Cognitive Radio Communication and Networking: Principles and Practice

Chapter 5

Large Random Matrices

5.1 Large Dimensional Random Matrices: Moment Approach, Stieltjes Transform and Free Probability

The necessity of studying the spectra of large dimensional random matrices, in particular, the Wigner matrices, arose in nuclear physics in the 1950s. In quantum mechanics, the energy levels of quantum are not directly observable (very similar to many problems in today's wireless communications and the Smart Grid), but can be characterized by the eigenvalues of a matrix of observations [10].

Let X_ij be i.i.d. standard normal variables of n × p matrix X

The sample covariance matrix is defined as

where n vector samples of a p-dimensional zero-mean random vector with population matrix I.

The classical limit theorem are no longer suitable for dealing with large dimensional data analysis. In the early 1980s, major contributions on the existence of the limiting spectral distribution (LSD) were made. In recent years, research on random matrix theory has turned toward second-order limiting theorems, such as the central limit theorem for linear spectral statistics, the limiting distributions of spectral spacings, and extreme eigenvalues.

Many applied problems require an estimate of a covariance matrix and/or of its inverse, where the matrix dimension is large compared ...