representation of type (7.5) and (7.6) in random matrix theory was used for the first time by Pastur for investigation of the Stieltjes transform of random matrices.
LIMIT THEOREMS FOR SPECTRA OF RANDOM MATRICES WITH MARTINGALE STRUCTURE 1,3 , A.TIKHOMIROV1,2,3 ¨ F. GOTZE
University of Bielefeld1 Syktyvkar State University and the Mathematical Department of IMM of the Ural Branch of Russian Academy of Sciences2
Abstract. We study classical ensembles of real symmetric random matrices introduced by Eugene Wigner. We discuss Stein’s method for the asymptotic approximation of expectations of functions of the normalized eigenvalue counting measure of high dimensional matrices. The method is based on a differential equation for the density of the semi-circular law.
1. Introduction Let Xjk , 1 ≤ j ≤ k < ∞, be triangular array of random variables with EXjk = 0 2 2 and EXjk = σjk , and let Xkj = Xjk , for 1 ≤ j < k < ∞. For a fixed n ≥ 1, denote by λ1 ≤ . . . ≤ λn the eigenvalues of the symmetric n × n matrix 1 (1.1) Wn = (Wn (jk))nj,k=1 , Wn (jk) = √ Xjk , for 1 ≤ j ≤ k ≤ n, n and define its empirical spectral distribution function by n 1X (1.2) Fn (x) = I{λj ≤x} , n j=1 where I{B} denotes the indicator of an event B. We investigate the convergence of the expected spectral distribution function EFn (x) to the distribution function of Wigner’s semi-circular law. Let g(x) and G(x) denote the density and the distribution function of the standard semi-circular law, that is Z x 1 p 4 − x2 I{|x|≤2} , G(x) = g(u)du. (1.3) g(x) = 2π −∞ The goal of this paper is to illustrate the possibilities of Stein’s method for the investigation of Stieltjes transform of empirical spectral distribution function of random matrices and some of its applications. A similar approach based on the representation of type (7.5) and (7.6) in random matrix theory was used for the first time by Pastur for investigation of the Stieltjes transform of random matrices of the Wigner ensemble (see, for instance [12], [7], [10]). Date: December, 10, 2003. 1 Research supported by the DFG- Forschergruppe FOR 399/1-1 at Bielefeld. 2 Partially supported by Russian Foundation for Fundamental Research. Grant N 02-01-00233, 3 Partially supported by INTAS grant N 03-51-5018. 1
2
F. G¨ otze, A.Tikhomirov
We state some general conditions of the convergence of the expected distribution function of random matrices to the semi circular law (of not necessarily independent entries) and give some applications of the main result. In Section 2 we derive Stein’s type equation for the semi circular law and provide on this basis some criteria of convergence to the semi-circular law. 2 2 We shall assume that EXjl = 0 and σjl := EXjl , for 1 ≤ j ≤ l ≤ n. Introduce jl σ-algebras F = σ{Xkm : 1 ≤ k ≤ m ≤ n, {k, m} = 6 {j, l}}, 1 ≤ j ≤ l ≤ n, and F j = σ{Xkm : 1 ≤ k ≤ m ≤ n, k 6= j and m 6= j}, 1 ≤ j ≤ n. We introduce as well Lindeberg’s ratio for random matrices, that is for any τ > 0, (1.4)
Ln (τ ) =
n 1 X 2 EXjl I{|Xjl |>τ √n} . n2 j,l=1
Theorem 1.1. Assume that the random variables Xjl , 1 ≤ j ≤ l ≤ n, n ≥ 1 satisfy the following conditions E{Xjl |F jl } = 0,
(1.5) (1.6)
ε(1) n :=
1 n2
X
2 2 E|E{Xjl |F j } − σjl |→0
as
n → ∞,
1≤j≤l≤n
(1.7) there exists σ 2 > 0, such that ε(2) n :=
1 n2
X
2 |σjl − σ2 | → 0
as
n → ∞,
1≤j≤l≤n
and (1.8)
for any fixed τ > 0,
Ln (τ ) → 0
as
n → ∞.
Then (1.9)
∆n := sup |EFn (x) − G(xσ −1 )| → 0
as
n → ∞.
x
By kWk we shall denote the Frobenius norm of a matrix W (1.10)
kWk2 =
n X
n X
|λj |2 =
j=1
|W (j, k)|2 .
j,k=1
Remark 1.2. Note that condition (1.7) implies that (1.11)
lim
n→∞
1 EkWn k2 = σ 2 < ∞. n
2 Corollary 1.3. Let Xlj , 1 ≤ l ≤ j < ∞ be independent and EXlj = 0, EXlj = σ2 . Assume that, for any fixed τ > 0,
Ln (τ ) → 0,
(1.12)
as
n → ∞.
Then the expected spectral distribution function of matrix W convergence to the distribution function of the semi-circular law, also (1.13)
∆n := sup |EFn (x) − G(xσ −1 )| → 0, x
as
n → ∞.
Spectra of random matrices with martingale structure
3
The investigation of the convergence the spectral distribution functions of real symmetric random matrices with independent entries has a long history. For details see for example [1]. Convergence assuming Lindeberg’s condition (Corollary 1.3) was proved in Pastur [13]. An other application of Theorem 1.1 is the distribution of spectra for the unitary (n) invariant ensemble of real symmetric n × n matrices Wn = (Xlj ) induced by the √ uniform distribution on the sphere of the radius N in RN with N = n(n+1) , that 2 is X (n) (1.14) (Xlj )2 = N. 1≤l≤j≤n
Rosenzweig in [14] has defined this class of random matrices as a fixed trace ensemble and proved the semi-circular law using Wigner’s method of moments. This class is described in Mehta [9], Ch. 19, as well. (n)
Corollary 1.4. Let Xlj , 1 ≤ l ≤ j ≤ n be distributed as above, for any n ≥ 1. Then (1.15)
∆n := sup |Fn (x) − G(x)| → 0 as n → ∞. x
We may consider the ensemble of real symmetric √ n × n matrices determined by the uniform distribution in the ball of the radius N in RN with N = n(n+1) , that 2 means that X (n) (1.16) (Xlj )2 ≤ N. 1≤l≤j≤n
This class of random matrices was introduced by Bronk in [2] as a bounded trace ensemble. The eigenvalue density for the bounded trace ensemble is identical to the density of zeros of Hermite-like polynomials. Using this fact, Bronk proved the semi-circular law for such matrices. See also [9], Ch. 19. (n)
Corollary 1.5. Let Xlj , 1 ≤ l ≤ j ≤ n be distributed uniformly in the ball of the √ , for any n ≥ 1. Then radius N in RN with N = n(n+1) 2 (1.17)
∆n → 0,
as
n → ∞.
2. Proofs of the Corollaries Proof of Corollary 1.3. It is easy to check that all conditions of Theorem 1.1 hold. Proof of Corollary 1.4. To prove the corollary we have to examine the conditions (1.5), (1.6), (1.7) and (1.8) of Theorem 1.1 only. Since all one-dimensional (n) distributions are symmetric and the random variables Xlj , 1 ≤ l ≤ j ≤ n, are ex2 changeable, the condition (1.7) holds with σ = 1. It is easy to see that conditional distribution of Xlj given Flj is symmetric. This yields condition (1.5). To check condition (1.6) we note that, since Xlj are exchangeable, X 1 2 2 (2.1) ε(1) E E{Xjl | F j } − σ 2 = E E{X11 | F (1) } − 1 . n := 2 n 1≤j≤l≤n
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F. G¨ otze, A.Tikhomirov
Furthermore, using equality (1.16), we get (2.2)
n n o X 2 2 (1) E{X1l | F (1) } − E X11 |F =N− l=2
X
2 Xlj .
2≤l≤j≤n
This implies that n X
(2.3)
2 E{X1l | F (1) } =
n X
l=1
2 X1l .
l=1
Since X1l , l = 1, . . . , n are conditionally exchangeable given F (1) , we have n
2 E{X1l |F (1) } =
(2.4)
1X 2 X1l . n l=1
Applying Cauchy’s inequality, we get n o 1 1 2 E E X1l | F (1) − 1 ≤ E 2 n
(2.5)
n 2 X 2 (X1l − 1) . l=1
Direct calculations show that, for l 6= j, 2 2 E(X1l − 1)(X1j − 1) =
(2.6)
2 . N
The last relation yields C 2 E E{X1l | F (1) } − 1 ≤ √ . n
(2.7)
The Lindeberg condition (1.8) follows from the symmetry of distribution and boundness of random variables. This concludes the proof. Proof of Corollary 1.5. We have to check the conditions of Theorem 1.1. The conditions (1.5), (1.7) and (1.8) hold by the same reasons as in Corollary 1.4. We now check condition (1.6). Since X11 , . . . , X1n are conditionally exchangeable distributed given F (1) , we get ( n ) n o X 1 2 (1) 2 (1) (2.8) E X1l | F = E X1l |F , n l=1
This implies that (2.9)
n n 1 X o 2 2 2 2 E E X11 | F (1) − EX11 (X1l − EX11 ) . ≤ E n l=1
A simple calculation shows that (2.10)
2 2 2 2 EX11 X12 − EX11 EX12 =−
2N 2 . (N + 4)(N + 2)2
This implies that (2.11)
C 2 2 E|E{X11 |F (1) } − EX11 |≤ √ , n
which completes the proof.
Spectra of random matrices with martingale structure
5
3. Stein’s equation for the semi-circular law We start from a simple characterization of the semi-circular law. Introduce a class of bounded functions without discontinuity of second order C1{−2,2} = {f : R → R : f ∈ C1 (R \ {−2, 2}); lim|y|→∞ |yf (y)| < ∞; lim sup |4 − y 2 ||f 0 (y)| < C}. y→±2
By C(R) we denote a class of continuous functions on R, by C1 (B), B ⊂ R, we denote a class of all functions f : R → R differentiable on B . At first we prove the following Lemma 3.1. Assume that a bounded function ϕ(x) without discontinuity of second order satisfies the following conditions ϕ(x) is continuity in the points x = ±2
(3.1) and
Z
2
(3.2)
p ϕ(u) 4 − u2 du = 0.
−2
Then there exists a function f ∈ C1{−2,2} such that, for any x 6= ±2, (4 − x2 )f 0 (x) − 3xf (x) = ϕ(x)
(3.3)
If ϕ(±2) = 0 then there exists a continuity solution of the equation (3.3). Proof. We define the following function f (x). Let x < −2. Then Z x Z x Z u 3udu 3vdv ϕ(u)du (3.4) f (x) := − exp − exp . 2−4 2−4 u v u2 − 4 −2 −2 −2 After integrating we get f (x) := −
(3.5)
Z
1
x
3 2
p ϕ(u) u2 − 4 du.
(x2 − 4) −2 Analogously, for x > 2 we define Z x p 1 ϕ(u) u2 − 4 du. (3.6) f (x) := − 3 (x2 − 4) 2 2 For x ∈ (−2, 2), (3.7)
f (x) :=
Z
1 |x2 − 4|
3 2
x
p ϕ(u) 4 − u2 du.
−2
It is straightforward to show that the above function f (x) satisfies the equation (3.3). We calculate now (3.8)
a := lim f (x), x&2
b := lim f (x), x%−2
c := lim f (x), x%2
d := lim f (x) x&−2
According to de l’Hˆ opital’s rule, we get 1 1 (3.9) a = −c = − ϕ(2) b = −d = ϕ(−2). 6 6 The function f (x) has springs in the point x = ±2 if ϕ(±2) 6= 0 only. The representation (3.5)–(3.7) together imply that lim supy→∞ |yf (y)| < ∞ and lim supy→±2 |(y 2 − 4)f 0 (y)| < ∞. This concludes Lemma.
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F. G¨ otze, A.Tikhomirov
Proposition 3.2. The random variable ξ has distribution function G(x) if and only if the following equality holds, for any function f ∈ C1{−2,2} , E(4 − ξ 2 )f 0 (ξ) − 3Eξf (ξ) = 0.
(3.10)
Proof. At first we proof the necessity. Assume that ξ has distribution function 1 G(x). By definition of class C1{−2,2} , the function (4 − y 2 )g(y)f 0 (y) = 2π (4 − 3
y 2 ) 2 f 0 (y)I[−2,2] (y) is continuous and E|(4 − ξ 2 )f 0 (ξ)| < ∞. Integrating by parts, we get Z 2 2 0 (3.11) E(4 − ξ )f (ξ) = (4 − y 2 )g(y)f 0 (y)dy −2
Z
2
=−
f (y)[−2yg(y) + (4 − y 2 )g 0 (y)]dy.
−2
Using definition (1.3), it is easy to check that, for y ∈ [−2, 2], (4 − y 2 )g 0 (y) = −yg(y).
(3.12)
Substituting (3.12) in (3.11) we obtain (3.10). Now we prove the sufficiency of condition (3.10). According to lemma 3.1, for any fixed number x, there exists a function f (x) (y) ∈ C1{−2,2} such that following equation holds (3.13)
(4 − y 2 )(f (x) (y))0y − 3yf (x) (y) = hx (y) − G(x).
Evaluating the expectation of (4 − ξ 2 )(f (x) (y))0x (ξ) − 3ξf (x) (ξ), we get that, for any x, P{ξ ≤ x} = G(x),
(3.14) which proves the proposition.
3.1. A Stein equation for random matrices. Let W denote a symmetric random matrix with eigenvalues λ1 ≤ λ2 ≤ · · · ≤ λn . If W = U−1 ΛU, where U is an orthogonal matrix and Λ is a diagonal matrix, one defines f (W) = U−1 f (Λ)U, where f (Λ) = diag(f (λ1 ), . . . , f (λn )). We can now formulate the convergence to the semi circular law for the spectral distribution function of random matrices. Theorem 3.3. Let Wn denote a sequence of random matrices of order n × n such that, for any function f ∈ C1{−2,2} . (3.15)
1 3 ETr(4In − Wn2 )f 0 (Wn ) − ETrWn f (Wn ) → 0, n n
as n → ∞.
Then (3.16)
∆n := sup |EFn (x) − G(x)| → 0, as n → ∞. x 0
Proof. For any function f ∈ C{−2,2} , introduce the operator 1 3 ETr(4In − Wn2 )f 0 (Wn ) − ETrWn f (Wn ). n n There exists, for any x, a solution of equation (3.13) f (x) ∈ C0{−2,2} such that (3.17)
(3.18)
Ln (f ) :=
Ln (f (x) ) = Gn (x) − G(x),
Spectra of random matrices with martingale structure
7
where Gn (x) := EFn (x). To prove (3.18) we introduce a random variable J,which is uniformly distributed on {1, . . . , n} and which is independent on Xlj , 1 ≤ l ≤ j ≤ n. Note that n
(3.19)
P{λJ ≤ x} =
n
1X 1X P{λj ≤ x} = EI{λj ≤x} = EFn (x). n j=1 n j=1
Consider equality (3.13) for y = λJ (3.20)
(4 − λ2J )(f (x) (λJ ))0y − 3λJ f (x) (λJ ) = hx (λJ ) − G(x).
Taking mathematical expectations of the both parts of the equality (3.20), we obtain (3.18). Relations (3.15), (3.18) together complete the proof. For any L > 0, consider the class of function C1{−2,2,L} := {f ∈ C1{−2,2} : f (y) = 0, for y ∈ / [−L, L]}. Corollary 3.4. Assume that, for some positive constant C, (3.21)
lim sup n→∞
1 EkWn k2 ≤ C. n
Assume that, for any L > 0 and for any f ∈ C1{−2,2,L} , Ln (f ) → 0, as n → ∞.
(3.22) Then
∆n = sup |EFn (x) − G(x)| → 0, as n → ∞.
(3.23)
x
Proof. Let L > 0 be fix. Consider the function fex (y) = f (x) (y)I{[−L,L]} (y). According to (3.13) we have n
(3.24)
Ln (fex ) =
1X E(I{λj ≤x} − G(x))I{|λj |≤L} n j=1 n
= Gn (x) − G(x) −
1X E(I{λj ≤x} − G(x))I{|λj |≥L} . n j=1
This equality implies that n
(3.25)
lim sup |Gn (x) − G(x)| ≤ lim sup n→∞
n→∞
1X EI{|λj |≥L} n j=1
n
≤
1 1X 1 1 lim sup E|λj |2 ≤ 2 lim sup EkWn k2 . L2 n→∞ n j=1 L n→∞ n
Since the last limit on the right hand side of (3.25) is finite and L is arbitrary, we get (3.26)
lim sup |Gn (x) − G(x)| = 0. n→∞
This concludes the proof.
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F. G¨ otze, A.Tikhomirov
Let Wn = U0n Λn Un where Λn = diag(λ1 , . . . , λn ). We introduce the trun(L) (L) (L) (L) (L) (L) cated matrix Wn := U0n Λn Un , where Λn = diag(λ1 , . . . , λn ) and λj = λj I{|λj |≤L} . The expected empirical spectral distribution function of the matrix (L)
Wn
is defined by n
EFn(L) (x)
(3.27)
1X EI (L) . := n j=1 {λj ≤x}
It is easy to see that n
(3.28)
|EFn (x) − EFn(L) (x)| ≤
1X 1 1 EI{|λj |≥L} ≤ 2 EkWn k2 . n j=1 L n
4. The Stieltjes transform Introduce the Stieltjes transform of a random variable ξ with distribution function F (x), for any z = u + iv, v 6= 0, Z ∞ 1 1 = dF (x). (4.1) T (z) = E ξ−z x − z −∞ In random matrix theory the Stieltjes transform of random variables was used for the first time by Marchenko, Pastur [8]. Note that T (z) is analytic for non-real z and satisfies the conditions (4.2)
Im T · Im z > 0,
Im z 6= 0,
sup y|T (iy)| = 1. y≥1
It can be shown that for any continuous function ϕ(λ) with compact support Z ∞ Z 1 ∞ ϕ(λ)Im T (λ + iv)dλ (4.3) ϕ(λ)dF (λ) = lim v→0 π −∞ −∞ Furthermore the one-to-one correspondence between distribution functions and their Stieltjes transforms is continuous with respect to the weak convergence of distribution functions as well as with respect to the uniform convergence on compacts sets in C\R . 4.1. The Stieltjes transform of a semi-circular law. Introduce the function 1 1 , for any non-real z. Note that (fz (x))0x = −(fz (x))0z = − (x−z) fz (x) = x−z 2. Denote by S(z) the Stiltjes transform of the semi-circular law. By definition of the Stieltjes transform, we have Z ∞ 1 1 (4.4) S(z) = dG(x) = Efz (ξ), S 0 (z) = −E(fz (ξ))0x = E , (ξ − z)2 −∞ x − z where ξ is a random variables with the distribution function G(x). Applying now equation (3.10), we obtain (4.5)
E
4 − ξ2 ξ + 3E = 0. 2 (ξ − z) ξ−z
Combining equations (4.5), (4.4) and the obvious relations (4.6)
4 − ξ2 4 − z2 2z = −1− , (ξ − z)2 (ξ − z)2 ξ−z
Spectra of random matrices with martingale structure
9
and ξ z =1+ , ξ−z ξ−z
(4.7)
we get a differential equation for the Stieltjes transform of the semi-circular law (z 2 − 4)S 0 (z) − zS(z) − 2 = 0.
(4.8)
Solving this differential equation we get, for z and z0 such that Im z · Im z0 > 0, 1
(4.9)
(z 2 − 4) 2
S(z) =
1
(z02 − 4) 2
1
S(z0 ) + 2(z02 − 4) 2
Z
z
du 3
z0
(u2 − 4) 2
,
√ √ √ where u2 − 4 = u − 2 u + 2 (principal argument). Since S(z) = S(z), it is sufficient to calculate this integral for Im z > 0 only. Putting u = (ζ + ζ −1 ) and √ 1 ζ(u) = 2 (u + u2 − 4), we obtain Z
z
(4.10)
Z
du 3
z0
(u2 − 4) 2
ζ(z)
= ζ(z0 )
ζdζ 1 1 1 = − . (ζ 2 − 1)2 2 (ζ 2 (z0 ) − 1) (ζ 2 (z) − 1)
It is easy to check that (ζ 2 (u) − 1)−1 =
(4.11)
√ u − u2 − 4 √ . 2 u2 − 4
Substituting (4.11) in (4.10), we obtain 1
(z 2 − 4) 2
(4.12)
S(z) =
Since Im
p z02 − 4 > 0,
(4.13)
1
(z02 − 4) 2
1 |z0 − 2
q
1 S(z0 ) + 2
z02 − 4| =
z0 −
q
z02
p 1 −4 − z − z2 − 4 . 2
2 2 p p ≤ ≤ 2v0−1 . 2 2 |z0 + z0 − 4| Im {z0 + z0 − 4}
According to (4.2), for z0 = u0 + iv0 , we have (4.14)
|S(z0 )| ≤ v0−1 .
Passing to the limit in equality (4.12) in z0 = u0 + iv0 as v0 → ∞, we get the following formula for the Stieltjes transform of the semi-circular law, for all z such that Im z > 0, (4.15)
S(z) = −
p 1 z − z2 − 4 2
This formula implies a simple algebraic equation for the Stieltjes transform of the semi-circular law, namely (4.16)
S 2 (z) + zS(z) + 1 = 0.
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F. G¨ otze, A.Tikhomirov
5. The Stieltjes transform of the spectral distribution function of a random matrix In the what follows we shall omit the sub-index n in the notation of matrices. Recall that the spectral distribution function of a random symmetric matrix W is defined as n 1X (5.1) Fn (x) = I{λj ≤x} , n j=1 where λ1 , . . . , λn are the eigenvalues of a n × n real symmetric matrix W. 5.1. The Stieltjes transform of the spectral distribution function of random matrix. We introduce the resolvent matrix for a symmetric matrix W , for any non-real z, (5.2)
R(z) = (W − zI)−1 ,
where I denote the identity matrix of order n × n. Introduce the Stieltjes transform of the spectral distribution function as Z ∞ n 1X 1 1 1 dFn (x) = = TrR(z). (5.3) Mn (z) := x − z n λ − z n j −∞ j=1 We shall consider also the Stieltjes transform of expected spectral distribution function Z ∞ 1 1 dEFn (x) = ETrR(z). (5.4) Sn (z) := EMn (z) = n −∞ x − z 5.1.1. Estimation of the variance of the Stieltjes transforms of spectral distribution functions of random matrices. In this section we give a general bound for the variance of Mn (z) without restrictions on the moments of matrix entries and without assumption on independence. Let 1 (5.5) Vn2 = E|Mn (z) − Sn (z)|2 = 2 E|TrR(z) − ETrR(z)|2 . n To bound the last quantity we repeat the martingale decomposition of TrR − ETrR developed in [3], p. 9. Let Ek denote the conditional expectation given a σ-algebra Fk = σ{Xij , k+1 ≤ i ≤ j ≤ n}. Introduce the (n−1)×(n−1) matrix W(k) obtained from W by deleting the kth row and column. Set R(k) = (W(k) − zIn−1 )−1 . Let (5.6)
γk = Ek−1 TrR − Ek TrR = Ek−1 κk − Ek κk ,
where (5.7)
κk = TrR − TrR(k) .
The equation (5.6) follows since Ek−1 TrR(k) = Ek TrR(k) . To bound κk we need some auxiliary lemmas. Let A be an n × n symmetric matrix and A(k) denote a principal sub-matrix, the obtained from A by deleting k-th row and column.
Spectra of random matrices with martingale structure
11
Lemma 5.1. The following bound holds, for z = u + iv, v > 0, −1 −1 (k) ≤ v −1 . (5.8) Tr (A − zIn ) − Tr A − zIn−1 Proof. Applying Schur’s complements formula (see [6], Ch. 08, p. 21) with a0k = (Xk1 , . . . , Xkk−1 , Xkk+1 , . . . , Xkn ), we get 1 1 0 (k) −1 (5.9) det(A−zIn ) = √ Xkk − z − ak (A − zIn−1 ) ak det(A(k) −zIn−1 ) n n Taking the logarithm of both parts of equation (5.9) and taking derivatives we obtain Tr(A − zIn )−1 − Tr(A(k) − zIn−1 )−1
(5.10)
=
1 + a0k (A(k) − zIn−1 )−2 ak . akk − z − a0k (A(k) − zIn−1 )−1 ak
Let T be an orthogonal transformation which transforms A into diagonal form. Denote by µ1 ≤ · · · ≤ µn−1 the eigenvalues of Ak and let (y1 , . . . , yn−1 ) = a0k T0 . Then n−1 X 0 (k) −2 2 −2 yl (µl − z) (5.11) |1 + ak (A − zIn−1 ) ak | = 1 + l=1 Xn−1 −1 ≤1+ yl2 (µl − u)2 + v 2 l=1 −1 2 ≤ 1 + a0k A(k) − uIn−1 + v 2 In−1 ak . Since for any commuting matrices A, B such that A2 + B2 is non-degenerate (A + iB)−1 = (A − iB)(A2 + B2 )−1 ,
(5.12)
we can directly verify that (5.13) −1 −1 0 (k) 0 (k) 2 2 Im akk − z − ak A − zIn−1 ak = −v 1 + ak (A − uIn−1 ) + v In−1 ak . The last two relations yield the result.
Applying Lemma 5.1 with A = W and A we get |κk | ≤
(5.14)
(k)
=W
(k)
for symmetric matrices ,
1 . v
This immediately implies that 2 . v Since the martingale differences γk are uncorrelated, for k = 1, . . . , n, and n P TrR − E TrR = γd , we get |γk | ≤
(5.15)
k=1
(5.16)
Vn2 =
1 4 E|TrR − ETrR|2 ≤ . n2 nv 2
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F. G¨ otze, A.Tikhomirov
Proposition 5.2. Assume that, for any v 6= 0, 1 3 (5.17) Rn (W)(z) := ETr(4I − W2 )R2 (z) + ETrWR(z) → 0, as n → ∞ n n uniformly on compacts sets in C\R. Then ∆n → 0, as n → ∞
(5.18)
Proof. Without loss of generality we consider the case Im z > 0 only. In view of relations (4.4)–(4.8) we have Rn (W)(z) = (z 2 − 4)Sn0 (z) − zSn (z) − 2,
(5.19)
where Sn (z) := n1 ETr (W − zI)−1 . Write ηn (z) := Rn (W)(z). Rewriting the representation (5.19) in integral form we get, for any fixed z0 such that Im z0 > 0, Z z 1 1 (2 + ηn (u))du (z 2 − 4) 2 2 2 . (5.20) Sn (z) = 2 Sn (z0 ) + (z0 − 4) 1 3 (u2 − 4) 2 (z0 − 4) 2 z0 Let S(z) denote the Stieltjes transform of the semi-circular law. According to Section 4.1, S(z) satisfies an equality (z 2 − 4)S 0 (z) − zS(z) − 2 = 0
(5.21)
The equations (5.21) and (5.20) together imply that 1
(5.22)
Sn (z) − S(z) =
(z 2 − 4) 2
1
2 2 1 (Sn (z0 ) − S(z0 )) + (z − 4)
Z
(z02 − 4) 2
z
ηn (ζ) 3
z0
(ζ 2 − 4) 2
dζ.
By definition of the Stieltjes transform, we have, for z = x + iv, 1 (5.23) max{|Sn (z)|, |S(z)|} ≤ . v Let ε > 0. Using (5.23), we choose z0 with =z0 > 2ε such that |Sn (z0 ) − S(z0 )| ≤ ε.
(5.24)
Without loss of generality we assume that |z02 − 4| > 1. Let z belong to some compact set K in the upper half plane. It is easy to check that there exists some constant C(K) depending on compact set K only such that, for all n ≥ 1, the first summand in (5.22) satisfies the following inequality (z 2 − 4) 12 (5.25) 2 1 (Sn (z0 ) − S(z0 )) ≤ C(K)ε. (z0 − 4) 2 In order to bound the second term on Rthe right hand side of (5.22) we consider z z such that |z 2 − 4| ≥ 1. Let the integral z0 denote integration along to the any path Γ(z0 , z) from z0 to z in the upper half plane. Without loss of generality we assume that, for all ζ ∈ Γ(z0 , z), the inequality |ζ 2 − 4| > 1 holds. Then there exists a constant C(K) depending on the compact set K only such that Z z 2 1 ηn (ζ) 2 (5.26) ≤ C(K) sup |ηn (ζ)|. 3 (z − 4) (ζ 2 − 4) 2 z0
2
ζ∈Γ(z0 ,z)
If 0 < |z − 4| ≤ 1 we have Z Z z Z z0 z η (ζ) 0 η (ζ) η (ζ) n n n (5.27) 3 ≤ 3 + 3 , 2 2 2 z0 (ζ − 4) 2 z0 (ζ − 4) 2 z00 (ζ − 4) 2
Spectra of random matrices with martingale structure
13 2
where z00 denotes some point in the upper half plane such that |z00 − 4| = 1. It is straightforward to check that there exists some absolute constant C1 such that, for |z 2 − 4| ≤ 1, Z z 1 3 2 2 − (5.28) |z − 4| 2 |ζ − 4| 2 dζ ≤ C1 . z00 The relations (5.21) and (5.28) together imply that Z z 1 ηn (ζ) ≤ C(K) sup |ηn (ζ)| + C1 (5.29) |z 2 − 4| 2 3 0 (ζ 2 − 4) 2 ζ∈Γ(z0 ,z0 )
z0
sup
|ηn (ζ)|.
ζ: |z 2 −4|≤1
Furthermore, the relations (5.23)–(5.29) imply that (5.30)
lim sup |Sn (z) − S(z)| ≤ C(K)ε n→∞
uniformly on z ∈ K. That means that Sn (z) → S(z) as n → ∞ uniformly on the all compact sets in C \ R. Note that for any distribution function F (x) with Stieltjes transform T (z) we have, for z = x + iv, (5.31)
Im{T (z)} = F ∗ Fv (x),
where Fv denotes Cauchy’s distribution function with parameter v and Fv0 (x) = v 1 π x2 +v 2 . This concludes the proof. 6. A truncation of random variables In this Section we prove the following useful statement. Consider some symmetric matrix D of order n × n. Put f = W + √1 D (6.1) W n and −1 e f − zI (6.2) R(z) = W . Define the Stieltjes transform of the expected spectral distribution function of maf by trix W 1 e (6.3) Sen (z) = ETrR(z). n Lemma 6.1. For any non-real z = u + iv with v > 0 the following inequality holds 1 1 (6.4) |Sen (z) − Sn (z)| ≤ E 2 TrD2 . nv 2 Proof. Note that 1 e e (6.5) R(z) = R(z) + √ R(z)DR(z). n Denote by k · ks spectral norm of a matrix. For resolvent matrices we have, for z = u + iv, v > 0, 1 e (6.6) max{kR(z)k s , kR(z)ks } ≤ . v Inequality (6.6) implies that 1 1 1 e √ |TrR(z)DR(z)| (6.7) ≤ (TrD2 ) 2 . nv 2 n n
14
F. G¨ otze, A.Tikhomirov
Relations (6.7) and (6.5) together conclude the proof.
(c)
For a random variable X, put X = XI{|X|≤c} − EXI{|X|≤c} , which is a truncated and recentred. Introduce the matrix W(c) with truncated entries: W (c) (j, l) = (c) (c) √1 X . Let R(c) (z) = (W(c) − zI)−1 and Sn (z) = n1 ETrR(c) (z). n jl Corollary 6.2. For any non-real z = u + iv with v > 0, 12 n X 1 2 (6.8) |Sn(c) (z) − Sn (z)| ≤ v −2 2 EXjl I{|Xjl |>c} . n j,l=1
f = W(c) and Proof. The result follows from Lemma 6.1 with W (τ )
(τ
√
We define Sn (z) := Sn
n)
√1 D n
= W(c) − W.
(z).
Corollary 6.3. For any non-real z = u + iv with v > 0, 1
|Sn(τ ) (z) − Sn (z)| ≤ v −2 Ln2 (τ ).
(6.9)
√ Proof. The result follows from Corollary (6.2) with c = τ n.
7. The proof of Theorem 1.1 Proof. Without loss of generality we shall assume that σ 2 = 1. We introduce the random variables, for fixed τ > 0, (τ )
Xjl = Xjl I{|Xjl |≤τ √n − EXjl I{|Xjl |≤τ √n} .
(7.1)
Define symmetric matrix W(τ ) of order n × n by W (τ ) (l, j) = (τ )
−1
(τ )
(τ ) Sn (z)
j ≤ n. Let R (z) = (W − zIn ) and = Corollary (6.3) we have, for any z = u + iv, v > 0,
1 n ETrR
(τ ) √1 X , n lj (τ )
for 1 ≤ l ≤
(z). According to
1
|Sn(τ ) (z) − Sn (z)| ≤ v −2 Ln2 (τ ).
(7.2) Note that
2 1 ETrR(τ ) , n using Cauchy’s integral formula, we get
(7.3)
0
Sn(τ ) (z) =
and Sn0 (z) =
0
1 ETrR2 . n
1
|Sn(τ ) (z) − Sn0 (z)| ≤ 4v −3 Ln2 (τ ).
(7.4)
Introduce the matrices E(lj) , for 1 ≤ l ≤ j ≤ n, with entries E (lj) (m, k) = 1 if and only if, {m, k} = {l, j} or {m, k} = {j, l} and E (lj) (m, k) = 0, otherwise. Define (lj)
(lj)
(lj)
− zIn )−1 . the matrices W(τ ) = W(τ ) − √1n Xlj E(lj) and R(τ ) (z) = (W(τ ) In what follows we shall omit z in the notation of resolvent matrices. The resolvent matrix R(τ ) may be represented as (lj) (lj) (lj) (lj) 1 (τ ) 1 (τ ) 2 (7.5) R(τ ) = R(τ ) − √ Xlj R(τ ) E(lj) R(τ ) + Xlj R(τ ) E(lj) R(τ ) . n n For the matrix R(τ ) , we have notice that X 1 1 (τ ) (7.6) ETrW(τ ) R(τ ) = √ (2 − δjl )EXjl R(τ ) (j, l). n n n 1≤j≤l≤n
Spectra of random matrices with martingale structure
15
Using equality (7.5), we obtain 1 ETrW(τ ) R(τ ) = A1 − A2 + A3 , n
(7.7) where 1 A1 = √ n n A2 = A3 =
1 n2
X
(τ )
(2 − δjl )EXjl R(τ )
(j,l)
(j, l),
1≤j≤l≤n (τ ) 2
X
(2 − δlj )EXlj
R(τ )
(lj)
E(lj) R(τ )
(lj)
(lj),
1≤l≤j≤n
1 √
n2
(τ ) 3
X n
(2 − δlj )EXlj
R(τ )
(lj)
E(lj) R(τ ) (l, j).
1≤l≤j≤n
Here and in what follows δkj denotes, δkj = 1, k = j, and δkj = 0 otherwise. We estimate A3 at first . It is easy to see that (lj) (lj) (lj) 1 (7.8) max{kR(τ ) E(lj) R(τ ) ks , kR(τ ) E(lj) R(τ ) ks } ≤ 2 . v (τ )
According to the definition of Xlj and inequality (7.8), we get X 2τ 2τ (τ ) 2 EXlj ≤ 2 (1 + ε(2) (7.9) |A3 | ≤ 2 2 n ). n v v 1≤l≤j≤n
By condition (1.5), (7.10)
A1 = 0.
Using relations (7.8) and (τ )
(7.11)
R(τ )
(lj)
Xjk (lj) = R(τ ) + √ R(τ ) E(lj) R(τ ) , n
we get |A2 − A4 | ≤ C(1 + ε(2) n )
(7.12)
τ , v3
where (7.13)
A4 =
1 n2
(τ ) 2
X
(2 − δlj )EXlj
R(τ ) E(lj) R(τ ) (lj).
1≤l≤j≤n
A simple calculation shows that (7.14)
2
R(τ ) E(lj) R(τ ) (lj) = R(τ ) (j, j)R(τ ) (l, l) + δlj R(τ ) (l, j).
Substituting (7.14) in (7.13), we get (7.15)
A4 = A5 + A6 ,
where A5 =
1 n2
2 A6 = 2 n
X
(τ ) 2
(2 − δlj )EXlj
R(τ ) (j, j)R(τ ) (l, l),
1≤l≤j≤n
X 1≤l
