# An Introduction to Statistical Analysis of Random Arrays

Vāčeslav Leonidovič Girko
VSP, 1998 - Mathematics - 673 pages
This 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 References 649 Index 669 Copyright