Related papers: Eigenvalue statistics for the sum of two complex W…
Let A be a p-variate real Wishart matrix on n degrees of freedom with identity covariance. The distribution of the largest eigenvalue in A has important applications in multivariate statistics. Consider the asymptotics when p grows in…
Identifying the spectrum of the sum of two given Hermitian matrices with fixed eigenvalues is the famous Horn's problem.In this note, we investigate a variant of Horn's problem, i.e., we identify the probability density function (abbr. pdf)…
Random matrix theory has become a cornerstone in modern statistics and data science, providing fundamental tools for understanding high-dimensional covariance structures. Within this framework, the Wishart matrix plays a central role in…
The Wishart distribution and its generalizations are among the most prominent probability distributions in multivariate statistical analysis, arising naturally in applied research and as a basis for theoretical models. In this paper, we…
We compute analytically the probability distribution and moments of the sum and product of the non-zero eigenvalues and singular values of random matrices with (i) non-negative entries, (ii) fixed rank, and (iii) prescribed sums of the…
Quantifying the eigenvalue spectra of large random matrices allows one to understand the factors that contribute to the stability of dynamical systems with many interacting components. This work explores the effect that the interaction…
Estimating the eigenvalues of a population covariance matrix from a sample covariance matrix is a problem of fundamental importance in multivariate statistics; the eigenvalues of covariance matrices play a key role in many widely…
Finding the eigenvalues connected to the covariance operator of a centred Hilbert-space valued Gaussian process is genuinely considered a hard problem in several mathematical disciplines. In statistics this problem arises for instance in…
We develop a formalism to compute the statistics of the top eigenpair of weighted sparse graphs with finite mean connectivity and bounded maximal degree. Framing the problem in terms of optimisation of a quadratic form on the sphere and…
We study concentration inequalities for structured weighted sums of random data, including (i) tensor inner products and (ii) sequential matrix sums. We are interested in tail bounds and concentration inequalities for those structured…
We study sample covariance matrices of the form $W=\frac 1n C C^T$, where $C$ is a $k\times n$ matrix with i.i.d. mean zero entries. This is a generalization of so-called Wishart matrices, where the entries of $C$ are independent and…
The classical methods of multivariate analysis are based on the eigenvalues of one or two sample covariance matrices. In many applications of these methods, for example to high dimensional data, it is natural to consider alternative…
We initiate the study of goodness-of-fit testing when the data consist of positive definite matrices. Motivated by the recent appearance of the cone of positive definite matrices in numerous areas of applied research, including diffusion…
Recently, D. Wang has devised a new contour integral based method to simplify certain matrix integrals. Capitalizing on that approach, we derive a new expression for the probability density function (p.d.f.) of the joint eigenvalues of a…
The Wishart model of random covariance or correlation matrices continues to find ever more applications as the wealth of data on complex systems of all types grows. The heavy tails often encountered prompt generalizations of the Wishart…
We consider a versatile matrix model of the form ${\bf A}+i {\bf B}$, where ${\bf A}$ and ${\bf B}$ are real random circulant matrices with independent but, in general, nonidentically distributed Gaussian entries. For this model, we derive…
We derive concentration inequalities for the spectral measure of large random matrices, allowing for certain forms of dependence. Our main focus is on empirical covariance (Wishart) matrices, but general symmetric random matrices are also…
The variance--covariance matrix plays a central role in the inferential theories of high-dimensional factor models in finance and economics. Popular regularization methods of directly exploiting sparsity are not directly applicable to many…
The correlated Wishart model provides the standard benchmark when analyzing time series of any kind. Unfortunately, the real case, which is the most relevant one in applications, poses serious challenges for analytical calculations. Often…
We propose a technique for calculating and understanding the eigenvalue distribution of sums of random matrices from the known distribution of the summands. The exact problem is formidably hard. One extreme approximation to the true density…