero, variance one i.i.d. entries), and its limiting eigenvalue distribution, namely the semi-circular distribution, have attracted much attention. The 2kth moment of the limit equals the number of non-crossing pair-partitions of the set {1, 2,…, 2k}. There are several extensions of this result in the literature. In this paper, we consider a unifying
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Joint CLT for top eigenvalues of sample covariance matrices of separable high dimensional long memory processes Random Matrices Theory Appl. (IF 1.209) Pub Date : 2022-03-08 Peng Tian
For N,n ∈ ℕ, consider the sample covariance matrix SN(T) = 1 NXX∗ from a data set X = CN1/2ZT n1/2, where Z = (Zi,j) is a N × n matrix having i.i.d. entries with mean zero and variance one, and CN,Tn are deterministic positive semi-definite Hermitian matrices of dimension N and n, respectively. We assume that (CN)N is bounded in spectral norm, and Tn is a Toeplitz matrix with its largest eigenvalues
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Some strong convergence theorems for eigenvalues of general sample covariance matrices Random Matrices Theory Appl. (IF 1.209) Pub Date : 2021-11-27 Yanqing Yin
The aim of this paper is to investigate the spectral properties of sample covariance matrices under a more general population. We consider a class of matrices of the form Sn = 1 nBnXnXn∗B n∗, where Bn is a p × m nonrandom matrix and Xn is an m × n matrix consisting of i.i.d standard complex entries. p/n → c ∈ (0,∞) as n →∞ while m can be arbitrary but no smaller than p. We first prove that under some
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A dynamical version of the SYK model and the q-Brownian motion Random Matrices Theory Appl. (IF 1.209) Pub Date : 2021-11-22 Miguel Pluma, Roland Speicher
We extend recent results on the asymptotic eigenvalue distribution of the SYK model to the multivariate case and relate the limit of a dynamical version of the SYK model with the q-Brownian motion, a non-commutative deformation of classical Brownian motion. Furthermore, we extend the results for fluctuations to the multivariate setting and treat also higher correlation functions. The structure of our
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A dynamical version of the SYK model and the q-Brownian motion Random Matrices Theory Appl. (IF 1.209) Pub Date : 2021-11-22 Miguel Pluma, Roland Speicher
We extend recent results on the asymptotic eigenvalue distribution of the SYK model to the multivariate case and relate the limit of a dynamical version of the SYK model with the q-Brownian motion, a non-commutative deformation of classical Brownian motion. Furthermore, we extend the results for fluctuations to the multivariate setting and treat also higher correlation functions. The structure of our
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On the empirical spectral distribution for certain models related to sample covariance matrices with different correlations Random Matrices Theory Appl. (IF 1.209) Pub Date : 2021-11-17 Alicja Dembczak-Kołodziejczyk, Anna Lytova
Given n,m ∈ ℕ, we study two classes of large random matrices of the form ℒn =∑α=1mξ αyαyαTand𝒜 n =∑α=1mξ α(yαxαT + x αyαT), where for every n, (ξα)α are iid copies of a random variable ξ = ξ(n) ∈ ℝ, (xα)α, (yα)α ⊂ ℝn are two (not necessarily independent) sets of independent random vectors having different covariance matrices and generating well concentrated bilinear forms. We consider two main asymptotic
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On the asymptotic behavior of the eigenvalue distribution of block correlation matrices of high-dimensional time series Random Matrices Theory Appl. (IF 1.209) Pub Date : 2021-10-22 Philippe Loubaton, Xavier Mestre
We consider linear spectral statistics built from the block-normalized correlation matrix of a set of M mutually independent scalar time series. This matrix is composed of M2 blocks. Each block has size L×L and contains the sample cross-correlation measured at L consecutive time lags between each pair of time series. Let N denote the total number of consecutively observed windows that are used to estimate
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Partial isometries, duality, and determinantal point processes Random Matrices Theory Appl. (IF 1.209) Pub Date : 2021-10-22 Makoto Katori, Tomoyuki Shirai
A determinantal point process (DPP) is an ensemble of random nonnegative-integer-valued Radon measures Ξ on a space S with measure λ, whose correlation functions are all given by determinants specified by an integral kernel K called the correlation kernel. We consider a pair of Hilbert spaces, Hℓ,ℓ=1,2, which are assumed to be realized as L2-spaces, L2(Sℓ,λℓ), ℓ=1,2, and introduce a bounded linear
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Random Toeplitz matrices: The condition number under high stochastic dependence Random Matrices Theory Appl. (IF 1.209) Pub Date : 2021-10-22 Paulo Manrique-Mirón
In this paper, we study the condition number of a random Toeplitz matrix. As a Toeplitz matrix is a diagonal constant matrix, its rows or columns cannot be stochastically independent. This situation does not permit us to use the classic strategies to analyze its minimum singular value when all the entries of a random matrix are stochastically independent. Using a circulant embedding as a decoupling
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On the asymptotic behavior of the eigenvalue distribution of block correlation matrices of high-dimensional time series Random Matrices Theory Appl. (IF 1.209) Pub Date : 2021-10-22 Philippe Loubaton, Xavier Mestre
We consider linear spectral statistics built from the block-normalized correlation matrix of a set of M mutually independent scalar time series. This matrix is composed of M2 blocks. Each block has size L × L and contains the sample cross-correlation measured at L consecutive time lags between each pair of time series. Let N denote the total number of consecutively observed windows that are used to
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Partial isometries, duality, and determinantal point processes Random Matrices Theory Appl. (IF 1.209) Pub Date : 2021-10-22 Makoto Katori, Tomoyuki Shirai
A determinantal point process (DPP) is an ensemble of random nonnegative-integer-valued Radon measures Ξ on a space S with measure λ, whose correlation functions are all given by determinants specified by an integral kernel K called the correlation kernel. We consider a pair of Hilbert spaces, Hℓ,ℓ = 1, 2, which are assumed to be realized as L2-spaces, L2(S ℓ,λℓ), ℓ = 1, 2, and introduce a bounded
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Random Toeplitz matrices: The condition number under high stochastic dependence Random Matrices Theory Appl. (IF 1.209) Pub Date : 2021-10-22 Paulo Manrique-Mirón
In this paper, we study the condition number of a random Toeplitz matrix. As a Toeplitz matrix is a diagonal constant matrix, its rows or columns cannot be stochastically independent. This situation does not permit us to use the classic strategies to analyze its minimum singular value when all the entries of a random matrix are stochastically independent. Using a circulant embedding as a decoupling
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Spectral properties for the Laplacian of a generalized Wigner matrix Random Matrices Theory Appl. (IF 1.209) Pub Date : 2021-10-14 Anirban Chatterjee, Rajat Subhra Hazra
In this paper, we consider the spectrum of a Laplacian matrix, also known as Markov matrices where the entries of the matrix are independent but have a variance profile. Motivated by recent works on generalized Wigner matrices we assume that the variance profile gives rise to a sequence of graphons. Under the assumption that these graphons converge, we show that the limiting spectral distribution converges
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The limit empirical spectral distribution of complex matrix polynomials Random Matrices Theory Appl. (IF 1.209) Pub Date : 2021-09-25 Giovanni Barbarino, Vanni Noferini
We study the empirical spectral distribution (ESD) for complex n × n matrix polynomials of degree k under relatively mild assumptions on the underlying distributions, thus highlighting universality phenomena. In particular, we assume that the entries of each matrix coefficient of the matrix polynomial have mean zero and finite variance, potentially allowing for distinct distributions for entries of
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Gap probabilities in the bulk of the Airy process Random Matrices Theory Appl. (IF 1.209) Pub Date : 2021-09-11 Elliot Blackstone, Christophe Charlier, Jonatan Lenells
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.
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Global and local scaling limits for the β = 2 Stieltjes–Wigert random matrix ensemble Random Matrices Theory Appl. (IF 1.209) Pub Date : 2021-08-11 Peter J. Forrester
The eigenvalue probability density function (PDF) for the Gaussian unitary ensemble has a well-known analogy with the Boltzmann factor for a classical log-gas with pair potential −log |x − y|, confined by a one-body harmonic potential. A generalization is to replace the pair potential by −log |sinh(π(x − y)/L)|. The resulting PDF first appeared in the statistical physics literature in relation to non-intersecting
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Regression conditions that characterize free-Poisson and free-Kummer distributions Random Matrices Theory Appl. (IF 1.209) Pub Date : 2021-07-31 Agnieszka Piliszek
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.
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Joint global fluctuations of complex Wigner and deterministic matrices Random Matrices Theory Appl. (IF 1.209) Pub Date : 2021-07-16 Camile Male, James A. Mingo, Sandrine Péché, Roland Speicher
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.
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Asymptotic distribution of correlation matrix under blocked compound symmetric covariance structure Random Matrices Theory Appl. (IF 1.209) Pub Date : 2021-07-01 Shin-ichi Tsukada
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
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Moderate deviations for linear eigenvalue statistics of β-ensembles Random Matrices Theory Appl. (IF 1.209) Pub Date : 2021-07-01 Fuqing Gao, Jianyong Mu
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.
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CLT with explicit variance for products of random singular matrices related to Hill’s equation Random Matrices Theory Appl. (IF 1.209) Pub Date : 2021-07-01 Phanuel Mariano, Hugo Panzo
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
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Applications in random matrix theory of a PIII′ τ-function sequence from Okamoto’s Hamiltonian formulation Random Matrices Theory Appl. (IF 1.209) Pub Date : 2021-06-26 Dan Dai, Peter J. Forrester, Shuai-Xia Xu
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
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Noncentral complex Wishart matrices: Moments and correlation of minors Random Matrices Theory Appl. (IF 1.209) Pub Date : 2021-06-21 Velio Tralli, Andrea Conti
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
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Large-dimensional random matrix theory and its applications in deep learning and wireless communications Random Matrices Theory Appl. (IF 1.209) Pub Date : 2021-06-18 Jungang Ge, Ying-Chang Liang, Zhidong Bai, Guangming Pan
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
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Additivity violation of quantum channels via strong convergence to semi-circular and circular elements Random Matrices Theory Appl. (IF 1.209) Pub Date : 2021-06-18 Motohisa Fukuda, Takahiro Hasebe, Shinya Sato
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
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Critical edge behavior in the singularly perturbed Pollaczek–Jacobi type unitary ensemble Random Matrices Theory Appl. (IF 1.209) Pub Date : 2021-06-14 Zhaoyu Wang, Engui Fan
We investigate the orthogonal polynomials associated with a singularly perturbed Pollaczek–Jacobi type weight wPJ2(x,t; α,β) = xα(1 − x)βe−t x(1−x), where t ∈ [0,∞), α > 0, β > 0 and 0 < x < 1. Based on our observation, we find that this weight includes the symmetric constraint wPJ2(x,t; α,β) = wPJ2(1 − x,t; β,α). Our main results obtained here include two aspects: (1) Strong asymptotics: we deduce
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Sampling distributions of optimal portfolio weights and characteristics in small and large dimensions Random Matrices Theory Appl. (IF 1.209) Pub Date : 2021-06-07 Taras Bodnar, Holger Dette, Nestor Parolya, Erik Thorsén
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
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Ridgelized Hotelling’s T2 test on mean vectors of large dimension Random Matrices Theory Appl. (IF 1.209) Pub Date : 2021-06-07 Gao-Fan Ha, Qiuyan Zhang, Zhidong Bai, You-Gan Wang
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
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Orthogonal polynomials, Hankel determinants and small eigenvalues associated with a deformed octic Freud weight Random Matrices Theory Appl. (IF 1.209) Pub Date : 2021-06-04 Mengkun Zhu,Jie Hu,Yang Chen,Xiaoli Wang
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Sample canonical correlation coefficients of high-dimensional random vectors: Local law and Tracy–Widom limit Random Matrices Theory Appl. (IF 1.209) Pub Date : 2021-05-31 Fan Yang
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,C 21/2y), we study the sample correlation between these two vectors using canonical correlation analysis. Under the high-dimensional
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High-dimensional regimes of non-stationary Gaussian correlated Wishart matrices Random Matrices Theory Appl. (IF 1.209) Pub Date : 2021-05-21 Solesne Bourguin, Thanh Dang
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
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Improved composite quantile regression and variable selection with nonignorable dropouts Random Matrices Theory Appl. (IF 1.209) Pub Date : 2021-05-18 Wei Ma, Lei Wang
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
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Upper bounds for the maximum deviation of the Pearcey process Random Matrices Theory Appl. (IF 1.209) Pub Date : 2021-05-10 Christophe Charlier
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→∞ℙ supx>s N(x) −33 4π x4 3 −3ρ 2πx2 3 log x ≤ 42 3π + 𝜖 = 1, where 𝜖 > 0 is arbitrary. We also obtain
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Expressing the largest eigenvalue of a singular beta F-matrix with heterogeneous hypergeometric functions Random Matrices Theory Appl. (IF 1.209) Pub Date : 2021-05-10 Koki Shimizu, Hiroki Hashiguchi
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
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On a generalization of the CLT for linear eigenvalue statistics of Wigner matrices with inhomogeneous fourth moments Random Matrices Theory Appl. (IF 1.209) Pub Date : 2021-05-04 Zhenggang Wang, Jianfeng Yao
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.
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Marchenko–Pastur law with relaxed independence conditions Random Matrices Theory Appl. (IF 1.209) Pub Date : 2021-02-19 Jennifer Bryson, Roman Vershynin, Hongkai Zhao
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
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A degenerate Gaussian weight connected with Painlevé equations and Heun equations Random Matrices Theory Appl. (IF 1.209) Pub Date : 2020-12-26 Pengju Han, Yang Chen
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
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Pair dependent linear statistics for CβE Random Matrices Theory Appl. (IF 1.209) Pub Date : 2020-12-17 Ander Aguirre, Alexander Soshnikov, Joshua Sumpter
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.