Asymptotics of the distribution of the spectrum of random matrices

CONTENTS Introduction § 1. A new inversion formula for the Stieltjes transforms of spectral functions § 2. An inequality for the difference of the Stieltjes transforms of spectral functions § 3. An estimate of the deviation of sums of smoothed densities of distributions of eigenvalues from the semicircle law § 4. An estimate of the deviation of sums of smoothed densities of distributions of eigenvalues from the Marchenko-Pastur density § 5. The asymptotics of sums of smoothed densities of eigenvalue distributions of symmetric random matrices whose diagonal elements have non-zero mathematical expectations § 6. The asymptotics of Stieltjes transforms of spectral functions of symmetric random matrices whose elements have different dispersions § 7. The asymptotics of Stieltjes transforms of spectral functions of empirical covariance matrices § 8. Limit theorems for boundary eigenvalues of symmetric random matrices § 9. Limit theorems for boundary singular eigenvalues of random matrices References