Spectral analysis of large dimentional random matrices

Abstract

Abstract: The thesis studies the limiting spectral distributionsfor three classes of large dimensional random matrices. Theyinclude the Wigner type random matrices, the general samplecovariance matrices and the sparse random matrices expressed byHadamard products.The Wigner type random matrices generalize the well known Wignerrandom matrices with independent entries to the case where theentries in the matrices are statistically correlated. The generalsample covariance matrices taking the form:$$A_n=\frac{1}{N}T_{2n}^{1/2}X_nT_{1n}X_n^*T_{2n}^{1/2},$$where $T_{2n}$ is nonnegative definite and $T_{1n}$ is Hermitian.Existing works are only available for some special cases of $A_n$,{\it e.g.} when $T_{1n}=I_n$ with $T_{2n}$ nonnegative definite orwhen $T_{1n}$ and $T_{2n}$ are both diagonal matrices. This classof random matrices of significance in many application fieldsalso. The sparse random matrices, or more specifically, theHadamard products of a (normalized) sample covariance randommatrix and a sparsing matrix, are studied under a non-homogeneousand non-zero-one sparseness assumption. The results covers most ofthe usually seen works and provides new results to the randommatrix theory.For the first two classes of random matrices, we mainly rely onthe Stieltjes transform method, while for the last class ofmatrices, we employ the moment method.