## An Introduction to Statistical Analysis of Random ArraysThis book contains the results of 30 years of investigation by the author into the creation of a new theory on statistical analysis of observations, based on the principle of random arrays of random vectors and matrices of increasing dimensions. It describes limit phenomena of sequences of random observations, which occupy a central place in the theory of random matrices. This is the first book to explore statistical analysis of random arrays and provides the necessary tools for such analysis. This book is a natural generalization of multidimensional statistical analysis and aims to provide its readers with new, improved estimators of this analysis. The book consists of 14 chapters and opens with the theory of sample random matrices of fixed dimension, which allows to envelop not only the problems of multidimensional statistical analysis, but also some important problems of mechanics, physics and economics. The second chapter deals with all 50 known canonical equations of the new statistical analysis, which form the basis for finding new and improved statistical estimators. Chapters 3-5 contain detailed proof of the three main laws on the theory of sample random matrices. In chapters 6-10 detailed, strong proofs of the Circular and Elliptic Laws and their generalization are given. In chapters 11-13 the convergence rates of spectral functions are given for the practical application of new estimators and important questions on random matrix physics are considered. The final chapter contains 54 new statistical estimators, which generalize the main estimators of statistical analysis. |

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### Contents

Introduction to the theory of sample matrices of fixed dimension | 1 |

Measurability and nondegeneracy of eigenvalues of random matrices | 3 |

Maximum likelihood estimates of the parameters of a multivariate normal distribution | 4 |

Symmetric random real matrices | 5 |

Random Hermitian matrices | 6 |

Distribution of roots of a random polynomial with real coefficients | 8 |

Expected number of real roots of a random polynomial with real coefficients | 9 |

Expected number of real roots of a random real analytic function | 11 |

Inequality for the heavy tails of normalized spectral functions outside of interval x 2 + с о О | 471 |

REFORM and martingale differences methods of deriving the first main inequality | 473 |

The invariance principle for symmetric random matrices with independent entries | 476 |

Main equation for the resolvent of random symmetric matrix | 480 |

Inequalities for the coefficients of the main equation | 481 |

Important lemma | 482 |

Equation for the sum of smoothed distribution functions of eigenvalues of random symmetric matrix | 483 |

The double Fmethod for finding the second main inequality for spectral functions | 484 |

Average density of complex roots of a random polynomial with complex coefficients | 12 |

Expected normalized counting functions of the roots of the random polynomial obtained by Ftransform | 13 |

Fdistribution of the roots of a random polynomial | 16 |

Calculation of expected logarithm of the absolute value of a random polynomial | 18 |

Distribution function of the eigenvalues of a random nonsymmetric real matrix when its eigenvalues are real | 20 |

Random nonsymmetric real matrices | 23 |

Canonical Vequation A2e The Vdensity of eigenvalues | 26 |

Spectral stochastic differential equations for random symmetric | 30 |

Random nonsymmetric real matrices with special form of selected eigenvectors | 35 |

Random complex matrices | 36 |

Random complex symmetric matrices | 37 |

Distribution of triangular decomposition of random complex matrices | 38 |

Distribution of triangular decomposition of unitary invariant random complex matrices | 39 |

Random unitary matrices | 40 |

Orthogonal random matrices | 41 |

The method of triangular matrices for rigorous derivation of the distribution function of eigenvalues of Gaussian nonsymmetric real matrix Girko 19... | 44 |

Straight and back spectral Kolmogorov equations for densities of eigenvalues of random symmetric matrix processes with independent increments | 47 |

Spectral stochastic differential equations for random matrixvalued | 49 |

Canonical equation Kn for normalized spectral functions | 100 |

Canonical equation As2 for normalized spectral functions | 114 |

Stochastic canonical equation KM for normalized spectral | 127 |

Canonical equation K50 for normalized spectral functions | 137 |

Limit theorem for eigenvalues and eigenvectors of random | 154 |

Invariance principle for the traces of resolvents of random matrices | 167 |

Double Fmethod for finding limits of eigenvalues for the case | 180 |

The Second Law for the singular values and eigenvectors of random | 201 |

Invariance principle for the entries of resolvents of random matrices | 209 |

Limit theorem for eigenvalues and eigenvectors of random | 215 |

Canonical equation 12 and C for the boundary points of Gdensity | 222 |

Existence of the solution of accompanying spectral equation LI | 228 |

Calculations of coefficients of the main equation MS | 236 |

Invariance principle for the traces of resolvents of random matrices | 247 |

Equation for the sum of smoothed distribution functions | 261 |

The Third Law for the eigenvalues and eigenvectors of empirical | 277 |

Invariance principle for the entries of resolvents of random matrices | 285 |

Boundedness of the boundary points of spectral density | 293 |

Calculations of the coefficients of the main equation Л5 for the case | 299 |

Invariance principle for random matrices for the case of convergence | 311 |

The first proof of the Strong Circular Law | 325 |

Strong Law for normalized spectral functions of nonselfadjoint | 349 |

The method of perpendiculars for proving the Strong | 359 |

The rigorous proof of the Strong Circular Law | 367 |

equation Ki | 394 |

Invariance principle for random matrices | 396 |

Canonical equation Limit theorem for Gfunctions | 398 |

Strong Elliptic Law | 402 |

The Circular and Uniform Laws for eigenvalues of random nonsymmetric complex matrices with independent entries | 405 |

Canonical spectral equation and the boundary points of spectral density | 406 |

Strong law for Gfunctions цпхт | 407 |

Limit theorems for Gfunctions Canonical equation K26 | 408 |

Invariance principle for random matrices | 409 |

The equation for the sum of smoothed distribution functions of eigenvalues of random matrices | 413 |

Method of Fourier and inverse Fourier transforms for finding boundaries of eigenvalues | 414 |

Limit theorem for the singular values of random matrices | 420 |

Method of perpendiculars for proving the Strong Circular Law | 421 |

Substitution of entries of normally distributed random variables for random matrix | 423 |

Substitution of determinant of Gram matrix for determinant of random matrix | 424 |

Regularized Vstransform | 427 |

Regularized transform Circular Law | 429 |

Limit theorems for eigenvalues of random nonsymmetric matrices | 432 |

Strong VLaw for eigenvalues of nonsymmetric random matrices | 435 |

Strong Law for normalized spectral functions of nonselfadjoint random matrices with independent row vectors | 436 |

Substitution of the determinant of Gram matrix for the determinant of random matrix | 437 |

Regularized modified Vtransform for the spectral function | 442 |

Limit theorem for Stieltjes transform of spectral functions Canonical spectral equation if 7 | 446 |

The Strong FLaw | 447 |

The Fdensity of eigenvalues of random matrices with independent entries | 450 |

One example of the Vdensity for eigenvalues of random matrices with independent entries | 451 |

Convergence rate of the expected spectral functions of symmetric random matrices is equal to On12 | 453 |

Hermite polynomials | 454 |

ChristoffelDarboux formula | 455 |

Integral formula for the Hermite polynomials | 456 |

Method of steepest descent The first main asymptotic formula for Hermite polynomials | 457 |

The second main asymptotic formula for Hermite polynomials | 464 |

Structure of the Semicircle Law | 465 |

Imaginary logarithmic inequality for the expected normalized spectral functions | 466 |

Inequality for random normalized spectral functions | 470 |

Convergence rate of expected normalized spectral functions of the Hermitian Gaussian matrices outside of the interval x 2 en13 О 0 | 485 |

The fourth main inequality | 486 |

Inequality for the distance between expected normalized spectral function and Semicircle Law | 489 |

Inequality for the expected maximal distance between normalized spectral function and Semicircle Law | 491 |

Convergence rate for eigenvalues of random symmetric matrices | 492 |

Convergence rate of expected normalized spectral functions outside of the interval x 2 + с с 0 | 493 |

Convergence rate of expected spectral functions of the sample covariance matrix Rmnn is equal to O7i12 under the condition cl 495 1 Introduction a... | 495 |

Laguerre polynomials The basic properties | 497 |

Orthonormalized Laguerre functions | 499 |

Asymptotics of Laguerre polynomials in scheme of series | 500 |

Method of steepest descent The first main asymptotic formula for Laguerre polynomials | 502 |

The second main asymptotic formula for Laguerre polynomials | 513 |

Convegence rate for normalized spectral functions of Gaussian Gram matrices | 516 |

REFORM and martingaledifferences methods of deriving the first | 518 |

Invariance principle for symmetric random matrices with independent entries | 523 |

The second main inequality | 527 |

Double Fmethod for finding the boundaries of eigenvalues | 529 |

Convergence rate of normalized spectral function of empirical covariance matrices is equal to 0n12 | 531 |

The main fifth inequality for the expected maximal distance between a normalized spectral function and the limit law | 532 |

Convergence rate for eigenvalues of sample covariance matrices | 533 |

The First Spacing Law for random symmetric matrices | 535 |

Spacings and their spectral functions | 537 |

Expected spacing spectral functions | 538 |

Expected truncated spacing spectral functions | 539 |

Integral representation of the expected normalized spacing spectral functions | 540 |

Limits for Hermite polynomials | 541 |

Semicircle Law for Gaussian Hermitian matrices | 544 |

Limit theorem for normalized spacing spectral functions | 545 |

Limit of expected normalized spectral function of the spacings of the squares of eigenvalues | 546 |

Wigner surmise | 547 |

Mean level distance Dnmax mn | 548 |

Local limit average density for spacings of Hermitian Gaussian matrices | 549 |

Histograms of eigenvalues of random matrices | 551 |

Histograms of a part of eigenvalues of random matrices | 552 |

Ten years of General Statistical Analysis The main Gestimators of General Statistical Analysis | 553 |

G i estimator of generalized variance | 556 |

GI estimator of the real Stieltjes transform of the normalized spectral function of covariance matrices | 557 |

Gsestimator of inverse covariance matrix | 560 |

Class of Gjestimators for the traces of the powers of covariance matrices | 561 |

Gsestimator of smoothed normalized spectral function of symmetric matrices | 562 |

Geestimator of Stieltjes transform of covariance matrix pencil | 563 |

Gyestimator of the states of discrete control systems | 566 |

Class of Ggestimators of the solutions of systems of linear algebraic equations SLAB | 571 |

Ggestimator of the solution of the discrete KolmogorovWiener filter | 575 |

Gioestimator of the solution of a regularized discrete KolmogorovWiener filter with known free vector | 576 |

GII estimator of the Mahalanobis distance | 579 |

Gi2regularized Mahalanobis distance estimator | 580 |

Discrimination of two populations with common unknown covariance matrix GiaAndersonFisher statistics estimator | 581 |

Gnestimator of regularized discriminant function | 582 |

Gisestimator of the nonlinear discriminant function obtained by observation of random vectors with different covariance matrices | 583 |

Class of Gieestimators in the theory of experimental design when the design matrix is unknown | 584 |

Gi7estimate of T2statistics | 589 |

Quasiinversion method for solving Gequations | 590 |

Estimator Gao of regularized function of unknown parameters | 595 |

G2i estimator in the likelihood method | 596 |

G22estimator in the classification method | 598 |

G23estimator in the method of stochastic approximation | 600 |

Class of estimators G24 which minimizes the certain meansquare risks | 601 |

G25estimator of the Stieltjes transform of the principal components | 602 |

G26estimator of eigenvalues of the covariance matrix | 603 |

G27estimators of eigenvectors corresponding to extreme eigenvalues of the covariance matrix | 607 |

G28 consistent estimator of the trace of the resolvent of the Gram matrix | 608 |

G29 consistent estimator of singular values of the matrix | 609 |

Gaoconsistent estimator of eigenvectors corresponding to extreme singular values of the matrix | 610 |

Gsiestimator of the resolvent of a symmetric matrix | 611 |

G32estimator of eigenvalues of a symmetric matrix | 616 |

Gs4estimator of the transform | 620 |

Gasestimators of eigenvalues of random matrices with independent pairs of entries | 621 |

Gseestimator of eigenvectors of matrices with independent pairs of entries | 622 |

Gayestimator of the statistic | 623 |

Gas estimators of symmetric functions of eigenvalues | 624 |

G47estimator for solution of the system of linear differential | 635 |

GSIestimator of solutions of LPP | 642 |

649 | |

669 | |

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### Common terms and phrases

analytic functions blocks calculations canonical equation class of analytic column vectors common probability space complex matrix components conditions of Theorem Consider covariance matrix Denote distribution function eigenvalues eigenvalues of random eigenvectors empirical covariance matrices estimator Euler angles exists a unique fc-th Fn(x formula function whose Stieltjes FUNCTIONS OF RANDOM Girl2 Girl9 Haar measure Hermitian Hermitian matrix independent observations inequality integral Jacobian Lemma lim lim lim sup limit theorems Lindeberg's condition matrix Hn matrix Rmn normalized spectral function obtain orthogonal orthogonal matrix polynomial probability density probability one lim probability space proof of Theorem proved random complex random entries random matrices RANDOM SYMMETRIC MATRICES random variables random vectors row vectors satisfy the canonical satisfying the condition Stieltjes transform sup max sup sup symmetric matrix system of equations Theorem 3.1 transform is equal unique solution