sample covariance matrices Y = ( Y 1 , . . . , Y N ) M × N matrix Y = ( Y ij ) 1 ≤ i ≤ M , 1 ≤ j ≤ N Y ij independent identically distributed (real or complex) E ( Y 2 E ( Y ij ) = 0 , ij ) = 1 R M Wishart model : Y j standard Gaussian in numerous extensions
sample covariance matrices M × N Y = ( Y 1 , . . . , Y N ) matrix E ( Y ij ) = 0 , E ( Y 2 Y = ( Y ij ) 1 ≤ i ≤ M , 1 ≤ j ≤ N iid ij ) = 1
sample covariance matrices M × N Y = ( Y 1 , . . . , Y N ) matrix E ( Y ij ) = 0 , E ( Y 2 Y = ( Y ij ) 1 ≤ i ≤ M , 1 ≤ j ≤ N iid ij ) = 1 0 ≤ λ N 1 ≤ · · · ≤ λ N center of interest : eigenvalues M Y Y t ( M × M of non-negative symmetric matrix)
sample covariance matrices M × N Y = ( Y 1 , . . . , Y N ) matrix E ( Y ij ) = 0 , E ( Y 2 Y = ( Y ij ) 1 ≤ i ≤ M , 1 ≤ j ≤ N iid ij ) = 1 0 ≤ λ N 1 ≤ · · · ≤ λ N center of interest : eigenvalues M Y Y t ( M × M of non-negative symmetric matrix) � λ N singular values of Y k
sample covariance matrices M × N Y = ( Y 1 , . . . , Y N ) matrix E ( Y ij ) = 0 , E ( Y 2 Y = ( Y ij ) 1 ≤ i ≤ M , 1 ≤ j ≤ N iid ij ) = 1 0 ≤ λ N 1 ≤ · · · ≤ λ N center of interest : eigenvalues M Y Y t ( M × M of non-negative symmetric matrix) � λ N singular values of Y k k = λ N 1 � λ N k N Y Y t eigenvalues of N
sample covariance matrices M × N Y = ( Y 1 , . . . , Y N ) matrix E ( Y ij ) = 0 , E ( Y 2 Y = ( Y ij ) 1 ≤ i ≤ M , 1 ≤ j ≤ N iid ij ) = 1 0 ≤ λ N 1 ≤ · · · ≤ λ N center of interest : eigenvalues M Y Y t ( M × M of non-negative symmetric matrix) � λ N singular values of Y k k = λ N 1 � λ N k N Y Y t eigenvalues of N M � 1 spectral measure δ � λ N M k k =1
sample covariance matrices M × N Y = ( Y 1 , . . . , Y N ) matrix E ( Y ij ) = 0 , E ( Y 2 Y = ( Y ij ) 1 ≤ i ≤ M , 1 ≤ j ≤ N iid ij ) = 1 0 ≤ λ N 1 ≤ · · · ≤ λ N center of interest : eigenvalues M Y Y t ( M × M of non-negative symmetric matrix) � λ N singular values of Y k k = λ N 1 � λ N k N Y Y t eigenvalues of N M � 1 spectral measure δ � λ N M k k =1 asymptotics M = M ( N ) ∼ ρ N N → ∞
Marchenko-Pastur theorem (1967) N ( � k = λ N asymptotic behavior of the spectral measure λ k / N )
Marchenko-Pastur theorem (1967) N ( � k = λ N asymptotic behavior of the spectral measure λ k / N ) M � 1 δ � → ν Marchenko-Pastur distribution λ N M k k =1
Marchenko-Pastur theorem (1967) N ( � k = λ N asymptotic behavior of the spectral measure λ k / N ) M � 1 δ � → ν Marchenko-Pastur distribution λ N M k k =1 � � � 1 − 1 1 d ν ( x ) = + δ 0 + ( b − x )( x − a ) 1 [ a , b ] dx ρ ρ 2 π x
Marchenko-Pastur theorem (1967) N ( � k = λ N asymptotic behavior of the spectral measure λ k / N ) M � 1 δ � → ν Marchenko-Pastur distribution λ N M k k =1 � � � 1 − 1 1 d ν ( x ) = + δ 0 + ( b − x )( x − a ) 1 [ a , b ] dx ρ ρ 2 π x � � 2 � � 2 1 − √ ρ 1 + √ ρ a = a ( ρ ) = b = b ( ρ ) =
Marchenko-Pastur theorem (1967) N ( � k = λ N asymptotic behavior of the spectral measure λ k / N ) M � 1 δ � → ν Marchenko-Pastur distribution λ N M k k =1 � � � 1 − 1 1 d ν ( x ) = + δ 0 + ( b − x )( x − a ) 1 [ a , b ] dx ρ ρ 2 π x � � 2 � � 2 1 − √ ρ 1 + √ ρ a = a ( ρ ) = b = b ( ρ ) =
Marchenko-Pastur theorem M � � � 1 δ � → ν on a ( ρ ) , b ( ρ ) M ∼ ρ N λ N M k k =1 global regime
Marchenko-Pastur theorem M � � � 1 δ � → ν on a ( ρ ) , b ( ρ ) M ∼ ρ N λ N M k k =1 global regime large deviation asymptotics of the spectral measure
Marchenko-Pastur theorem M � � � 1 δ � → ν on a ( ρ ) , b ( ρ ) M ∼ ρ N λ N M k k =1 global regime large deviation asymptotics of the spectral measure fluctuations of the spectral measure
Marchenko-Pastur theorem M � � � 1 δ � → ν on a ( ρ ) , b ( ρ ) M ∼ ρ N λ N M k k =1 global regime large deviation asymptotics of the spectral measure fluctuations of the spectral measure M � � �� � � � λ N − R f d ν → G Gaussian variable f k k =1 f : R → R smooth
Marchenko-Pastur theorem � M � � 1 → ν M ∼ ρ N δ � on a ( ρ ) , b ( ρ ) λ N M k k =1 local regime
Marchenko-Pastur theorem � M � � 1 → ν M ∼ ρ N δ � on a ( ρ ) , b ( ρ ) λ N M k k =1 local regime behavior of the individual eigenvalues
Marchenko-Pastur theorem � M � � 1 → ν M ∼ ρ N δ � on a ( ρ ) , b ( ρ ) λ N M k k =1 local regime behavior of the individual eigenvalues spacings (bulk behavior)
Marchenko-Pastur theorem � M � � 1 → ν M ∼ ρ N δ � on a ( ρ ) , b ( ρ ) λ N M k k =1 local regime behavior of the individual eigenvalues spacings (bulk behavior) extremal eigenvalues (edge behavior)
extremal eigenvalues λ N M = max 1 ≤ k ≤ M λ N largest eigenvalue k
extremal eigenvalues λ N M = max 1 ≤ k ≤ M λ N largest eigenvalue k M = λ N � λ N M N
extremal eigenvalues λ N M = max 1 ≤ k ≤ M λ N largest eigenvalue k M = λ N � � 2 1 + √ ρ � λ N M → b ( ρ ) = M ∼ ρ N N
Marchenko-Pastur theorem (1967) N ( � k = λ N asymptotic behavior of the spectral measure λ k / N ) M � 1 δ � → ν Marchenko-Pastur distribution λ N M k k =1 � � � 1 − 1 1 d ν ( x ) = + δ 0 + ( b − x )( x − a ) 1 [ a , b ] dx ρ ρ 2 π x � � 2 � � 2 1 − √ ρ 1 + √ ρ a = a ( ρ ) = b = b ( ρ ) =
extremal eigenvalues λ N M = max 1 ≤ k ≤ M λ N largest eigenvalue k M = λ N � � 2 1 + √ ρ � λ N M → b ( ρ ) = M ∼ ρ N N
extremal eigenvalues λ N M = max 1 ≤ k ≤ M λ N largest eigenvalue k M = λ N � � 2 1 + √ ρ � λ N M → b ( ρ ) = M ∼ ρ N N fluctuations around b ( ρ )
extremal eigenvalues λ N M = max 1 ≤ k ≤ M λ N largest eigenvalue k M = λ N � � 2 1 + √ ρ � λ N M → b ( ρ ) = M ∼ ρ N N fluctuations around b ( ρ ) complex or real Gaussian (Wishart matrices)
extremal eigenvalues λ N M = max 1 ≤ k ≤ M λ N largest eigenvalue k M = λ N � � 2 1 + √ ρ � λ N M → b ( ρ ) = M ∼ ρ N N fluctuations around b ( ρ ) complex or real Gaussian (Wishart matrices) M 2 / 3 � � � λ N M − b ( ρ ) → C ( ρ ) F TW
extremal eigenvalues λ N M = max 1 ≤ k ≤ M λ N largest eigenvalue k M = λ N � � 2 1 + √ ρ � λ N M → b ( ρ ) = M ∼ ρ N N fluctuations around b ( ρ ) complex or real Gaussian (Wishart matrices) M 2 / 3 N − 1 � � λ N M − b ( ρ ) N → C ( ρ ) F TW
extremal eigenvalues λ N M = max 1 ≤ k ≤ M λ N largest eigenvalue k M = λ N � � 2 1 + √ ρ � λ N M → b ( ρ ) = M ∼ ρ N N fluctuations around b ( ρ ) complex or real Gaussian (Wishart matrices) M 2 / 3 N − 1 � � λ N M − b ( ρ ) N → C ( ρ ) F TW C. Tracy, H. Widom (1994) distribution F TW
extremal eigenvalues λ N M = max 1 ≤ k ≤ M λ N largest eigenvalue k M = λ N � � 2 1 + √ ρ � λ N M → b ( ρ ) = M ∼ ρ N N fluctuations around b ( ρ ) complex or real Gaussian (Wishart matrices) M 2 / 3 N − 1 � � λ N M − b ( ρ ) N → C ( ρ ) F TW C. Tracy, H. Widom (1994) distribution F TW K. Johansson (2000), I. Johnstone (2001)
F TW C. Tracy, H. Widom (1994) distribution � � ∞ � ( x − s ) u ( x ) 2 dx (complex) F TW ( s ) = exp − , s ∈ R s u ′′ = 2 u 3 + xu Painlev´ e II equation
F TW C. Tracy, H. Widom (1994) distribution � � ∞ � ( x − s ) u ( x ) 2 dx (complex) F TW ( s ) = exp − , s ∈ R s u ′′ = 2 u 3 + xu Painlev´ e II equation density
mean ≃ − 1 . 77 F TW ( s ) ∼ e − s 3 / 12 as s → −∞ 1 − F TW ( s ) ∼ e − 4 s 3 / 2 / 3 as s → + ∞ density (similar for real case)
extremal eigenvalues λ N M = max 1 ≤ k ≤ M λ N largest eigenvalue k M = λ N � � 2 1 + √ ρ λ N � M → b ( ρ ) = M ∼ ρ N N fluctuations around b ( ρ ) complex or real Gaussian (Wishart matrices) M 2 / 3 � � � λ N M − b ( ρ ) → C ( ρ ) F TW F TW C. Tracy, H. Widom (1994) distribution K. Johansson (2000), I. Johnstone (2001)
Gaussian (Wishart matrices)
Gaussian (Wishart matrices) completely solvable models
Gaussian (Wishart matrices) completely solvable models determinantal structure orthogonal polynomial analysis
Gaussian (Wishart matrices) completely solvable models determinantal structure orthogonal polynomial analysis asymptotics of Laguerre orthogonal polynomials
Gaussian (Wishart matrices) completely solvable models determinantal structure orthogonal polynomial analysis asymptotics of Laguerre orthogonal polynomials C. Tracy, H. Widom (1994) K. Johansson (2000), I. Johnstone (2001)
extension to non-Gaussian matrices
extension to non-Gaussian matrices A. Soshnikov (2001-02) � � ( YY t ) p �� moment method E Tr
extension to non-Gaussian matrices A. Soshnikov (2001-02) � � ( YY t ) p �� moment method E Tr L. Erd¨ os, H.-T. Yau (2009-12) (and collaborators) local Marchenko-Pastur law T. Tao, V. Vu (2010-11) Lindeberg comparison method
extension to non-Gaussian matrices A. Soshnikov (2001-02) � � ( YY t ) p �� moment method E Tr L. Erd¨ os, H.-T. Yau (2009-12) (and collaborators) local Marchenko-Pastur law T. Tao, V. Vu (2010-11) Lindeberg comparison method symmetric matrices
(brief) survey of recent approaches to non-asymptotic exponential inequalities
(brief) survey of recent approaches to non-asymptotic exponential inequalities quantify the limit theorems
(brief) survey of recent approaches to non-asymptotic exponential inequalities quantify the limit theorems spectral measure
(brief) survey of recent approaches to non-asymptotic exponential inequalities quantify the limit theorems spectral measure extremal eigenvalues ( mean ) 1 / 3 catch the new rate
(brief) survey of recent approaches to non-asymptotic exponential inequalities quantify the limit theorems spectral measure extremal eigenvalues ( mean ) 1 / 3 catch the new rate from the Gaussian case to non-Gaussian models
two main questions and objectives
two main questions and objectives tail inequalities for the spectral measure � M � � f ( � λ N k ) ≥ t P k =1
Marchenko-Pastur theorem M � � � 1 δ � → ν on a ( ρ ) , b ( ρ ) M ∼ ρ N λ N M k k =1 global regime large deviation asymptotics of the spectral measure fluctuations of the spectral measure M � � �� � � � λ N − R f d ν → G Gaussian variable f k k =1 f : R → R smooth
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