On the concentration of high dimensional matrices

On the concentration of high dimensional matrices with randomly signed entries S. G. Bobkov1,2 , F. G¨otze1 , and A. N. Tikhomirov1,3 January 21, 2006

Abstract Results on the concentration and asymptotic behavior of large dimensional random matrices with random signs are shown which extend corresponding results for sums with random signs and limiting Gaussian mixture families to Wigner distributions.

1

Introduction

Consider a family X = {Xjk }, 1 ≤ j ≤ k ≤ n, of random variables defined on some probability space (Ω, F, P). Put Xjk = Xkj , for 1 ≤ k < j ≤ n, and introduce the symmetric matrices   ε11 X11 ε12 X12 · · · ε1n X1n  1   ε21 X21 ε22 X22 · · · ε2n X2n  W(ε) = √  , .. .. .. ..  n . . . . εn1 Xn1 εn2 Xn2 · · · εnn Xnn where ε = {εjk } denotes an arbitrary family of signs ±1, satisfying the symmetry condition εjk = εkj . This constitutes an ensemble of 2n(n+1)/2 random matrices. Each of them has a random spectrum {λ1 (ε), . . . , λn (ε)} and an associated spectral distribution function Fn,ε (x, ω) =

1 card {j ≤ n : λj (ε) ≤ x}, n

1

x ∈ R, ω ∈ Ω.

Faculty of Mathematics, University of Bielefeld, Germany School of Mathematics. University of Minnesota. Research partially supported by NSF grant 3 Research supported by the DFG-Forschergruppe FOR 399/1. Partially supported by INTAS grant N 03-51-5018, and by RFBR–DFG grant N 04-01-04000 Key words: Distribution of spectrum, typical distributions, Wigner’s theorem, the semi-circle law, Bernoullian random variables, randomized models of matrices 2

1

CONCENTRATION AND LIMIT THEOREMS

2

Averaging over the random values Xij (ω), define the expected empirical distribution function Fn,ε (x) = E Fn,ε (x, ω). We shall study the general question about the possible limit laws for the distribution of these spectra. In particular, in this randomized model, it is natural to ask whether or not there is for growing n a certain universal limit distribution which approximates Fn,ε for most of ε’s. Our aim is to show that, under some regularity hypotheses on the distribution of the X’s, such a limit distribution exists and may be characterized as a mixture of the Wigner’s semi-circle laws analogously to the case of sums and Gaussian distributions. (τ ) To be more precise, define for the truncation random variables Xjk = Xjk I{|Xjk |≤τ √n} with truncated level τ > 0, n

(τ ) (σn,j )2

1 X (τ ) 2 (X ) = n k=1 jk

and

(σn(τ ) )2

n n 1 X X (τ ) 2 = 2 (X ) . n j=1 k=1 jk

We impose the following conditions: for any τ > 0, as n → ∞, 2 1) supj,k EXjk = O(1); P P √ 2 2) Ln (τ ) = n12 nj=1 nk=1 EXjk I(|Xjk | > τ n) = o(1); 2  Pn (τ ) 2 (τ ) 2 1 2 = o(1). 3) ∆1 = n j=1 E (σn,j ) − (σn )

The second assumption is the usual Lindeberg-type condition, introduced by L. A. Pastur for random matrices with independent entries [P1], often called Wigner ensemble. The third condition may be described as a kind of variance stabilization. To measure closeness of distributions on the real line, we shall use the L´evy metric: Given distribution functions F and G, the distance L(F, G) is defined as the infimum over all δ > 0 such that for all x ∈ R, F (x − δ) − δ ≤ G(x) ≤ F (x + δ) + δ. We write L(η) for the distribution of a random variable η. Let µn denote the n(n+1)/2 standard Bernoulli measure on the discrete cube {−1, Pn1} Pn . 2To each point it 1 −n(n+1)/2 2 assigns the mass 2 . Finally, define σn = n2 j=1 k=1 Xjk (σn ≥ 0). Theorem 1.1. Assume σn ⇒ σ, weakly in distribution, as n → ∞, for some random variable σ ≥ 0. Then, assuming the conditions 1) − 3), for any δ > 0, we have µn {ε : L(Fn,ε , L(σξ)) > δ} → 0, where ξ is a random variable independent of σ and√ distributed according to the semi1 circle law with variance 1 (that is, with density 2π 4 − x2 I(|x| ≤ 2) ). The statement may formally be sharpened by replacing δ with a sequence δn → 0. Thus, most of the distribution functions Fn,ε are close to L(σξ) in the the metric L.

CONCENTRATION AND LIMIT THEOREMS

3

Note that L(σξ) may be viewed as a mixture of the semi-circle laws with mixing 2 , so that σn2 converges measure L(σ). If a weak law of large numbers applies to Xjk in probability to a constant σ 2 , then the universal limit distribution L(σξ) is the semi-circle law with variance σ 2 . Note as well, when the distribution of X in Rn(n+1)/2 is symmetric with respect to all coordinate axes, the distribution of W(ε) will be independent of ε, so all distribution functions Fn,ε are identical. Therefore, taking, for example, εjk = 1, the model reduces to the non-randomized case and yields the Wigner theorem for a certain family of random matrices with dependent entries. As for the randomized model, there is one interesting concentration aspect, which in essence does not require any hypotheses on the joint distribution of the entries of X. Define the average distribution function with respect to the ε’s via X Fn (x) = 2−n(n+1)/2 Fn,ε (x). ε

It turns out that already assumption 1), suffices to guarantee that most of the distribution functions Fn,ε are very close to Fn . 2 Theorem 1.2. If EXjk ≤ α2 , for all j, k ≤ n, then for any δ ∈ (0, 1), 2

µn {ε : L(Fn,ε , Fn ) ≥ δ} ≤ C e−cn ,

(1.1)

where C and c are positive constants depending on δ and α, only. An important feature of the bound (1.1) is that with respect to the “dimension” the exponent is proportional to n2 . This is an analogue to results on the concentration of distributions of randomized sums for dependent summands. Recall that given a sequence of random variables, say, ξ1 , . . . , ξn , one considers the sums Sn (θ) = θ1 ξ1 + · · · + θn ξn ,

θ = (θ1 , . . . , θn )

with coefficients from the unit sphere S n−1 : θ12 + . . . θn2 = 1. In 1978 V. N. Sudakov discovered [S] that under a mild spectral assumption on the correlation operator of ξi ’s, most of the distributions of Sn (θ) are close to a certain “typical” distribution Fn which is the average of the L(Sn (θ))’s with respect to the uniform Lebesgue measure µn (dθ) on S n−1 . For the proof, he applied the L´evy-Schmidt isoperimetric theorem on the sphere and the measure concentration phenomenon related to it. Moreover, Pn 2 ξ and Z is a standard normal Fn itself is close to the law of ρn Z, where ρ2n = n1 i=1 i random variable independent of ρn . Thus, most of L(Sn (θ))’s are close to a mixture of symmetric Gaussian measures on the line. A different approach to Sudakov’s theorem, involving the case of independent coefficients, was later developed by H. von Weizs¨acker [W]. Various extensions and refinements were intensively discussed in the literature, cf. e.g. [A-B-P], [B1], [N-R]. One of the results was that, the

CONCENTRATION AND LIMIT THEOREMS

4

√ coefficients may have a special structure. For example, if θi = εi / n with εi = ±1, and ξi ’s are orthonormal in L2 (Ω, F, P), it was shown in [B2] that νn {ε : L(L(Sn ), Fn ) ≥ δ} ≤ C e−cn , where now νn denotes the normalizing counting measure on the discrete cube {−1, 1}n . (In case of the trigonometric system, the problem goes back to the work by R. Salem and A. Zygmund [S-Z].) Thus, the role of the semi-circle law in the framework of randomized matrices, as stated in Theorems 1.1–1.2, is similar to that of the normal distribution in the framework of randomized sums. The paper is organized as follows. In Section 2 we collect some useful identities and bounds for resolvent matrices. Section 3 deals with the concentration property of the family of characteristic functions associated with the expected empirical distributions Fn,ε . In section 4 we convert this property in terms of the L´evy distance and prove Theorem 1.2. In Section 5 we show that the average distribution Fn is close to a mixture of semi-circle laws. For the readers convenience, this section is divided into three subsections. The proof of Theorem 1.1 is completed in Section 6. In the Appendix we prove some auxiliary bounds for resolvent matrices.

2

General identities and bounds for resolvent

Let Ln denote the collection of all square n × n matrices A with complex entries ajk , 1 ≤ j, k ≤ n. Also, denote by Hn the collection of all n × n symmetric P matrices with real entries. Given a matrix A in Ln with entries ajk , Tr (A) = nj=1 ajj defines its P trace. The Hilbert-Schmidt norm of A is given by kAk2HS = nj,k=1 |ajk |2 . As usual, kAk denotes the spectral norm. Denote by I the unit matrix of size n × n. For a complex number z = u + iv, the value of the resolvent of A ∈ Hn at z is defined by R(z) = (A − zI)−1 . It is well-defined, whenever v > 0. For short, we also write R = R(z). The next following lemmas 2.1–2.5are easily obtained by diagonalization of symmetric matrices. Lemma 2.1. For any matrix A in Hn and z = u + iv with v > 0, √ √ n n n 2 kRkHS ≤ , kR kHS ≤ 2 . |Tr (R)| ≤ , v v v Lemma 2.2. For any A in Hn and z = u + iv with v > 0, we have kRk ≤ v1 . In particular, for any j = 1, . . . , n, n X k=1

|Rjk |2 ≤

1 . v2

CONCENTRATION AND LIMIT THEOREMS R +∞ Lemma 2.3. For any A in Hn and v > 0, −∞ kR(u + iv)k2HS du = precisely, for any j = 1, . . . , n, Z +∞ X n π |R(u + iv)jk |2 du = . v −∞ k=1

5 nπ . v

More

(2.1)

Lemma 2.4. Given A ∈ Hn and M ∈ Ln , for any v > 0, √ Z +∞ nπ 2 |Tr (MR (u + iv)| du ≤ kMkHS . v −∞ We will also use the following elementary identity: Lemma 2.5. Given matrices A and B in Hn , let R = RA (z) be the resolvent of A and R0 = RA−B (z) be the resolvent of A − B. Then, R0 = R + RBR0 = R + RBR + RBRBR0 .

3

Concentration of characteristic functions

To study the concentration problem, we mainly work with the Stieltjes transform of the expected spectral distributions Fn,ε , defined by Z +∞ dFn,ε (x) 1 Sn (z, ε) = = Tr (R(z, ε)), x−z n −∞ where R(z, ε) = E (W(ε) − zI)−1 represents the expected resolvent of the random matrix W(ε) at the point z = u + iv. We always assume that v > 0. The Stieltjes transform is related to the characteristic function fn (t, ε) of Fn,ε through the simple relation Z ev|t| +∞ itu e Im Sn (z, ε) du, t ∈ R. fn (t, ε) = π −∞ Correspondingly, the µn -average of such characteristic functions represents the characteristic function of Fn (x): Z ev|t| +∞ itu fn (t) = e Im Sn (z) du. (3.1) π −∞ Note that the imaginary part of the Stieltjes transform as a function of the real variable u is always integrable with respect to Lebesgue measure. As a first step, we show that, for any z = u + iv, v > 0, the values of the function ε → Sn (z, ε) are strongly concentrated about its µn -mean Z X Sn (z) = Sn (z, ε) dµn (ε) = 2−n(n+1)/2 Sn (x, ε). ε

CONCENTRATION AND LIMIT THEOREMS

6

This can be seen by virtue of the so-called concentration phenomenon on the discrete cube. Namely, given a complex-valued function g = g(ε) on {−1, 1}N , consider its discrete gradient with modulus given by |∇g(ε)|2 =

N X

|g(ε) − g(εk )|2 ,

k=1

where εk represents the ±1-sequence which is obtained from the sequence ε by replacing εk with −εk on the k-th place. The distribution of the modulus of the gradient is occurs in a number of deviation inequalities. As the simplest example, one may consider the so-called Poincar´e-type inequality: Lemma 3.1. For any function g : {−1, 1}N → C such that to the normalized counting measure µN , we have Z Z 1 2 |∇g|2 dµN . |g| dµN ≤ 4

R

g dµN = 0 with respect

Moreover, there is an inequality of Gaussian-type in terms of the `∞ -norm of the modulus of the gradient (cf. [B-G], [L]): Lemma 3.2. If |∇g(ε)| ≤ α, for all ε ∈ {−1, 1}N , then   Z 2 2 µN ε : g(ε) − g dµN ≥ h ≤ 4 e−h /4α ,

h ≥ 0.

We need to bound the modulus of the gradient |∇fn (t, ε)| for the specific function g(ε) = fn (t, ε) uniformly over all points ε in {−1, 1}n(n+1)/2 . In our setting, X |∇fn (t, ε)|2 = |fn (t, ε) − fn (t, εjk )|2 , 1≤j≤k≤n

where, εjk represents the symmetric ±1-matrix of size n × n, which is obtained from the matrix ε by replacing the entry εjk with −εjk on the jk-th and kj-th places. 2 Lemma 3.3. If EXjk ≤ α2 , for all j, k, then for all ε ∈ {−1, 1}n(n+1)/2 and t ∈ R,

|∇fn (t, ε)| ≤

Cα|t| max{1, α|t|}, n

where C is an absolute constant. By the inequalities in Lemmas 3.1-3.2, we therefore obtain:

(3.2)

CONCENTRATION AND LIMIT THEOREMS

7

2 ≤ α2 , for all j, k. Then we get, with respect to the norTheorem 3.1. Let EXjk malized counting measure µn on {−1, 1}n(n+1)/2 , for all t ∈ R, Z C 2 b2 |fn (t, ε) − fn (t)|2 dµn (ε) ≤ . (3.3) n2

where b = α|t| max{1, α|t|}. Moreover, with some universal c > 0, for all h ≥ 0, µn {ε : |fn (t, ε) − fn (t)| ≥ h} ≤ 4 e−c n

2 h2 /b2

.

Proof of Lemma 3.3. Let t 6= 0. We use an elementary bound X √
jk ajk fn (t, ε) − fn (t, ε ) , |∇fn (t, ε)| ≤ 2 sup A

(3.4)

(3.5)

1≤j≤k≤n

where the sup is taken over all matrices A in Ln with real entries {ajk } such that X a2jk = 1. (3.6) 1≤j≤k≤n

Here we assume that ajk = 0 for j > k. Fix such a matrix A. According to the relation between characteristic functions and Stieltjes transform, we have for z = u + iv, v > 0, and ajk real, X
ajk fn (t, ε) − fn (t, εjk ) = 1≤j≤k≤n

ev|t| π

Z

+∞

eitu Im

−∞

X


ajk Sn (z, ε) − Sn (z, εjk ) du,

1≤j≤k≤n

hence, X


ajk fn (t, ε) − fn (t, εjk ) ≤ 1≤j≤k≤n Z
ev|t| +∞ X jk a S (z, ε) − S (z, ε ) du. jk n n π −∞ 1≤j≤k≤n

(3.7)

Given 1 ≤ j ≤ k ≤ n, write W(εjk ) = W(ε) −

2εjk Xjk jk √ D , n

where we Djk denotes the symmetric matrix which has zero entries everywhere except at the jk-th and kj-th entries, where it equals 1. In particular, Djj has zero

CONCENTRATION AND LIMIT THEOREMS

8

entries everywhere except in the jj-th entry, where it is 1. By Lemma 2.5 with 2ε Xjk √ B = jk Djk , the resolvents R of W(ε) and Rjk of W(εjk ) are related by n R

2 4Xjk 2εjk Xjk jk =R+ √ RD R + RDjk RDjk Rjk . n n

jk

Taking the trace of the both sides and using commutativity of the trace, we obtain that 2 4Xjk 2εjk Xjk jk 2 Tr (R ) = Tr (R) + √ Tr (Djk RDjk Rjk R). Tr (D R ) + n n jk

Taking expectations and dividing by n, we get Sn (z, εjk ) − Sn (z, ε) =

2 n3/2

E εjk Xjk Tr (Djk R2 ) +

where bjk = Tr (Djk RDjk Rjk R). Hence, X
2 4 ajk Sn (z, εjk ) − Sn (z, ε) = 3/2 E Tr (MR2 ) + 2 n n 1≤j≤k≤n and Z +∞ X −∞

ajk

1≤j≤k≤n

2 n3/2

Z

4 2 E bjk Xjk , n2

X

2 ajk E bjk Xjk ,

1≤j≤k≤n


Sn (z, εjk ) − Sn (z, ε) du ≤

+∞

E −∞

4 |Tr (MR )| du + 2 n 2

 Z 2 |ajk | E Xjk

X 1≤j≤k≤n

+∞

 |bjk | du , (3.8)

−∞

where M is the matrix with entries Mjk = ajk εjk Xjk . By Lemma 2.4, √ Z +∞ nπ 2 |Tr (MR )| du ≤ kMkHS . v −∞ P 2 Since kMk2HS = j,k |ajk |2 Xjk , by (3.6), we have, E kMk2HS ≤ α2 , so E kMkHS ≤ α. Thus, √ Z +∞ nπ 2 E |Tr (MR )| du ≤ α. (3.9) v −∞ Now, let’s turn to the last term in (3.7) and recall the definition of bjk . Note that, for all symmetric matrices G, H in Ln , we have in general Tr (Djk GDjk H) = Gjj Hkk + Gkk Hjj + 2 Gjk Hjk ,

for j < k.

Also, Tr (Djj GDjj H) = Gjj Hjj . We apply these identities to G = R and H = Rjk R. By Lemma 2.2, |Rjk | ≤ v1 , for all k, so whether j < k or j = k, we have |Tr (Djk GDjk H)| ≤

4 |Hjk |. v

(3.10)

CONCENTRATION AND LIMIT THEOREMS Write jk

Hjk = (R R)jk =

n X

9

(Rjk )jl Rlk .

l=1

By the usual Cauchy inequality (for numbers), v u n uX |Hjk | ≤ t |(Rjk )jl |2 l=1

v u n uX t |Rlk |2 , l=1

so integrating over u and using Cauchy-Bunyakovski’s inequality, we get v v uZ +∞ n uZ +∞ n Z +∞ X X u u |(Rjk )jl |2 du t |Rlk |2 du. |Hjk | du ≤ t −∞

−∞

l=1

−∞

l=1

By Lemma 2.3 applied to the matrices Rjk and R, and in view of the symmetry of R, the right hand above side is equal to πv . Hence, from (3.9), Z +∞ 4π |bjk | du ≤ 2 , v −∞ 4π 2 and thus the second expectation Pin (3.7) is bounded by v2 α . In addition, by the Cauchy’s inequality and (3.6), 1≤j≤k≤n |ajk | ≤ n. Therefore, the second term in 16π 2 (3.7) is bounded by nv 2 α . Together with (3.8), this bound yields via (3.7) that Z +∞ X n αo
18πα 2πα 16πα2 jk + ≤ max 1, . a du ≤ S (z, ε ) − S (z, ε) jk n n nv nv 2 nv v −∞ 1≤j≤k≤n

Recalling (3.5) and (3.7), we obtain that, for any v > 0, n αo √ α |∇fn (t, ε)| ≤ 18 2 ev|t| max 1, . nv v

√ Choosing v = 1/|t| finishes the proof of the lemma with C = 18πe 2.

4

Concentration of distributions in L´ evy metric

We shall now study the closeness of the randomized distribution functions Fn,ε to the mean distribution functions Fn in terms of the L´evy metric. Note that this metric is related to the usual Kolmogorov’s supremum distance by 0 ≤ L(F, G) ≤ kF − Gk ≤ 1. Conversely, if G has a Lipschitz semi-norm bounded by a constant C, then we get an opposite bound kF − Gk ≤ (1 + C) L(F, G). The inequality (3.4) of Theorem 3.1 about the concentration property of the characteristic functions may be transformed into a concentration property of the distribution functions by virtue of Zolotarev’s general bound, cf. [Z]:

CONCENTRATION AND LIMIT THEOREMS

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Lemma 4.1. If f and g are the characteristic functions of the distribution functions F and G, then Z T  |f (t) − g(t)| C log T L(F, G) ≤ C dt + , T ≥ T0 , (4.1) t T 0 where C and T0 > 1 are absolute constants. We shall use as well the following general elementary result. R R Lemma 4.2. If x2 dF (x) ≤ α2 , and x2 dG(x) ≤ α2 , then the function ψ(t) = f (t)−g(t) has derivative satisfying |ψ 0 (t)| ≤ α2 , for any t > 0. t Indeed, write 0

Z

+∞

ψ (t) = −∞

eitx (1 − itx) − 1 2 x d(F (x) − G(x)). (tx)2

The function ξ(s) = eis (1 − is) − 1 has derivative ξ 0 (s) = s eis , so |ξ(s)| ≤ 21 s2 , for all s ∈ R, and the conclusion follows. Now fix a number δ, such that 0 < δ < 1, fix a number h > 0, and take T ≥ T0 to be specified later on. Consider a partition of [0, T ] into m consecutive intervals ∆i , i = 0, 1, . . . , m − 1, of equal length, not exceeding h. These intervals have endpoints T i, and ti = m T ti+1 − ti = ≤ h, 0 ≤ i ≤ m − 1. (4.2) m Introduce the subsets of the discrete cube   |fn (ti , ε) − fn (ti )| Ωi = ε : < h , 1 ≤ i ≤ m. ti By Theorem 3.1, 1 − µn (Ωi ) ≤ 4 exp{−c n2 h2 /B 2 }, where B = α max{1, αT }, and c is a positive universal constant. Hence, (m ) \ 1 − µn Ωi ≤ 4m exp{−c n2 h2 /B 2 }. i=1

  In (4.2) we may take m = Th + 1 (the integer part), so (m ) \ 4 (T + h) 1 − µn Ωi ≤ exp{−c n2 h2 /B 2 }. h i=1

(4.3)

CONCENTRATION AND LIMIT THEOREMS Recall that fn (t, ε) = fn00 (0, ε)

1 n

11

E Tr (eitW(ε) ). Therefore,

n n 1 1 XX 2 2 = − E Tr (W(ε) ) = − 2 . E Xjk n n j=1 k=1

This identity holds for all ε on the discrete cube, so the same is true for fn00 (0). Thus, |fn00 (0, ε)| ≤ α2 and the same is true for fn00 (0). Hence, by Lemma 4.2, the function ψ(t, ε) =

|fn (t, ε) − fn (t)| t

2 has a Lipschitz Tm semi-norm satisfying kψ(t, ε)kLip ≤ α on the half-axis t > 0. Let ε ∈ i=1 Ωi , so that |ψ(ti )| < h. for all i. Any point t ∈ [0, T ] belongs to some interval ∆i , so |t − ti | ≤ h, for some i ≤ m. By the Lipschitz property shown above, we obtain that

ψ(t, ε) ≤ ψ(ti , ε) + kψ(t, ε)kLip |t − ti | < (1 + σ 2 ) h. Hence,

RT 0

ψ(t, ε) dt ≤ (1 + α2 ) T h, and by Lemma 4.1,   log T 2 . L(Fn (x, ε), Fn (x)) ≤ C (1 + α ) T h + T

Thus, by (4.3), if   log T 2 δ ≥ C (1 + α ) T h + , T

(4.4)

then   4 (T + h) c n2 h2 µn {L(Fn (x, ε), Fn (x)) ≥ δ} ≤ exp − 2 . h α max{1, α2 T 2 }

(4.5)

The next step is to minimize the right hand side of (4.5) for all (T, h) satisfying (4.4) together with the assumption T ≥ T0 . In fact, as an almost optimal choice, we may take log 2δ T = β , δ where β is large enough. Then,   log T log β 2 δ ≤ δ + ≤ , T β log 2 β 2C where C is the constant in Zolotarev’s inequality (4.1) and (4.4). Furthermore, if we take δ h = , 2C (1 + α2 ) T

CONCENTRATION AND LIMIT THEOREMS

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(4.4) holds. As for the right hand side of the (4.5), we get with some absolute constant C 0 that h h δ δ3 ≥ ≥ 0 . = max{1, αT } 1 + αT C (1 + α3 ) T 2 β 2 C 0 (1 + α3 ) log2 2δ In addition, 2β 2 C(1 + α2 ) log2 2δ + δ 3 T +h C 00 (1 + α2 ) = ≤ . h δ3 δ4 Thus we conclude: Theorem 4.1. For any δ ∈ (0, 1), 2

µn {ε : L(Fn,ε , Fn ) ≥ δ} ≤ C e−cn ,

(4.6)

where C = C(α, δ) and c = c(α, δ) depend on δ and α, only. Here we may choose for some absolute positive constants C and c, C(1 + α2 ) , C(α, δ) = δ4

c δ6 c(α, δ) = 2 . α (1 + α6 ) log4 2δ

Thus, we get a more precise version of Theorem 1.2. The bound (4.6) seems to be correct with respect to the “dimension” n2 . However, we do not know the optimal n order of the function c(σ, δ) for small δ. Choosing δ = δn of order log for large n n1/3 yields a right hand side smaller than any power of n1 . Hence: Corollary 4.3. Up to some constant C = C(α) we have, for “most” of ε, L(Fn,ε , Fn ) ≤ C

log n . n1/3

(4.7)

In particular, Z L(Fn (x, ε), Fn (x)) dµn (ε) ≤ C

log n . n1/3

(4.8)

Remark. The log n term in (4.8) may be removed by using a Poincar´e-type inequality (3.3).

5

Asymptotic behavior of weighted random spectral distributions

In this section we consider the behavior of the spectral distribution functions that are averaged with respect to ε’s, X Fen (x) = Fn (x, ω) ≡ 2−n(n+1)/2 Fn,ε (x, ω), (5.1) ε

CONCENTRATION AND LIMIT THEOREMS

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under the conditions that, for any τ > 0, as n → ∞, en (τ ) = 12 Pn Pn X 2 I(|Xjk | > τ √n ) = o(1); 1) L jk j=1 k=1 n 2  (τ ) 2 (τ ) 2 e 21 := 1 Pn (σ ) − (σ ) = o(1). 2) ∆ n n,j j=1 n The study of the behavior of the spectral distribution function Fn (x, ω) for large n is essentially equivalent to the study of the spectra of a random matrix whose entries are independent and symmetrically distributed random variables, taking at most two opposite values. Note this is a non-i.i.d. model. First, we need to prove a (random) bound for the L´evy distance between the distribution function Fn (x, ω) and L(σn ξ), where ξ is a random variable having a standard semi-circle law (with variance 1). Recall the notations introduced in Section 1: n n 1 X (τ ) 2 1 X (τ ) 2 (τ ) (σnj )2 = (Xjk ) , (σn(τ ) )2 = (σ ) . n k=1 n j=1 nj (τ )

As before, Xjk stands for Xjk , truncated at level τ . Consider the random symmetric matrices, subject to a truncation procedure, n 1  (τ ) (τ ) √ W = εjk Xjk , R(τ ) = (W(τ ) − zI)−1 , n j,k=1 (τ )

and introduce the Stieltjes transform Sn (z) = n1 Tr (R(τ ) ). Let q (τ ) en (τ ) e2 L 1 σn e e 1 + ∆1 . √ + 2τ + + B= + ∆ (τ ) (τ ) n n (σn )2 (σn )2 Theorem 5.1. Given τ > 0 and ω ∈ Ω, assume that e ≤ σ (τ ) . B n

(5.2)

Then, there exist absolute constants C and T0 > 1 such that, for any T > T0 , e e (τ ) )−1 exp{2T σ (τ ) } log T + C log T + T Ln (τ ) . L(Fen , L(σn(τ ) ξ)) ≤ C B(σ n n (τ ) T 2σn Proof. Let fen and f denote the characteristic functions of Fen and the semi-circle law, respectively. By Zolotarev’s Lemma 4.1 we have, for all T ≥ T0 , Z T e (τ ) |fn (t) − f (tσn )| C log T (τ ) e L(Fn , L(σn ξ)) ≤ C dt + . t T 0 (τ )

Note that |σn − σn | ≤

e n (τ ) L (τ )

2σn

. This easily implies the bound

|f (tσn(τ ) ) − f (tσn )| ≤ |t| |σn − σn(τ ) | ≤ |t|

en (τ ) L (τ )

2σn

.

(5.3)

CONCENTRATION AND LIMIT THEOREMS

14

Then, applying equality (3.1) and inequality (5.3) together, we get, for any v > 0, Z ∞ e L(Fn , L(σn ξ)) ≤ C exp{vT } log T |Sn (u + iv) − (σn(τ ) )−1 S((u + iv)(σn (τ ) )−1 )|du −∞

+

en (τ ) log T L +T , (τ ) T 2σn

(5.4)

where Sn (z) and S(z) denote the Stieltjes transforms of the distribution function Fen (x) and the semi-circle law, respectively. To bound the right hand side of this inequality, we shall investigate the Stieltjes transform Sn (z) =

1 Eε Tr (R(z)). n

Theorem 5.2. Under the condition of Theorem 5.1, there exists an absolute positive constant C such that Z ∞ e B −1 −1 Sn (u + iv) − (σn(τ ) ) S((u + iv)(σn(τ ) ) ) du ≤ C . v −∞ Theorem 5.2 and inequality 5.4 together will complete the proof of Theorem 5.1. Proof of Theorem 5.2 First we bound the difference between Stieltjes transforms of the spectrum of the matrix W and the one of the spectrum of the matrix W(τ ) with truncated entries. Lemma 5.1. For any v > 0, we have Z

q

∞

Sn (u + iv) − Sn(τ ) (u + iv) du ≤

−∞

(τ )

2(σn )2 + v 2 p q Ln (τ ). 3 (τ ) 2 v σn

The proof of this Lemma is postponed to the Appendix. √ Now, assuming |Xjk | ≤ τ n, we prove: Theorem 5.3. Under the condition of Theorem 5.1, there exists an absolute positive (τ ) constant C such that, for all v ≥ 2σn , Z +∞ e CB (τ ) (τ ) −1 (τ ) −1 . Sn (u + iv) − (σn ) S((u + iv)(σn ) ) du ≤ v −∞ The proof of this Theorem is given in Section 5.1. Lemma 5.1 and Theorem 5.3 together imply the result of Theorem 5.2. 

CONCENTRATION AND LIMIT THEOREMS

5.1

15

Proof of Theorem 5.3. Uniform bound

In the sequel we assume that

√ |Xjk | ≤ τ n.

We will omit the index τ in the notation. In this Section we show that the Stieltjes transform of the spectral distribution of the matrix W satisfies a certain approximate equation that characterizes the semi-circle law. Furthermore, we give a bound for the error of this approximation which is uniform in u. Then, in Section 5.3, using the obtained representation, we derive an integral bound for the difference between the Stieltjes transforms of the semi-circle law and the spectral distribution of the matrix. 5.1.1

The main equation

We recall the following notations. Let X = (Xjk )nj,k=1 denote a symmetric matrix of order n. Let εjk , 1 ≤ j ≤ k ≤ n, denote the Bernoulli i.i.d. random variables. Consider a symmetric random matrix W of order n with entries 1 Wjk = √ εjk Xjk , n

for 1 ≤ j ≤ k ≤ n.

Introduce the resolvent matrix R = (W − zIn )−1 , where In is the identity matrix of order n and z = u + vi with v > 0. Let jν denote the multi-index jν = (j1 , . . . , jν ) with distinct numbers j1 , . . . , jν from {1. . . . , n}. Introduce the matrix W(jν ) obtained from W by deleting the j1 th,...,jν -th both rows and columns. Consider also the resolvent matrix R(jν ) = (W(jν ) − zIn−ν )−1 . If ν = 0, the set of indices {j1 , . . . , jν } is empty, and we may also write W = W(j0 ) (j ) and R = R(j0 ) . Let aj ν denote the j-th column without j-th elements of the matrix W(jν ) . According to Lemma 7.1 below, we have the following equality for the diagonal entries of the resolvent matrices: For all j ∈ {1, . . . , n}\{j1 , . . . , jν }, ν = 0, 1, . . . , n− 1. 1

(j )

Rjjν =

εjj Xjj √ n

−z−

1 n

P(jν+1 ) l6=k

(j

)

εjk εjl Xjk Xjl Rklν+1 −

where jν+1 = (j1 , . . . , jν , j) and where from {1, . . . , n} \ {j1 , . . . , jν+1 }.

P(jν+1 )

1 n

P(jν+1 ) l

(j

Xjl2 Rll ν+1

)

,

(5.5)

indicates summation over all indices

CONCENTRATION AND LIMIT THEOREMS

16

Introduce the following notations: 1 1 X(jν+1 ) (j ) (j ) (j ) γj,1ν = √ εjj Xjj , γj,2ν = − εjk εjl Xjl Xjk Rlkν+1 , n l6=k n   1 X(jν+1 ) 2 1 (jν ) (jν+1 ) (jν+1 ) , γj,3 = − Xjl Rll − Tr R n l n !  1 X(jν+1 ) 2 1 1 (jν ) (jν+1 ) (jν ) γj,4 = − Tr R − Tr R Xjl n l n n !   n 1X 2 1 1 (jν ) (jν ) (jν ) γj,5 = − , Tr R − Eε Tr R X n l=1 jl n n ! ! ν+1 n X X 1 1 1 1 (j ) (j ) 2 γj,6ν = Xjj Tr R(jν ) , γj,7ν = − Xjl2 − σn2 Eε Tr R(jν ) . p n p=1 n n l=1 n Using these notations, we may rewrite the equality (5.5) in the form 1

(j )

Rjjν = −

z+

(j ,τ ) σn2 Sn ν (z)

1

+ z+

(j )

(j ) σn2 Sn ν (z)

where Sn(jν ) (z)

1 = Eε Tr R(jν ) , n

(j ) Γj ν

=

7 X

Γj Rjjν ,

(5.6)

(j )

γj,sν .

s=1

Taking the mean value of the equation (5.6) with respect to both j and ε, we get 1 1 1 X(jν ) (j ) (j ) (5.7) + Sn(jν ) (z) = − Eε Γj ν Rjjν . (j ) (j ) ν ν z + σn2 Sn (z) z + σn2 Sn (z) n j In particular, for ν = 0, we have Sn (z) = −

1 + δn (z), z + σn2 Sn (z)

where

n

δn (z) =

X 1 Eε Γj Rjj . n(z + σn2 Sn (z)) j=1

√ Assuming that |Xjk | ≤ τ n, we prove the following

Theorem 5.4. For v ≥ 2σn , the following inequality holds |δn (z)| ≤ where

e1 B , σn2

p 2 2 σ n e1 = 2τ + √ + ∆1 + ∆1 + √1 . B σn σn3 n n

(5.8)

CONCENTRATION AND LIMIT THEOREMS

17

e1 ≤ 1 v. Then, for v ≥ max{1, 2σn }, Corollary 5.2. Assume that B 2 Sn (z) = σn−1 S((z + δn (z))σn−1 ) + δn (z) where S(z) = − z2 +

√

z 2 −4 . 2

Proof. Note that, for v ≥ max{1, 2σn }, the assumption implies Im {z + σn2 δn (z)} > 0. Solving equation (5.8) with respect to Sn (z), we arrive at the desired representation.

5.2

Proof of Theorem 5.4

We start with straightforward bounds for the γj,s with s 6= 3. For s = 3, the bound will be based on a certain recurrence procedure. The bounds uniformly in u = 0 and for any 1 ≤ j, k ≤ n, √ |Xjk | ≤ τ n. We shall estimate the γj,s , s = 1, . . . , 7, error terms of the approximation of the Stieltjes transform of the distribution function Fn (x) by the Stieltjes transform of the semi-circle law using several auxiliary Lemmas. We start with obvious estimates. Lemma 5.3. We have (j )

A1 ν ≡

1 X(jν ) (j ) Eε |γj,1ν | ≤ τ. n j (j )

Proof. It is straightforward to check that A1 ν ≤ Lemma 5.4. (j )

A2 ν ≡

1 3 n2

Pn

j=1

|Xjj | ≤ τ .

1 X(jν ) σn τ (j ) Eε |γj,2ν | ≤ . n j v

CONCENTRATION AND LIMIT THEOREMS

18

(j )

Proof. Applying Cauchy’s inequality and the definition of γj,2ν , we obtain that 1 X(jν ) 1 X(jν ) 12 (jν ) 2 (j ) Eε |γj,2ν | ≤ Eε |γj,2 | n j n j 1 X(jν ) ≤ 2 n j ≤

X(jν+1 )

! 12

1 X(jν+1 )

n

n l6=k

τ X(jν ) ≤ nv j

)

l6=k

τ X(jν ) j

(j

2 Xjl2 Xjk Eε |Rklν+1 |2

! 21 (j

)

Xjl2 Eε |Rklν+1 |2

1 X(jν+1 ) 2 Xjl n l

! 12 ≤

τ σn . v

Here we have used that, for all l ∈ {1, . . . , n}\{j1 , . . . , jν+1 } and for all ν = 0, 1, . . . , n, X(jν+1 )

|Rlk |2 ≤

k

Lemma 5.5. (j )

A4 ν ≡

1 . v2

σ2 1 X(jν ) (j ) Eε |γj,4ν | ≤ n . n j nv

Proof. Applying Lemma 7.2, we obtain (j )

A4 ν

1 X(jν ) ≤ 2 nv j

Lemma 5.6. (j )

A5 ν ≡

1 X(jν+1 ) 2 Xjk n k

!

σn2 . nv

≤

1 X(jν ) σn (j ) Eε |γj,5ν | ≤ √ . n j nv

Proof. Applying Lemma 7.3, we get (j ) A5 ν

1 X(jν ) 1 X(jν+1 ) 2 12 ≤ Xjk Eε n j n k

2 1 2σn2 (j ) (j ) (Tr R ν − Eε Tr R ν ) ≤ √ . n nv

Lemma 5.7. (j )

A6 ν

1 X(jν ) (ν + 1)σn2 (j ) ≡ Eε |γj,6ν | ≤ + n j nv

p

∆21 . v

CONCENTRATION AND LIMIT THEOREMS (j )

Proof. The definition of γj,6ν and the inequality | n1 Tr R(jν ) | ≤ v1 together imply ν n ν X n 2 X X X 1 (ν + 1)σn 1 1 1 (jν ) 2 2 2 + A6 ≤ Xjjp ≤ Xjjp − σn . nv p=1 n nv nv p=1 n j=1

j=1

(j )

Finally, we state a simple bound for A7 ν ≡ Lemma 5.8. (j )

A7 ν

1 n

P(jν ) j

(j )

E|γj,7ν |.

n p n X X ∆21 1 1 2 2 . X − σn ≤ ≤ nv j=1 n k=1 jk v (j )

The proof follows from the definition of γj,7ν . 5.2.2

The bound on A3

We prove the following: Lemma 5.9. The inequality n

1X e1 , A3 = E|γj,3 | ≤ 4B n j=1 holds for v ≥ 2σn with  e1 = 2τ + B

 q σn 1 1 1 √ + 1 √ + 3 ∆21 + ∆21 . σn n n σn

Proof. Introduce the following quantity (ν)

βj

=

max

(j )

jν : j ∈{j / 1 ,...,jν }

Eε |γj,3ν |.

For ν = 0 we define (0)

βj = Eε |γj,3 |. Using equality (5.6), we get (j )

1 X(jν+1 ) 2 1 X(jν+1 ) jν+1 j Xjk Rkk − Rllν+1 n k n l  1 1 X(jν+1 ) 2 1 X(jν+1 )  (jν+1 ) (j ) ≤ 2 Xjk Eε Γl + Eε Γk ν+1 . v n k n l

Eε |γj,3ν | ≤

Note that 1 X(jν ) (jν+1 ) Eε Γl max ≤ jν : j ∈{j / 1 ,...,jν } n l

(

7 X

s=1, s6=3

) 1 X(jν ) 1 X(jν ) (ν+1) (j ) E|γj,sν+1 | + βl . n l n l

19

CONCENTRATION AND LIMIT THEOREMS

20

According to Lemmas 5.3–5.8, we obtain ( ) X 1 X(jν ) (j ) max E|γj,sν+1 | ≤ Bν , jν : j ∈{j / 1 ,...,jν } n s6=3 l where σn  σn Bν = 1 + τ+√ v nv 



 p 2 ∆1 (ν + 2)σn √ +1 + . v n

Then 1 X(jν ) (ν) σn2 1 1 X(jν ) X(jν+1 ) 2 1 X(jν+1 ) (ν+1) Xjk βj ≤ 2 Bν + 2 2 βl n j v v n j n k l X X (j ) (j ) ν ν+1 1 1 2 (ν+1) Xjk βk , + 2 2 v n j k

(5.9)

In (5.9) we note the estimates 1 X(jν ) (ν) σn2 1 1 X(jν ) X(jν+1 ) 2 1 X(jν+1 ) (ν+1) Xjk βl βj ≤ 2 Bν + 2 2 n j v v n j n l k ! 1 1 X(jν ) 1 X(jν ) 2 (ν+1) + 2 Xjk βk . v n k n j6=k Thus we may rewrite the previous inequality as ) ( 1 1 X(jν ) X(jν+1 ) 2 1 X(jν ) (ν+1) 1 X(jν ) (ν) σn2 βj ≤ 2 Bν + 2 2 Xjk βl n j v v n j n k l ! 1 1 X(jν ) 1 X(jν ) 2 (ν+1) + 2 Xjk βk . v n k n j6=k (ν+1)

(5.10)

Note that βk in the second term on the right hand side of (5.10) does not depend P (ν+1) in the first term does not depend on j and k, on j and the quantity n1 l (jν ) βl respectively. We also note that n 1 X 2 1 σ (ν) 2 − σn2 . (5.11) βj ≤ n + Xkl v v n l=1 Inequalities (5.10) and (5.11) and the above remark imply that q 1 X(jν ) (ν) σn2 2σn2 1 X(jν ) (ν+1) 2σn2 2 βl + 3 ∆21 + 3 ∆21 . βj ≤ 2 Bν + 2 n j v v n l v v

CONCENTRATION AND LIMIT THEOREMS Put

21

p 2 ∆21 2 e + 2 ∆21 . Bν = Bν + v vσn

Using this notation, we have 1 X(jν ) (ν) σn2 e 2σ 2 1 X(jν ) (ν+1) βj ≤ 2 Bν + 2n βl . n j v v n l Since the last inequality does not depend on jν , we may write n n 2σn2 1 X (ν+1) 1 X (ν) σn2 e β ≤ 2 Bν + 2 β . n j=1 j v v n l=1 l

For ν = 0, 1, . . . , n, n ≥ 1, introduce the quantity Dnν = (5.9) may be rewritten as Dn,ν ≤

1 n

Pn

j=1

(ν)

βj . Then inequality

2σn2 e 2σn2 B + Dn,ν+1 . ν v2 v2

Note that Dn,n = 0. We may take v 2 ≥ 4σn2 . Then we get 1 1e Dn,ν ≤ B ν + Dn,ν+1 , 2 2 which, for v ≥ 2σn , implies that e1 , Dn,0 ≤ 4B where

 e1 = 2τ + B

 q σn 1 1 2 1 √ + 1 √ + 3 ∆1 + ∆21 . σn n n σn

Finally, we note that n

n

1X 1 X (0) E|γj,3 | ≤ β . n j=1 n j=1 j The last inequalities together imply that

5.3

1 n

Pn

j=1

e1 . E|γj,3 | ≤ 4B

Proof of Theorem 5.4. An integral bound

In this Section we prove Theorem 5.4. In particular, we obtain some integral bounds for the difference between the Stieltjes transforms of the spectral distribution function and of the semi-circle law. As a simple corollary, we get bounds for the distance between the corresponding characteristic functions.

CONCENTRATION AND LIMIT THEOREMS

22

(τ ) First we consider the matrix W √ , but for simplicity of notation, we omit the symbol τ , assuming that |Xjk | ≤ τ n for all j, k. Note that according to Corollary 5.2, we may write the solution of equation (5.8) in the form   z + σn2 δn (z) 1 S + δn (z). (5.12) Sn (z) = σn σn

Using |S 0 (z)| ≤ v1 , this implies that, for v ≥ 2σn , Sn (z) − 1 S( z ) ≤ (1 + σn−1 ) |δn (z)| σn σn and

Z

∞


Sn (z) − σn−1 S zσn −1 du ≤ (1 + σn−1 )

Z

∞

|δn (u + iv)|du. −∞

−∞

We return to equation (5.7) and to the definition δn (z) as n

X 1 δn (z) = Eε Γj Rjj , n(z + σn2 Sn (z)) j=1 P7

γj,p . Integrating, we get Z ∞ Z ∞ n 1X R jj Γj |δn (z)| du ≤ Eε z + σ 2 Sn (z) du n j=1 −∞ −∞ n n 7 Z 1 XX ∞ 1 ≤ Eε |γj,s | |Rjj (u + iv)|du. n j=1 s=1 −∞ |z + σn2 Sn (z)|

where Γj =

p=1

Introduce the quantities Z 1 X(jν ) ∞ 1 (jν ) e Aj,s = Eε |γj,s | |Rjj (u + iv)|du. 2 n j −∞ |z + σn Sn (z)| Note that Z ∞ −∞

|Rjj (u + iv)|du 1 ≤ 2 |z + σn Sn (z)| 2

Z

∞ 2

Z

∞

|Rjj (u + iv)| du + −∞

−∞

du |z + σn2 Sn (z)|2

 .

Using the representation of the diagonal entries of the matrix R via eigenvalues λ1 , . . . , λn and eigenvectors u1 , . . . , un of the matrix W, we obtain Z ∞ Z ∞ n X 1 π 2 2 |Rjj (z)| du ≤ ujk du ≤ . 2 2 v −∞ (λj − u) + v −∞ k=1 Equation (5.12) implies the following inequality Z ∞ Z ∞ Z ∞ e2 1 B 2 du ≤ 2 |S (z)| du + 2 du. n 2 2 2 2 2 −∞ |z + σn Sn (z)| −∞ −∞ v |z + σn Sn (z)|

CONCENTRATION AND LIMIT THEOREMS It is straightforward to check that Z ∞ Z ∞Z 2 |Sn (z)| du ≤ −∞

−∞

∞

−∞

23

1 π . dudF (x) ≤ n (u − x)2 + v 2 v e2

According to the condition 5.2 of Theorem 5.1, we have | Bv2 | ≤ 14 . It then follows that Z ∞ Cπ 1 , (5.13) du ≤ 2 2 v −∞ |z + σn Sn (z)| for z = u + iv with v ≥ 2σn . Using these bounds, similarly to Lemmas 5.3–5.8, we show that for s 6= 3 X e es ≤ B . (5.14) A v s6=3 Applying the same argument as in the proof of Lemma 5.9, we show that, for v ≥ 2σn , e e3 ≤ C B . A v From these inequalities it follows that Z

∞

|Sn (z) − σn−1 S(zσn−1 | du ≤

−∞

e CB . v

The last inequality completes the proof of Theorem 5.4.

6

Proof of Theorem 1.1

e By Theorem 1.2, it remains to show that Fn (x) → EX G(x), as n → ∞. Here −1 e e G(x) = G(xσ ). Recall that Fn (x) = E Fn (x). The latter expectation may be splitted into the three integrals such that, whenever 0 < m < M , E Fen (x) = E Fen (x)I{m≤σn ≤M } + E Fen (x)I{σn T0 , −1 e exp{2T M } log T lim sup E ψn (X)I{m≤σn ≤M } I{B≤σ lim sup EB e n } ≤ Cm n→∞

n→∞

+

C log T T en (τ ). + lim sup EL T m n→∞

Since e ≤ τ, lim sup EB n→∞

we obtain, for any τ > 0 and for any T > T0 , lim sup E ψn (X)I{m≤σn ≤M } I{B≤σ e n } ≤ C(T, M, m) τ + n→∞

C log T , T

where C(T, M ) = C log T exp{2T M }. The left hand side of the last inequality doesn’t depend on τ and T . In the limit with τ → 0 and T → ∞ we get lim sup E ψn (X)I{m≤σn ≤M } I{B≤σ e n } = 0.

(6.3)

n→∞

Relations (6.2) and (6.3) yield en (x)I{m≤σn ≤M } . lim inf E Fen (x) ≥ lim inf E G n→∞

n→∞

Since σn converges weakly in distribution to σ as n → ∞ we obtain e lim inf E Fen (x) ≥ E G(x)I {m≤σ≤M } , n→∞

In the limit with m → 0 and M → ∞ we get e lim inf E Fen (x) ≥ E G(x). n→∞

(6.4)

The representation (6.1) yields the following inequality Fn (x) ≤ E Fen (x)I{m≤σn ≤M } + E I{σn 0,   λ2 + v 2 u2 . sup ≤ (u − λ)2 + v 2 v2 u Proof. The function f (u) =

u2 (u−λ)2 +v 2

the first derivative

f 0 (u) = 2u

−λu + λ2 + v 2 , ((u − λ)2 + v 2 )2 2

2

with critical points at u0 = 0 and u1 = λ +v . It is easy to see that λ   u2 u2 + v 2 = f (u ) = . sup 1 (u − λ)2 + v 2 v2 u Thus the Lemma is proved. Lemma 7.5. Z

∞

−∞

r

1 π Eε Tr (|R|4 ) du ≤ 3 n v2

p 2σn2 + v 2 . √ σn

CONCENTRATION AND LIMIT THEOREMS

28

Proof. Consider the following equalities v Z ∞r Z ∞u X u1 n 1 1 t Eε Tr |R|4 = Eε du n n j=1 ((u − λj )2 + v 2 )2 −∞ −∞ v Z ∞u X u1 n du u2 + σn2 t p . Eε = 2 2 2 n j=1 ((u − λj ) + v ) u2 + σn2 −∞ Applying H¨older inequality, we get Z r ∞ 1 4 du ≤ E Tr |R| −∞ n ε

Z

∞

−∞

n

1X u2 + σn2 Eε du n j=1 ((u − λj )2 + v 2 )2

! 21 Z

∞

−∞

du u2 + σn2

 21 . (7.5)

By Lemma 7.4, we have Z

∞

−∞

n n 1X π X λ2j + v 2 + σn2 π 2σn2 + v 2 u2 + σn2 du ≤ = . Eε E ε n j=1 ((u − λj )2 + v 2 )2 nv j=1 v2 v v2

(7.6)

Inequalities (7.5) and (7.6) together imply the Lemma. We prove the following Lemma 7.6. For any v > 0, the following inequality holds, p Z ∞ 2 2p n+v (τ ) Sn (u + iv) − Sn du ≤ 2σ Ln (τ ). 3√ v 2 σn −∞ Proof. Applying the resolvent equality (A + B − zI)−1 = (A − zI)−1 + (A − zI)−1 B(A − zI)−1 , we get 1 1 Eε Tr R − Eε Tr R(τ ) ≤ 1 Eε |Tr R(τ ) W(τ ) R| n n n

(7.7)

Using Cauchy’s inequality, we get 1 1 (τ ) Eε Tr R − Eε Tr R ≤ 1 n n n

n X j,k=1

(τ ) 2 Eε Wjk

! 12

n X

2 Eε (RR(τ ) )jk

j,k=1

We rewrite the last inequality as follows q 1 p 1 (τ ) Eε Tr R − Eε Tr R ≤ Ln (τ ) Eε Tr RRR(τ ) R(τ ) . n n

! 21 .

CONCENTRATION AND LIMIT THEOREMS From the last bound it follows that   q p 1 p 1 (τ ) Eε Tr R − Eε Tr R ≤ Ln (τ ) Eε Tr |R|4 + Eε Tr |R(τ ) |4 . n n

29

(7.8)

Inequality (7.8) and Lemma 7.5 together imply s 2 2 1 1 π Eε Tr R − Eε Tr R(τ ) ≤ 3 σn + 2v . n σn v2 −∞ n

Z

∞

Lemma 7.5 and the last inequality together imply the result.

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S. G. B.: School of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church St. S.E., Minneapolis, MN 55455 USA F. G., A. N. T.: Department of Mathematics, Bielefeld University, Bielefeld 33501, Germany