On a surprising relation between rectangular and square free convolutions

Debbah and Ryan have recently proved a result about the limit empirical singular distribution of the sum of two rectangular random matrices whose dimensions tend to infinity. In this paper, we reformulate it in terms of the rectangular free convolution introduced in a previous paper and then we give a new, shorter, proof of this result under weaker hypothesis: we do not suppose the \pro measure in question in this result to be compactly supported anymore. At last, we discuss the inclusion of this result in the family of relations between rectangular and square random matrices.Comment: 8 page

On a surprising relation between the Marchenko-Pastur law, rectangular and square free convolutions

n this paper, we prove a result linking the square and the rectangular R-transforms, the consequence of which is a surprising relation between the square and rectangular versions the free additive convolutions, involving the Marchenko-Pastur law. Consequences on random matrices, on infinite divisibility and on the arithmetics of the square versions of the free additive and multiplicative convolutions are given.Comment: 11 pages, 1 figure. To appear in Ann. Inst. Henri Poincar\'e Probab. Sta

Fluctuations of an improved population eigenvalue estimator in sample covariance matrix models

This article provides a central limit theorem for a consistent estimator of population eigenvalues with large multiplicities based on sample covariance matrices. The focus is on limited sample size situations, whereby the number of available observations is known and comparable in magnitude to the observation dimension. An exact expression as well as an empirical, asymptotically accurate, approximation of the limiting variance is derived. Simulations are performed that corroborate the theoretical claims. A specific application to wireless sensor networks is developed.Comment: 30 p

Channel Capacity Estimation using Free Probability Theory

In many channel measurement applications, one needs to estimate some characteristics of the channels based on a limited set of measurements. This is mainly due to the highly time varying characteristics of the channel. In this contribution, it will be shown how free probability can be used for channel capacity estimation in MIMO systems. Free probability has already been applied in various application fields such as digital communications, nuclear physics and mathematical finance, and has been shown to be an invaluable tool for describing the asymptotic behaviour of many large-dimensional systems. In particular, using the concept of free deconvolution, we provide an asymptotically (w.r.t. the number of observations) unbiased capacity estimator for MIMO channels impaired with noise called the free probability based estimator. Another estimator, called the Gaussian matrix mean based estimator, is also introduced by slightly modifying the free probability based estimator. This estimator is shown to give unbiased estimation of the moments of the channel matrix for any number of observations. Also, the estimator has this property when we extend to MIMO channels with phase off-set and frequency drift, for which no estimator has been provided so far in the literature. It is also shown that both the free probability based and the Gaussian matrix mean based estimator are asymptotically unbiased capacity estimators as the number of transmit antennas go to infinity, regardless of whether phase off-set and frequency drift are present. The limitations in the two estimators are also explained. Simulations are run to assess the performance of the estimators for a low number of antennas and samples to confirm the usefulness of the asymptotic results.Comment: Submitted to IEEE Transactions on Signal Processing. 12 pages, 9 figure

Finite Dimensional Statistical Inference

In this paper, we derive the explicit series expansion of the eigenvalue distribution of various models, namely the case of non-central Wishart distributions, as well as correlated zero mean Wishart distributions. The tools used extend those of the free probability framework, which have been quite successful for high dimensional statistical inference (when the size of the matrices tends to infinity), also known as free deconvolution. This contribution focuses on the finite Gaussian case and proposes algorithmic methods to compute the moments. Cases where asymptotic results fail to apply are also discussed.Comment: 14 pages, 13 figures. Submitted to IEEE Transactions on Information Theor

Convolution operations arising from Vandermonde matrices

Different types of convolution operations involving large Vandermonde matrices are considered. The convolutions parallel those of large Gaussian matrices and additive and multiplicative free convolution. First additive and multiplicative convolution of Vandermonde matrices and deterministic diagonal matrices are considered. After this, several cases of additive and multiplicative convolution of two independent Vandermonde matrices are considered. It is also shown that the convergence of any combination of Vandermonde matrices is almost sure. We will divide the considered convolutions into two types: those which depend on the phase distribution of the Vandermonde matrices, and those which depend only on the spectra of the matrices. A general criterion is presented to find which type applies for any given convolution. A simulation is presented, verifying the results. Implementations of all considered convolutions are provided and discussed, together with the challenges in making these implementations efficient. The implementation is based on the technique of Fourier-Motzkin elimination, and is quite general as it can be applied to virtually any combination of Vandermonde matrices. Generalizations to related random matrices, such as Toeplitz and Hankel matrices, are also discussed.Comment: Submitted to IEEE Transactions on Information Theory. 16 pages, 1 figur