On the Empirical Spectral Distribution of Symmetric Matrices with Dependent Entries
Tuesday, April 28, 2015 - 2:00pm - 2:50pm
Magda Peligrad (University of Cincinnati)
In this talk, we will survey some recent results on the empirical eigenvalue distribution of symmetric matrices with dependent entries, selected from regular random fields. Emphasis will be put on the covariance matrix. It will be pointed out that, in many situations of interest, the limiting spectral measure always exists and depends only on the covariance structure of the field. The strength of the dependence is not important; the field can have long or short range memory and no rate of convergence to zero of the covariances is imposed. We characterize this limit in terms of the Stieltjes transform, via a certain equation involving the spectral density of the field. If the entries of the matrix are square integrable functions of an independent field, the results hold without any other additional assumptions. The talk is based on joint works with F. Merlevède.