List of basic notations and assumptions How the stochastic canonical equation was found Chapter 1. Canonical equation K I 1.1. Main assertion 1.2. Perturbation formulas for the entries of resolvent of a matrix 1.3. Strong Law for normalized spectral functions of random matrix. xv XIX 1 1 3The method of martingale differences 4 1.4. Limit theorem for random quadratic forms 8 1.5. Inequalities for the entries of the resolvents of random matrices 1.6. Limit theorem for a sum of random entries multiplied by diagonal entries of the resolvents of random matrices 1.7. Proof of the limit theorem for the sum of diagonal entries of the resolvents of random matrices by the method of martingale differences 1.8. Main inequality. Accompanying system of canonical equations Kl 1.9. Existence of solution of the system of canonical equations Kl 1.10. Uniqueness of the solution of the system of canonical equations KI 1.11. Existence of the densities of accompanying normalized spectral functions. The completion of the proof of Theorem 1.1 1.12. Limit theorem for individual spectral functions 1.13. Strong Law for individual spectral functions of random symmetric matrices 1.14. Weak Law for random matrices 1.15. Canonical equation Kl for sparse random symmetric matrices Chapter 2. Canonical equation K 2 . Necessary and sufficient modified Lindeberg's condition. The Wigner and Cubic laws 2.1. Formulation of the main assertion 2.2. Invariance principle for the entries of the resolvents of random matrices 2.3. Equation Ml for the trace of the resolvent of a random symmetric matrix 2.4. Solvability of the accompanying equation Ll 2.5. Proof of the existence of the density of the accompanying normalized spectral function based on the unique solvability of the spectral equation Ll 37 2.6. Uniform inequality for normalized spectral functions of random 3.1. Main theorem for ACE-matrices 3.2. Limit theorem for random nonnegative definite quadratic forms 3.3. Accompanying random infinitely divisible law for random quadratic forms 3.4. Self-averaging of accompanying random infinitely divisible law 3.5. Limit Theorem for perturbed diagonal entries of resolvents 3.6. Limit theorem for the sum of random entries multiplied by diagonal entries of a resolvents 3.7. Accompanying random infinitely divisible law for the sum of random entries 3.8. Method of martingale differences in the proof of the limit theorem for random quadratic forms 3.9. Method of the regularization of the resolvents of ACE-matrices 3.10. Vanishing of the imaginary parts of the entries of the resolvents of ACE-matrices 3.11. Accompanying regularized stochastic canonical equation K 3 3.12. Uniqueness of the solution of the accompanying regularized stochastic canonical equation K3 3.13. Method of successive approximations for the solution of the accompanying regularized stochastic canonical equation Chapter 4. Stochastic canonical equation K4 for symmetric random matrices with infinitely small entries. Necessary and sufficient conditions for the convergence of normalized spectral fun...
The method of random determinants for estimating and calculating the permanent of a matrix is considered. The following assertion is proved: if inf min α,· ; · > 0, sup max a,ij < oo, then
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