The principal objective of this thesis is : the study of the fluctuations of functionals of spectrum for large random matrices, the construction of consistent estimators and the study of their performances, in the situation where the dimension of observations is with the same order as the number of the available observations. There will be two parts in the report : the methodological contribution and the estimation in large-dimensional data. As to the methodological contribution, we will study the fluctuations for spectral linear statistics of the model 'information-plus-noise' for analytic functionals, and the extension for non-analytic functionals. The extension is based on the interpolation between random variables and Gaussian terms. This method can be applied to empirical covariance matrices. Another part consists in the estimation of the eigenvalues of the real covariance from the empirical covariance for high dimensional data and the study of its performance. We propose a new consistent estimator and the fluctuation of the estimator will be studied . In wireless communications, this procedure permits a secondary network to ensure the presence of the available spectral resources.
Estimation and fluctuations of functionals of large random matrices