| Preface | p. ix |
| Introduction | p. 1 |
| Kolmogorov Asymptotics in Problems of Multivariate Analysis | p. 2 |
| Spectral Theory of Large Covariance Matrices | p. 5 |
| Limit Formulas for Spectral Functions | p. 5 |
| Spectral Functions for Fixed Dimension and Sample Size | p. 11 |
| Method to Single out the Leading Terms | p. 13 |
| Approximately Unimprovable Essentially Multivariate Procedures | p. 14 |
| Spectral Properties of Large Wishart Matrices | p. 25 |
| Wishart Distribution | p. 27 |
| Limit Moments of Wishart Matrices | p. 32 |
| Limit Formula for the Resolvent of Wishart Matrices | p. 38 |
| Resolvents and Spectral Functions of Large Sample Covariance Matrices | p. 40 |
| Spectral Functions of Random Gram Matrices | p. 41 |
| Spectral Functions of Sample Covariance Matrices | p. 47 |
| Limit Spectral Functions of the Increasing Sample Covariance Matrices | p. 52 |
| Resolvents and Spectral Functions of Large Pooled Sample Covariance Matrices | p. 61 |
| Problem Setting | p. 61 |
| Spectral Functions of Pooled Random Gram Matrices | p. 63 |
| Spectral Functions of Pooled Sample Covariance Matrices | p. 68 |
| Limit Spectral Functions of the Increasing Pooled Sample Covariance Matrices | p. 73 |
| Normal Evaluation of Quality Functions | p. 76 |
| Measure of Normalizability | p. 77 |
| Spectral Functions of Large Covariance Matrices | p. 78 |
| Normal Evaluation of Sample Dependent Functionals | p. 79 |
| Discussion | p. 86 |
| Estimation of High-Dimensional Inverse Covariance Matrices | p. 87 |
| Shrinkage Estimators of the Inverse Covariance Matrices | p. 88 |
| Generalized Ridge Estimators of the Inverse Covariance Matrices | p. 89 |
| Asymptotically Unimprovable Estimators of the Inverse Covariance Matrices | p. 98 |
| Epsilon-Dominating Component-Wise Shrinkage Estimators of Normal Mean | p. 102 |
| Estimation Function for the Component-Wise Estimators | p. 102 |
| Estimators of the Unimprovable Estimation Function | p. 104 |
| Improved Estimators of High-Dimensional Expectation Vectors | p. 112 |
| Limit Quadratic Risk for a Class of Estimators of Expectation Vectors | p. 113 |
| Minimization of the Limit Quadratic Risk | p. 119 |
| Statistics to Approximate the Limit Risk Function | p. 124 |
| Statistics to Approximate the Extremal limit Solution | p. 126 |
| Quadratic Risk of Linear Regression with a Large Number of Random Predictors | p. 131 |
| Spectral Functions of Sample Covariance Matrices | p. 133 |
| Functionals Depending on the Statistics S and g0 | p. 135 |
| Functionals Depending on Sample Covariance Matrices and Covariance Vectors | p. 144 |
| The Leading Part of the Quadratic Risk and its Estimator | p. 148 |
| Special Cases | p. 153 |
| Linear Discriminant Analysis of Normal Populations with Coinciding Covariance Matrices | p. 156 |
| Problem Setting | p. 157 |
| Expectation and Variance of Generalized Discriminant Functions | p. 159 |
| Limit Probabilities of the Discrimination Errors | p. 166 |
| Population Free Quality of Discrimination | p. 169 |
| Problem Setting | p. 169 |
| Leading Parts of Functionals for Normal Populations | p. 171 |
| Leading Parts of Functionals for Arbitrary Populations | p. 173 |
| Discussion | p. 176 |
| Proofs | p. 177 |
| Theory of Discriminant Analysis of the Increasing Number of Independent Variables | p. 187 |
| Problem Setting | p. 187 |
| A Priori Weighting of Independent Variables | p. 193 |
| Minimization of the Limit Error Probability for a Priori Weighting | p. 201 |
| Weighting of Independent Variables by Estimators | p. 203 |
| Minimization of the Limit Error Probability for Weighting by Estimators | p. 210 |
| Statistics to Estimate Probabilities of Errors | p. 214 |
| Contribution of Variables to Discrimination | p. 217 |
| Selection of a Large Number of Independent Variables | p. 219 |
| Conclusions | p. 227 |
| References | p. 233 |
| Index | p. 239 |
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