Amalgamated Free Lévy Processes as Limits of Sample Covariance Matrices

We prove the existence of joint limiting spectral distributions for families of random sample covariance matrices modeled on fluctuations of discretized Lévy processes. These models were first considered in applications of random matrix theory to financial data, where datasets exhibit both strong multicollinearity and non-normality. When the underlying Lévy process is non-Gaussian, we show that the limiting spectral distributions are distinct from Marčenko–Pastur. In the context of operator-valued free probability, it is shown that the algebras generated by these families are asymptotically free with amalgamation over the diagonal subalgebra. This framework is used to construct operator-valued $$^*$$ ∗ -probability spaces, where the limits of sample covariance matrices play the role of non-commutative Lévy processes whose increments are free with amalgamation.

Steering Witness and Steering Criterion of Gaussian States

Quantum steering is an important quantum resource, which is intermediate between entanglement and Bell nonlocality. In this paper, we study steering witnesses for Gaussian states in continuous-variable systems. We give a definition of steering witnesses by covariance matrices of Gaussian states, and then obtain a steering criterion by steering witnesses to detect steerability of any (m+n)-mode Gaussian states. In addition, the conditions for two steering witnesses to be comparable and the optimality of steering witnesses are also discussed.

A Projection Method for the Estimation of Error Covariance Matrices for Variational Data Assimilation in Ocean Modelling

Data assimilation methods are an invaluable tool for operational ocean models. These methods are often based on a variational approach and require the knowledge of the spatial covariances of the background errors (differences between the numerical model and the true values) and the observation errors (differences between true and measured values). Since the true values are never known in practice, the error covariance matrices containing values of the covariance functions at different locations, are estimated approximately. Several methods have been devised to compute these matrices, one of the most widely used is the one developed by Hollingsworth and Lönnberg (H-L). This method requires to bin (combine) the data points separated by similar distances, compute covariances in each bin and then to find a best fit covariance function. While being a helpful tool, the H-L method has its limitations. We have developed a new mathematical method for computing the background and observation error covariance functions and therefore the error covariance matrices. The method uses functional analysis which allows to overcome some shortcomings of the H-L method, for example, the assumption of statistical isotropy. It also eliminates the intermediate steps used in the H-L method such as binning the innovations (differences between observations and the model), and the computation of innovation covariances for each bin, before the best-fit curve can be found. We show that the new method works in situations where the standard H-L method experiences difficulties, especially when observations are scarce. It gives a better estimate than the H-L in a synthetic idealised case where the true covariance function is known. We also demonstrate that in many cases the new method allows to use the separable convolution mathematical algorithm to increase the computational speed significantly, up to an order of magnitude. The Projection Method (PROM) also allows computing 2D and 3D covariance functions in addition to the standard 1D case.

BEST LINEAR UNBIASED LATENT VALUES PREDICTORS FOR FINITE POPULATION LINEAR MODELS WITH DIFFERENT ERROR SOURCES

We develop best linear unbiased predictors (BLUP) of the latent values of labeled sample units selected from a finite population when there are two distinct sources of measurement error: endogenous, exogenous or both. Usual target parameters are the population mean, the latent values associated to a labeled unit or the latent value of the unit that will appear in a given position in the sample. We show how both types of measurement errors affect the within unit covariance matrices and indicate how the finite population BLUP may be obtained via standard software packages employed to fit mixed models in situations with either heteroskedastic or homoskedastic exogenous and endogenous measurement errors.

Some strong convergence theorems for eigenvalues of general sample covariance matrices

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 [Formula: see text], where [Formula: see text] is a [Formula: see text] nonrandom matrix and [Formula: see text] is an [Formula: see text] matrix consisting of i.i.d standard complex entries. [Formula: see text] as [Formula: see text] while [Formula: see text] can be arbitrary but no smaller than [Formula: see text]. We first prove that under some mild assumptions, with probability 1, for all large [Formula: see text], there will be no eigenvalues in any closed interval contained in an open interval which is outside the supports of the limiting distributions for all sufficiently large [Formula: see text]. Then we get the strong convergence result for the extreme eigenvalues as an extension of Bai-Yin law.