An Introduction to Statistical Analysis of Random Arrays

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Vāčeslav Leonidovič Girko
VSP, 1998 - Mathematics - 673 pages
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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
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About the author (1998)

Vyacheslav L. Girko is Professor of Mathematics in the Department of Applied Statistics at the National University of Kiev and the University of Kiev Mohyla Academy. He is also affiliated with the Institute of Mathematics, Ukrainian Academy of Sciences. His research interests include multivariate statistical analysis, discriminant analysis, experiment planning, identification and control of complex systems, statistical methods in physics, noise filtration, matrix analysis, and stochastic optimization. He has published widely in the areas of multidimensional statistical analysis and theory of random matrices.

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