Spectral analysis of large dimensional random matrices = 大维随机矩阵的谱分析 /
副标题：无
作 者：Zhidong Bai, Jack W. Silverstein.
分类号：
ISBN：9787030177667
微信扫一扫,移动浏览光盘
简介
The aim of the book is to introduce basic concepts, main results, and widely applied mathematical tools in the spectral analysis of large dimensional random matrices. In it we will introduce many of the fundamental results, such as the semicircular law of Wigner matrices, the Marchenko-Pastur law, the limiting spectral distribution of the multivariate F matrix, limits of extremal eigenvalues, spectrum separation theorems, convergence rates of empirical spectral distributions, central limit theorems of linear spectral statistics and the partial solution of the famous circular law. While deriving the main results, the book will simultaneously emphasize the ideas and methodologies of the fundamental mathematical tools, among them being: truncation techniques, matrix transformations, moment convergence theorems, and the Stieltjes transform. Thus, its treatment is especially fitting to the needs of mathematics and statistic graduate students, and beginning researchers, who can learn the basic methodologies and ideas to solve problems in this area. It may also serve as a detailed handbook on results of large dimensional random matrices for practical users. ...
目录
1 introduction.
1.1 large dimensional data analysis
1.2 random matrix theory
1.2.1 spectral analysis of large dimensional random matrices
1.2.2 limits of extreme eigenvalues
1.2.3 convergence rate of esd
1.2.4 circular law
1.2.5 clt of linear spectral statisticslinear spectral statistics
1.2.6 limiting distributions of extreme eigenvalues and spacings
1.3 methodologies
1.3.1 moment method
1.3.2 stieltjes transform
1.3.3 orthogonal polynomial decomposition
2 wigner matrices and semicircular law
2.1 semicircular law by the moment method
2.1.1 moments of the semicircular law
2.1.2 some lemmas of combinatorics
2.1.3 semicircular law for iid case
2.2 generalizations to the non-lid case
2.2.1 proof of theorem 2.9
.2.3 semicircular law by stieltjes transform
2.3.1 stieltjes transform of semicircular law
2.3.2 proof of theorem 2.9
3 sample covariance matrices, marcenko-pastur law
3.1 mp law for iid case
3.1.1 moments of the mp law
3.1.2 some lemmas on graph theory and combinatorics
3.1.3 mp law for iid case
3.2 generalization to the non-iid case
3.3 proof of theorem 3.9 by stieltjes transform
3.3.1 stieltjes transform of mp law
3.3.2 proof of theorem 3.9
4 product of two random matrices
4.1 some graph theory and combinatoric results
4.2 existence of the esd of sntn
4.2.1 truncation of the esd of tn
4.2.2 truncation, centralization and rescaling of the x-variables
4.2.3 proof of theorem 4.3
4.3 lsd of f matrix
4.3.1 general formula for the product
4.3.2 lsd of multivariate f matrices
4.4 proof of theorem 4.5
4.4.1 truncation and centralization
4.4.2 proof by stieltjes transform
5 limits of extreme eigenvalues
5.1 limit of extreme eigenwalues of the wigner matrix
5.1.1 sufficiency of conditions of theorem 5.1
5.1.2 necessity of conditions of theorem 5.1
5.2 limits of extreme eigenvalues of sample covariance matrix
5.2.1 proof of theorem 5.10
5.2.2 the proof of theorem 5.11
5.2.3 necessity of the conditions
5.3 miscellanies
6 spectrum separation
6.1 what is spectrum separation
6.1.1 mathematical tools
6.2 proof of (1)
6.2.1 truncation and some simple facts
6.2.2 a preliminary convergence rate
6.2.3 convergence of sn — esn
6.2.4 convergence of expected value
6.2.5 completing the proof
6.3 proof of (2)
6.4 proof of (3)
6.4.1 convergence of a random quadratic form
6.4.2 spread of eigenvalues
6.4.3 dependence on y
6.4.4 completing the proof of (3)
7 semicircle law for hadamard products
7.1 renormalized sample covariance matrix
7.2 sparse matrix and hadamard product
7.3 proof of theorem 7.4..
7.3.1 truncation and centralization
7.4 proof of theorem 7.4 by moment approach
8 convergence rates of esd
8.1 some lemmas about integrals of stieltjes transforms
8.2 convergence rates of expected esd of wigner matrices
8.2.1 lemmas on truncation, centralization and rescaling
8.2.2 proof of theorem 8.6
8.2.3 some lemmas of preliminary calculation
8.3 further extensions
8.4 convergence rates of expected esd of sample covariance matrices
8.4.1 assumptions and results
8.4.2 truncation and centralization
8.4.3 proof of theorem 8.16
8.5 some elementary calculus
8.5.1 increment of m-p density
8.5.2 integral of tail probability
8.5.3 bounds of stieltjes transforms of m-p law
8.5.4 bounds for bn
8.5.5 integrals of squared absolute values of stieltjes
transforms
8.5.6 higher central moments of stieltjes transforms
8.5.7 integral of δ
8.6 rates of convergence in probability and almost surely
9 clt for linear spectral statistics
9.1 motivation and strategy
9.2 clt of lss for wigner matrix
9.2.1 strategy of the proof
9.2.2 truncation and renormalization
9.2.3 mean function of mn
9.2.4 proof of the nonrandom part of (9.2.13) for j = l, r
9.3 convergence of the process mn - emn
9.3.1 finite-dimensional convergence of mn - emn
9.3.2 limit of s1
9.3.3 completion of proof of (9.2.13) for j = l, r
9.3.4 tightness of the process mn(z) - emn(z)
9.4 computation of the mean and covariance function of g(f)
9.4.1 mean function
9.4.2 covariance function
9.5 application to linear spectral statistics and related results
9.5.1 tchebychev polynomials
9.6 technical lemmas
9.7 clt of lss for sample covariance matrices
9.7.1 truncation
9.8 convergence of stieltjes transforms
9.9 convergence of finite dimensional distributions
9.10 tightness of mn1(z)
9.11 convergence of mn2(z)
9.12 some derivations and calculations
9.12.1 verification of (9.8.8)
9.12.2 verification of (9.8.9)
9.12.3 derivation of quantities in example (1.1)
9.12.4 verification of quantities in jonsson's results
9.12.5 verification of (9.7.8) and (9.7.9)
10 circular law
10.1 the problem and difficulty.
10.1.1 failure of techniques dealing with hermitian matrices
10.1.2 revisit of stieltjes transformation
10.2 a theorem establishing a partial answer to the circular law
10.3 lemmas on integral range reduction
10.4 characterization of the circular law
10.5 a rough rate on the convergence of vn,(x, z)
10.5.1 truncation and centralization
10.5.2 a convergence rate of the stieltjes transform of
10.6 proofs of (10.2.3) and (10.2.4)
10.7 proof of theorem 10.3
10.8 comments and extensions
10.8.1 relaxation of conditions assumed in theorem 10.3
10.9 some elementary mathematics
11 appendix a. some results in linear algebra
11.1 inverse matrices and resolvent
11.1.1 inverse matrix formula
11.1.2 holing a matrix
11.1.3 trace of inverse matrix
11.1.4 difference of traces of a matrix a and its major submatrices
11.1.5 inverse matrix of complex matrices
11.2 inequalities involving spectral distributions
11.2.1 singular value inequalities
11.3 hadamard product and odot product
11.4 extensions of singular value inequalities
11.4.1 definitions and properties
11.4.2 graph-associated multiple matrices
11.4.3 fundamental theorem on graph-associated mm
11.5 perturbation inequalities
11.6 rank inequalities
11.7 a norm inequality
12 appendix b. moment convergence theorem and stieltjes transform
12.1 moment convergence theorem
12.2 stieltjes transform
12.2.1 preliminary properties
12.2.2 inequalities of distance between distributions in terms of their stieltjes transforms
12.2.3 lemmas concerning levy distance
references
index...
1.1 large dimensional data analysis
1.2 random matrix theory
1.2.1 spectral analysis of large dimensional random matrices
1.2.2 limits of extreme eigenvalues
1.2.3 convergence rate of esd
1.2.4 circular law
1.2.5 clt of linear spectral statisticslinear spectral statistics
1.2.6 limiting distributions of extreme eigenvalues and spacings
1.3 methodologies
1.3.1 moment method
1.3.2 stieltjes transform
1.3.3 orthogonal polynomial decomposition
2 wigner matrices and semicircular law
2.1 semicircular law by the moment method
2.1.1 moments of the semicircular law
2.1.2 some lemmas of combinatorics
2.1.3 semicircular law for iid case
2.2 generalizations to the non-lid case
2.2.1 proof of theorem 2.9
.2.3 semicircular law by stieltjes transform
2.3.1 stieltjes transform of semicircular law
2.3.2 proof of theorem 2.9
3 sample covariance matrices, marcenko-pastur law
3.1 mp law for iid case
3.1.1 moments of the mp law
3.1.2 some lemmas on graph theory and combinatorics
3.1.3 mp law for iid case
3.2 generalization to the non-iid case
3.3 proof of theorem 3.9 by stieltjes transform
3.3.1 stieltjes transform of mp law
3.3.2 proof of theorem 3.9
4 product of two random matrices
4.1 some graph theory and combinatoric results
4.2 existence of the esd of sntn
4.2.1 truncation of the esd of tn
4.2.2 truncation, centralization and rescaling of the x-variables
4.2.3 proof of theorem 4.3
4.3 lsd of f matrix
4.3.1 general formula for the product
4.3.2 lsd of multivariate f matrices
4.4 proof of theorem 4.5
4.4.1 truncation and centralization
4.4.2 proof by stieltjes transform
5 limits of extreme eigenvalues
5.1 limit of extreme eigenwalues of the wigner matrix
5.1.1 sufficiency of conditions of theorem 5.1
5.1.2 necessity of conditions of theorem 5.1
5.2 limits of extreme eigenvalues of sample covariance matrix
5.2.1 proof of theorem 5.10
5.2.2 the proof of theorem 5.11
5.2.3 necessity of the conditions
5.3 miscellanies
6 spectrum separation
6.1 what is spectrum separation
6.1.1 mathematical tools
6.2 proof of (1)
6.2.1 truncation and some simple facts
6.2.2 a preliminary convergence rate
6.2.3 convergence of sn — esn
6.2.4 convergence of expected value
6.2.5 completing the proof
6.3 proof of (2)
6.4 proof of (3)
6.4.1 convergence of a random quadratic form
6.4.2 spread of eigenvalues
6.4.3 dependence on y
6.4.4 completing the proof of (3)
7 semicircle law for hadamard products
7.1 renormalized sample covariance matrix
7.2 sparse matrix and hadamard product
7.3 proof of theorem 7.4..
7.3.1 truncation and centralization
7.4 proof of theorem 7.4 by moment approach
8 convergence rates of esd
8.1 some lemmas about integrals of stieltjes transforms
8.2 convergence rates of expected esd of wigner matrices
8.2.1 lemmas on truncation, centralization and rescaling
8.2.2 proof of theorem 8.6
8.2.3 some lemmas of preliminary calculation
8.3 further extensions
8.4 convergence rates of expected esd of sample covariance matrices
8.4.1 assumptions and results
8.4.2 truncation and centralization
8.4.3 proof of theorem 8.16
8.5 some elementary calculus
8.5.1 increment of m-p density
8.5.2 integral of tail probability
8.5.3 bounds of stieltjes transforms of m-p law
8.5.4 bounds for bn
8.5.5 integrals of squared absolute values of stieltjes
transforms
8.5.6 higher central moments of stieltjes transforms
8.5.7 integral of δ
8.6 rates of convergence in probability and almost surely
9 clt for linear spectral statistics
9.1 motivation and strategy
9.2 clt of lss for wigner matrix
9.2.1 strategy of the proof
9.2.2 truncation and renormalization
9.2.3 mean function of mn
9.2.4 proof of the nonrandom part of (9.2.13) for j = l, r
9.3 convergence of the process mn - emn
9.3.1 finite-dimensional convergence of mn - emn
9.3.2 limit of s1
9.3.3 completion of proof of (9.2.13) for j = l, r
9.3.4 tightness of the process mn(z) - emn(z)
9.4 computation of the mean and covariance function of g(f)
9.4.1 mean function
9.4.2 covariance function
9.5 application to linear spectral statistics and related results
9.5.1 tchebychev polynomials
9.6 technical lemmas
9.7 clt of lss for sample covariance matrices
9.7.1 truncation
9.8 convergence of stieltjes transforms
9.9 convergence of finite dimensional distributions
9.10 tightness of mn1(z)
9.11 convergence of mn2(z)
9.12 some derivations and calculations
9.12.1 verification of (9.8.8)
9.12.2 verification of (9.8.9)
9.12.3 derivation of quantities in example (1.1)
9.12.4 verification of quantities in jonsson's results
9.12.5 verification of (9.7.8) and (9.7.9)
10 circular law
10.1 the problem and difficulty.
10.1.1 failure of techniques dealing with hermitian matrices
10.1.2 revisit of stieltjes transformation
10.2 a theorem establishing a partial answer to the circular law
10.3 lemmas on integral range reduction
10.4 characterization of the circular law
10.5 a rough rate on the convergence of vn,(x, z)
10.5.1 truncation and centralization
10.5.2 a convergence rate of the stieltjes transform of
10.6 proofs of (10.2.3) and (10.2.4)
10.7 proof of theorem 10.3
10.8 comments and extensions
10.8.1 relaxation of conditions assumed in theorem 10.3
10.9 some elementary mathematics
11 appendix a. some results in linear algebra
11.1 inverse matrices and resolvent
11.1.1 inverse matrix formula
11.1.2 holing a matrix
11.1.3 trace of inverse matrix
11.1.4 difference of traces of a matrix a and its major submatrices
11.1.5 inverse matrix of complex matrices
11.2 inequalities involving spectral distributions
11.2.1 singular value inequalities
11.3 hadamard product and odot product
11.4 extensions of singular value inequalities
11.4.1 definitions and properties
11.4.2 graph-associated multiple matrices
11.4.3 fundamental theorem on graph-associated mm
11.5 perturbation inequalities
11.6 rank inequalities
11.7 a norm inequality
12 appendix b. moment convergence theorem and stieltjes transform
12.1 moment convergence theorem
12.2 stieltjes transform
12.2.1 preliminary properties
12.2.2 inequalities of distance between distributions in terms of their stieltjes transforms
12.2.3 lemmas concerning levy distance
references
index...
Spectral analysis of large dimensional random matrices = 大维随机矩阵的谱分析 /
- 名称
- 类型
- 大小
光盘服务联系方式: 020-38250260 客服QQ：4006604884
云图客服：
用户发送的提问，这种方式就需要有位在线客服来回答用户的问题，这种 就属于对话式的,问题是这种提问是否需要用户登录才能提问
Video Player
×
Audio Player
×
pdf Player
×
