Spectral Convergence of Large Block-Hankel Gaussian Random Matrices

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Random Matrices

Philippe Loubaton, Xavier Mestre

To cite this version:

Philippe Loubaton, Xavier Mestre. Spectral Convergence of Large Block-Hankel Gaussian Random

Matrices. Colombo F., Sabadini I., Struppa D., Vajiac M. (eds) Advances in Complex Analysis and

Operator Theory. Trends in Mathematics. Birkhäuser, Cham, 2017. �hal-01618531�

arXiv:1704.06651v1 [math.PR] 19 Apr 2017

ian Random Matrices

Philippe Loubaton and Xavier Mestre

Dedicated to Prof. Daniel Alpay on occasion of his 60th birthday

Abstract. This paper studies the behaviour of the empirical eigenvalue distri- bution of large random matricesWNW_N^H where WN is aM L×N matrix, whose M block lines of dimensions L×N are mutually independent Han- kel matrices constructed from complex Gaussian correlated stationary ran- dom sequences. In the asymptotic regime where M →+∞,N →+∞and

M L

N → c > 0, it is shown using the Stieltjes transform approach that the empirical eigenvalue distribution of WNW^H_N has a deterministic behaviour which is characterized.

Mathematics Subject Classification (2000).Primary 60B20; Secondary 15B52.

Keywords. Large random matrices, Stieltjes transform of positive matrix- valued measures, Hankel matrices.

1. Introduction

1.1. The addressed problem and summary of the main results.

In this paper, we consider aM L×N block-Hankel matrixW_N composed ofM Hankel matrices gathered on top of each other, namely

W_N =

W^T_1,N · · · W^T_M,N T

.

For each m = 1, . . . , M, W_m,N is a Hankel matrix of dimensions L×N, with (i, j)th entry equal to

{W_m,N}_i,j=wm,N(i+j−1)

for 1≤i≤L, 1≤j≤N, where the random variables (wm,N(n))m=1,...,M,n=1,...,N+L−1

are zero mean complex Gaussian random variables . The different blocksW_m,N

This work was supported by the Labex B´ezout under grant ANR-10-LABX-0058.

are independent, but we allow for some time invariant correlation structure within each Hankel matrix, namely

E

wm,N(k)w^∗_m′,N(k^′)

= rm(k−k^′) N δm−m^′

whererm(k),k∈Z, is a sequence of correlation coefficients defined as rm(k) =

Z 1

0 Sm(ν) e^2iπνkdν

where (S^m)m=1,...,M are positive functions. We denote by (ˆλk,N)k=1,...,ML the eigenvalues of random matrixW_NW^H_N, where (·)^Hstands for transpose conjugate.

The purpose of this paper is to study the asymptotic properties of the empirical eigenvalue distributiondˆµN(λ) =_ML¹ PML

k=1δ_λ−λˆk,N ofW_NW_N^H whenM →+∞, N →+∞ and L is such that cN = ^ML_N converges towards a non zero constant c >0.

It is well established that the asymptotic behaviour of the empirical eigen- value distribution of large Hermitian matrices can be evaluated by studying the behaviour of their Stieltjes transforms. In the context of the present paper, the Stieltjes transform ofdˆµN(λ) is the functionqN(z) defined onC\R⁺ by

qN(z) = Z

R⁺

1

λ−zdˆµN(λ) = 1 M L

MLX

k=1

1 ˆλk,N−z

and also coincides with qN(z) = _ML¹ trQN(z) where Q_N(z) is the resolvent of matrixWNW^H_N defined by

Q_N(z) = W_NW^H_N−zI−1

.

We denote bySML(R⁺) the set of allM L×M Lmatrix valued functions defined onC\R⁺by

S^ML(R⁺) = Z

R⁺

1

λ−zdµ(λ)

whereµis a positive M L×M L matrix-valued measure carried byR⁺ satisfying µ(R⁺) =I_ML. In this paper, we establish that there exists a function T_N(z) of SML(R⁺), defined as the unique element of SML(R⁺) satisfying a certain func- tional equation, that verifies

1

M Ltr ((QN(z)−T_N(z))AN)→0 (1.1) almost surely for each z ∈ C\R⁺, where (A_N)N≥1 is an arbitrary sequence of deterministicM L×M Lmatrices satisfying sup_NkA_Nk<+∞. Particularized to the case whereA_N =I, this property implies that

qN(z)−tN(z)→0 (1.2)

almost surely for each z ∈ C\R⁺, where tN(z) = _ML¹ tr(T_N(z)) is the Stieltjes transform of the probability measure µN = _ML¹ tr(µ_N). In the present context,

this turns out to imply that almost surely, for each bounded continuous function φdefined onR⁺, it holds that

1 M L

XML k=1

φ(ˆλk,N)− Z

R⁺

φ(λ)dµN(λ)→0. (1.3)

It is also useful to study the respective rates of convergence towards 0 of the variance and of the mean of _ML¹ PML

k=1φ(ˆλk,N)−R

R⁺φ(λ)dµN(λ) when φ is a smooth function. For this, it appears sufficient to restrict to the caseφ(λ) = _λ−¹_z wherez∈C\R⁺, i.e. to study the rate of convergence of var(qN(z)) =E|qN(z)− E(qN(z))|²and ofE(qN(z))−R

R⁺φ(λ)dµN(λ). More generally, if (AN)N≥1is any sequence of deterministic M L×M L matrices satisfying sup_NkA_Nk < +∞, we establish that

var 1

M Ltr (QN(z)AN)

=O( 1

M N) (1.4)

and that, provided ^L_MN^3/2 →0, 1

M Ltr ((E(QN(z))−T_N(z))A_N) =O( L

M N) (1.5)

for eachz∈C\R⁺.

In this paper, we concentrate on the characterization of the asymptotic be- haviour of the terms _ML¹ tr (QN(z)AN), and do not discuss on the behaviour of general linear statistics _ML¹ PML

k=1φ(ˆλk,N) of the eigenvalues. In order to establish our results, we follow the general approach introduced in [14] and developed in more general contexts in [15]. This approach takes benefit of the Gaussianity of the random variables wm(n), and use the Poincar´e-Nash inequality to evaluate the variance of various terms and the integration by parts formula to evaluate approximations of matrixE(Q_N(z)).

Apart large random matrix methods, the properties of Stieltjes transforms of positive matrix valued measures play a crucial role in this paper. The first author (in the alphabetic order) of this paper had the chance to learn these tools from Prof. D. Alpay at the occasion of past collaborations. The authors are thus delighted to dedicate this paper to Prof. D. Alpay on occasion of his 60th birthday.

1.2. Motivations

The present paper is motivated by the problem of testing whether M complex Gaussian zero mean times series (xm(n))n∈Z are mutually independent. For each m = 1, . . . , M, xm is observed from time n = 1 to n = N, and a relevant statistics depending on ((xm(n))n=1,...,N)m=1,...,M has to be designed and stud- ied. A reasonable approach can be drawn by noting that if the M time series are independent, then, for each integer L, the covariance matrix RL of M L- dimensional random vector xL(n) = (x1(n), . . . , x1(n+L−1), x2(n), . . . , x2(n+

L−1), . . . , xM(n), . . . , xM(n+L−1))^T is block diagonal, a property implying that κN = 1

M L log det(RL)− XM m=1

log det(R^m,m_L )

!

= 0 (1.6)

whereR^m,m_L represents them-thL×Ldiagonal block ofR_L. Therefore, it seems relevant to approximate matrixR_L by the standard estimator ˆR_L defined by

Rˆ_L= 1 M L

XN n=1

x_L(n) (xL(n))^H

so that we can evaluate the term ˆκN, obtained by replacingR_Lby ˆR_Lin (1.6), and compare it to 0. The present paper is motivated by the study of this particular test under the hypothesis that the series (xm)m=1,...,M are uncorrelated and assuming that M and N are both large. In this context, a crucial problem is to choose parameterL. On one hand, Lshould be chosen in such a way thatM L/N <<1 in order to make the estimation error kRˆL−RLk reasonably low, and thus ˆκN

close to 0 under the uncorrelation hypothesis. On the other hand, choosing a small value for Lis not satisfying because comparing ˆκN to 0 allows to test that E(xm(l)x^∗_m′(l^′)) = 0 for each (m, m^′) only for l, l^′ ∈ [0, . . . , L−1], a property that does not imply formally that the time series xm and xm^′ are independent.

Therefore, choosing L as large as possible is relevant. In this case, the ratio ^ML_N may no longer be very small, and ˆκN may not converge towards 0. It is thus of fundamental interest to evaluate the behaviour of ˆκN whenM andNare large and that ^ML_N is not negligible. This question is connected to the problem addressed in the present paper because the following results potentially allow to establish that

1

MLlog det( ˆRL) has a deterministic behaviour that can be characterized. This term can indeed be written as

1

M Llog det( ˆR_L) = 1 M L

MLX

k=1

φ(ˆλk,N)

where (ˆλk,N)k=1,...,ML are the eigenvalues of ˆRL and where φ(λ) = logλ. More- over, if we denote by wm,N(n) the random variablewm,N(n) = √¹

Nxm(n), then it is easily seen that matrix ˆR_L coincides with matrix W_NW^H_N where W_N is constructed from the wm,N(n) as above, up to end effects (because matrix W_N depends on random variables (wm(N+l))m=1,...,M,l=1,...,L−1while matrix ˆR_Ldoes not depend on these entries). However, these end effects can be shown to be neg- ligible. Therefore, the asymptotic behaviour of _ML¹ log det( ˆR_L) appears to be a consequence of the results of the present paper. We finally mention that under some reasonable assumptions,kRˆ^m,m_L −R^m,m_L k →0 so that it holds that

1 M L

XM m=1

log det( ˆR^m,m_L )− 1 M L

XM m=1

log det(R^m,m_L )→0

In summary, the asymptotic behaviour of ˆκN appears to be a consequence the study of the empirical eigenvalue distribution ofW_NW^H_N in the asymptotic regime whereM, N →+∞in such a way that ^ML_N converges towards a non zero constant.

1.3. On the literature.

The study of the asymptotic behaviour of large random Gram matrices has a long history. Since the pionneering work of Marcenko-Pastur in 1967 ([13]), a number of random matrix models have been considered (see e.g. [2], [15] and the references therein). In the following, we mention the previous works that are connected to the present paper. As matrixW_N is a block line matrix withL×N blocks, we mention the works of Girko ([7], chapter 16) as well as [4] devoted to the case where the blocks are i.i.d. We however mention that [7] and [4] did not characterize the rates of convergence and that the techniques used in these works do not allow to address the case where L →+∞. The works devoted to Hankel matrices are also relevant. [3] addressed the case where M = 1 andN, L→+∞ at the same rate, except that in [3], the random variablesw1,N(n) are forced to 0 forN < n≤N+L−1. Using the moments method, [3] showed that the empirical eigenvalue distribution ofWNW^∗_N converges towards a non compactly supported limit distribution. The random matrix model considered in [12] is similar to matrix WN of the present paper, except that in [12], for eachm= 1, . . . , M, the random variables (wm(n)) are uncorrelated, i.e. the spectral densities (S^m(ν))m=1,...,M

are reduced toS^m(ν) = 1 for eachν. Using the Poincar´e-Nash inequality and the integration by parts formula, [12] studied the asymptotic behaviour of the empirical eigenvalue distribution ˆµN in the asymptotic regimeM, N →+∞and ^ML_N →c, c >0. It was established that function tN(z) defined in (1.2) coincides with the Stieltjes transform of the Marcenko-Pastur distribution with parametercN, so that ˆ

µN converges weakly almost surely towards the Marcenko pastur distribution. The rates of convergence of the variance and of the expectation of qN(z)−tN(z) are both characterized. Finally, [12] proved that providedL=O(N^α) withα <2/3, then the extreme non zero eigenvalues ofW_NW^∗_N converge almost surely towards the end points of the support of the Marcenko-Pastur distribution. Therefore, the present paper is a partial generalization of [12].

1.4. Assumptions, general notations, and background on Stieltjes transforms of positive matrix valued measures.

Assumptions onL, M, N

Assumption 1.1. • All along the paper, we assume that L, M, N satisfy M → +∞, N → +∞ in such a way thatcN = ^ML_N → c, where 0 < c <+∞. In order to shorten the notations, N →+∞should be understood as the above asymptotic regime.

• In section 6,L, M, N also satisfy ^L_MN^3/2 →0 or equivalently _M^L4 →0.

Assumptions on sequences(rm)m=1,...,Mand spectral densities(Sm)m=1,...,M. We assume that sequences (rm)m=1,...,M satisfy the condition

sup

M

X

n∈Z

1 M

XM m=1

|rm(n)|²

!^1/2

<+∞. (1.7)

As it holds that

1 M

PM

m=1|rm(n)|2

≤ M¹

PM

m=1|rm(n)|², condition (1.7 ) implies that

sup

M

1 M

XM m=1

X

n∈Z

|rm(n)|<+∞ and that for eachm,P

n∈Z|rm(n)|<+∞. Therefore, each spectral densitySmis continuous. We also assume that

sup

M

sup

m=1,...,M

νmax∈[0,1]Sm(ν)<+∞ (1.8) infM inf

m=1,...,M min

ν∈[0,1]S^m(ν)>0. (1.9) In the following, we denote byR(k) the diagonal matrix

R(k) = diag(r1(k), . . . , rM(k)). (1.10) General notations.

In the following, we will often drop the indexN, and will denoteW_N,Q_N, . . . byW,Q, . . .in order to simplify the notations. The N columns of matrix Ware denoted (w_j)j=1,...,N. For 1≤l≤L, 1≤m≤M, and 1≤j≤N,W^m_i,jrepresents the entry (i+ (m−1)L, j) of matrix W.

If A is a M L×M L matrix, we denote by A^m_i ^1,m2

1,i2 the entry (i1+ (m1− 1)L, i2 + (m2−1)L) of matrix A, while A^m1,m2 represents the L×L matrix (A^m_i ^1,m2

1,i2 )1≤(i1,i2)≤L.

C⁺denotes the set of complex numbers with strictly positive imaginary parts.

The conjuguate of a complex numberzis denotedz^∗. Ifz∈C\R⁺, we denote by δz the term

δz= dist(z,R⁺). (1.11)

The conjugate transpose of a matrixA is denotedA^H while the conjugate of A (i.e. the matrix whose entries are the conjugates of the entries of A) is denoted A^∗.

kAk represents the spectral norm of matrixA. IfA and B are 2 matrices, A⊗Brepresents the Kronecker product ofAandB, i.e. the block matrix whose block (i, j) is A_i,jB. If A is a square matrix, Im(A) and Re(A) represent the Hermitian matrices

Im(A) = A−A^H

2i , Re(A) =A+A^H 2

If (AN)N≥1(resp. (bN)N≥1) is a sequence of matrices (resp. vectors) whose dimensions increase with N, (AN)N≥1 (resp. (bN)N≥1) is said to be uniformly bounded if sup_N≥1kA_Nk<+∞(resp. sup_N≥1kb_Nk<+∞).

Ifν∈[0,1] and ifRis an integer, we denote bydR(ν) theR–dimensional vec- tord_R(ν) = (1, e^2iπν, . . . , e^{2iπ(R−1)ν})^T, and bya_L(ν) the vectora_L(ν) = √¹

Rd_R(ν).

If x is a complex-valued random variable, the variance of x, denoted by Var(x), is defined by

Var(x) =E(|x|²)− |E(x)|² The zero-mean random variablex−E(x) is denotedx^◦.

Nice constants and nice polynomials

Definition 1.2. A nice constant is a positive constant independent of the dimensions L, M, N and complex variablez. A nice polynomial is a polynomial whose degree is independent fromL, M, N, and whose coefficients are nice constants.

In the following, P1 and P2 will represent generic nice polynomials whose values may change from one line to another, and C(z) is a generic term of the formC(z) =P1(|z|)P2(1/δz).

Background on Stieltjes transforms of positive matrix valued measures. In the following, we denote by S^K(R⁺) the set of all Stieltjes transforms of K×K positive matrix-valued measures µ carried by R⁺ verifying µ_K(R⁺) = I_K. The elements of the classS^K(R⁺) satisfy the following properties:

Proposition 1.3. Consider an element S(z) = R

R⁺ dµ(λ)

λ−z of SK(R⁺). Then, the following properties hold true:

• (i)Sis analytic on C\R⁺

• (ii) Im(S(z))≥0 andIm(zS(z))≥0 ifz∈C⁺

• (iii)limy→+∞−iyS(iy) =I_K

• (iv)S(z)S^H(z)≤ ^I_δ^K2_z for eachz∈C\R⁺

• (v) R

R⁺λ dµ(λ) = limy→+∞Re (−iy(IK+ iyS(iy))

Conversely, if a function S(z)satisfy properties (i), (ii), (iii), then S(z)∈ SK(R⁺)

While you have not been able to find a paper in which this result is proved, it has been well known for a long time (see however [9] for more details on (i), (ii), (iii), (v)), as well as Theorem 3 of [1] from which (iv) follows immediately) .

2. Toeplitzification operators.

In the following derivations, it will be useful to consider the following Toeplitzifi- cation operators, which inherently depend on the correlation function rm(·). Let

J_K denote the K×K shift matrix with ones in the first upper diagonal and ze- ros elsewhere, namely {J_K}i,j =δj−i−1, and let J⁻_K¹ denote its transpose. For a given squared matrixMwith dimensionsR×R, we define Ψ^(m)_K (M) as anK×K Toeplitz matrix with (i, j)th entry equal to

nΨ^(m)_K (M)o

i,j=

RX−1

l=−R+1

rm(i−j−l)τ(M) (l) (2.1) or, alternatively, as the matrix

Ψ^(m)_K (M) =

KX−1 n=−K+1

RX−1 l=−R+1

rm(n−l)τ(M) (l)

!

J⁻_Kⁿ (2.2) where the sequenceτ(M) (l),−R < l < R, is defined as

τ(M) (l) = 1 Rtrh

MJ^l_Ri

. (2.3)

We can express this operator more compactly using frequency notation, namely

Ψ^(m)_K (M) =

KX−1 n=−K+1

Z 1

0 Sm(ν)a^H_R(ν)Ma_R(ν) e^2πiνndν

J⁻_Kⁿ

= Z 1

0 Sm(ν)a^H_R(ν)Ma_R(ν)d_K(ν)d^H_K(ν)dν wherea_R(ν) =d_R(ν)/√

Randd_R(ν) = 1,e^2πiν, . . . ,e^{2πi(R−1)νT}

. In particular, whenrm(k) =σ²δk(white observations), we have

Ψ^(m)_K (M) =σ²

KX−1

n=−K+1

1

Rtr [MJⁿ_R]J⁻_Kⁿ=σ²

KX−1

n=−K+1

τ(M)J⁻_Kⁿ

=σ²TK,min(R,K)(M)

whereTK,R(X) is the classical Toeplitzation operator in [12]. The following prop- erties are easily checked.

• Given a square matrix A of dimension K ×K and a square matrix B of dimensionR×R, we can write

1 Ktrh

AΨ^(m)_K (B)i

= Z 1

0 S^m(ν)a^H_K(ν)AaK(ν)a^H_R(ν)BaR(ν)dν= 1 Rtrh

Ψ^(m)_R (A)Bi (2.4)

• Given the square matricesB,C(of dimensionK×K) andD,E(of dimension R×R), we have

1 Ktrh

BΨ^(m)_K

DΨ^(m)_R (C)Ei

= 1 Ktrh

CΨ^(m)_K

DΨ^(m)_R (B)Ei .

• Given a square matrixM and a positive integerK, we have

Ψ^(m)_K (M)≤ sup

ν∈[0,1]

Sm(ν)a^H_R(ν)Ma_R(ν)≤ sup

ν∈[0,1]|Sm(ν)| kMk.

• Given a square positive definite matrixMand a positive integerK, condition (1.9) implies that

Ψ^(m)_K (M)>0. (2.5)

We define here two other operators that will be used throughout the paper, which respectively operate on N×N and M L×M L matrices. In order to keep the notation as simple as possible, we will drop the dimensions in the notation of these operators.

• Consider an N ×N matrix M. We define Ψ (M) as an M L×M L block diagonal matrix withmth diagonal block given by Ψ^(m)_L (M).

• Consider an M L×M L matrix M, and let M^m,m denote its mth L×L diagonal block. We define Ψ (M) as theN×N matrix given by

Ψ (M) = 1 M

XM m=1

Ψ^(m)_N (M^m,m). (2.6)

Ψ (M) can also be expressed as Ψ (M) =

NX−1

n=−(N−1) LX−1

l=−(L−1)

τ^(M)(M(R(n−l)⊗I_L)) (l)J⁻_Nⁿ (2.7) whereτ^(M)(A)(l) is defined for anyM L×M LmatrixAby

τ^(M)(A)(l) = 1

M Ltr A(I_M ⊗J^l_L)

= 1 Ltr

"

1 M

XM m=1

A^m,m

! J^l_L

#

and whereR(m) is defined in (1.10). Note also that Ψ (M) can alternatively be written as

Ψ (M) = 1 M

XM m=1

Z 1

0 Sm(ν)a^H_L (ν)M^m,ma_L(ν)d_N(ν)d^H_N(ν)dν.

Given these two new operators, and ifA andB areM L×M Land N×N matrices, we see directly from (2.4) that

1 Ntr

Ψ (A)B

= 1

M Ltr [AΨ (B)]. (2.8)

3. Variance evaluations

In this section, we provide some estimates on the variance of certain quantities that depend on the resolventQ(z) =

WW^H−zIML

−1

and co-resolventQ(z) =e W^HW−zIN−1

. We express the result in the following lemma.

Lemma 3.1. Let(AN)N≥1 be a sequence of deterministicM L×M Lmatrices and (GN)N≥1 a uniformly bounded sequence of deterministic N×N matrices. Then

Var 1

M LtrANQ(z)

≤C(z) M N

1

M Ltr(ANA^H_N) (3.1) Var

1

M LtrANQ(z)WGW^H

≤C(z) M N

1

M Ltr(ANA^H_N) (3.2) whereC(z) =P1(|z|)P2(1/δz)for two nice polynomialsP1,P2(see Definition 1.2).

We devote the rest of this section to proving of this result. In order to short the notations, matrices AN and GN will be denoted by A and G. We will be using the Poincar´e-Nash inequality ([6], [5]), which, in the present context, can be formulated as follows ( [15, 10]).

Lemma 3.2. Letξ=ξ(W,W^∗)denote aC¹complex function such that both itself and its derivatives are polynomically bounded. Under the above assumptions, we can write

Varξ≤E X

m,i1,i2,j1,j2

∂ξ

∂ W_i^m

1,j1

∗

!∗

Eh

W^m_i1,j1 W^m_i2,j2∗i ∂ξ

∂ W^m_i

2,j2

∗

+E X

m,i1,i2,j1,j2

∂ξ

∂ W^m_i

1,j1

Eh W^m_i

1,j1 W^m_i

2,j2

∗i ∂ξ

∂W^m_i

2,j2

!∗

whereW^m_i,j is the ((m−1)L+i, j)th entry ofW.

We just check that the first term, denotedβ, on the right hand side of the upper bound of Varξis in accordance with the results claimed in Lemma 3.1 for ξ= _ML¹ Tr(AQ(z)) andξ= _ML¹ Tr(AQ(z)WGW^H)). For this, we establish that it is possible to be back to the case where the spectral densities (S^m(ν))m=1,...,M

all coincide with 1 which is covered by the results of [12]. More precisely, given the Hankel structure of the matricesW_m, we can state that

Eh

W^m_i1,j1 W_i^m_2,j2∗i

= 1

Nrm(i1−i2+j1−j2). (3.3) Using thatrm(i1−i2+j1−j2) =R1

0 e^{2πi(i1−i2+j1−j2)ν}S^m(ν)dν, we obtain imme- diately thatβ can be written asβ =E(α) whereαis defined by

α= 1 N

Z 1 0

XM

m=1

S^m(ν)

X

i2,j2

∂ξ

∂ W^m_i

2,j2

∗e^{−2πi(i2+j2)ν}

2

dν

Thus, (1.8) implies thatβ≤Cβ, where ˜˜ β =E(˜α) where ˜αis defined by

˜ α= 1

N Z 1

0

XM m=1

X

i2,j2

∂ξ

∂ W^m_i

2,j2

∗ e^{−2πi(i2+j2)ν}

2

dν

and whereC is a nice constant. It is clear that ˜αcoincides αwhenSm(ν) = 1 for eachm= 1, . . . , M and each ν ∈[0,1]. Whenξ= _ML¹ Tr(AQ(z)), it is proved in [12] that

˜ α≤ 1

M N 1

M Ltr QAQWW^HQ^HA^HQ^H .

As it holds thatQWW^H =I+zQand thatkQk ≤ _δ¹_z, we obtain that QWW^HQ^H≤ 1

δz

1 +|z|

δz

I_ML. Therefore,

˜ α≤ 1

δz

1 + |z|

δz

1 M N

1

M Ltr(QAA^HQ^H) and using againkQk ≤δ⁻_z¹,

β˜=E(˜α)≤ 1 δ³_z

1 + |z|

δz

1 M N

1

M Ltr(AA^H).

The conclusion follows from the observation that 1

δ_z³

1 +|z| δz

≤ 1

δ_z³ + 1 δ⁴_z

(|z|+ 1).

As for the caseξ=_ML¹ Tr(AQ(z)WGW^H), we refer to upper bound of the term equivalent to ˜αexpressed in Eq. (3.12-3.13) in [12], and omit further details.

4. Expectation of resolvent and co-resolvent

In this section, we analyze the expectation of the resolventQ(z) =

WW^H−zIML

−1

and co-resolventQ(z) =e W^HW−zIN

−1

. As a previous step, we need to ensure the properties of certain useful matrix valued functions. This is summarized in the following lemma.

Lemma 4.1. For z ∈C\R⁺, the matrix I_N +cNΨ (EQ(z)) is invertible, so that we can define

R(z) =e −1

z I_N+cNΨ (EQ(z))−1

. (4.1)

On the other hand, the matrixI_ML+ Ψ e R^T(z)

is also invertible, and we define R(z) = −1

z

I_ML+ Ψ e

R^T(z)−1

. (4.2)

Furthermore,R(z)e andR(z)are elements ofSN(R⁺)andSML(R⁺)respectively.

In particular, they are holomorphic onC\R⁺ and satisfy R(z)R^H(z)≤ I_ML

δ_z² , R(z)e Re^H(z)≤I_N

δ_z² (4.3)

Moreover, there exist two nice constants (see Definition 1.2) η andη˜such that R(z)R^H(z)≥ δ_z²

16(η²+|z|²)²IML (4.4) R(z)e Re^H(z)≥ δ_z²

16(˜η²+|z|²)²I_N. (4.5) Proof. Ifz∈R^−∗, the invertibility ofI_N+cNΨ (EQ(z)) is obvious. If z∈C⁺, it follows from the fact that

Im

I_N+cNΨ (EQ(z))

=cNΨ (EImQ(z))

and ImQ(z)>0. We now establish thatR(z) ande R(z) are elements of SN(R⁺) andS^ML(R⁺). By Proposition 1.3, we only need to prove that ImR(z)e ≥0,ImzR(z)e ≥ 0 when Imz >0, limy→+∞−iyRe(iy) =I_N, and similar properties for matrixR(z).

Clearly,

ImR(z) =e Re^H(z)

ImzIN+cNΨ (Im [zEQ(z)]) eR(z)>0 ImzRe(z) =cN|z|²Re^H(z)

Ψ (ImEQ(z)) eR(z)>0 whereas, noting thatEQ(iy)→0 asy→+∞, we see that

−iyRe(iy) = I_N+cNΨ (EQ(iy))−1

→I_N as y →+∞. In order to justify that IML+ Ψ

e R^T(z)

is invertible, we remark that Im

I_ML+ Ψ e R^T(z)

coincides with Ψ

Im(Re^T(z))

which is positive def- inite because ImR(z)e >0 (see (2.5)). Therefore, Im

IML+ Ψ

Re^T(z)

>0 and I_ML+ Ψ

e R^T(z)

is invertible. Finally, observing that ImR(z) =R^H(z)h

ImzIML+ Ψ Imh

zRe^T(z)ii

R(z)>0 ImzR(z) =|z|²R^H(z)h

Ψ

ImRe^T(z)i

R(z)>0 together with the fact that, sinceR(iy)e →0 asy→ ∞,

−iyR(iy) =

I_ML+ Ψ e

R^T(iy)−1

→I_ML

We eventually establish (4.4), and omit the proof of (4.5). For this, we notice that R(z) is a block-diagonal matrix, and that measureνdefined byR(z) =R

R⁺ dν(λ)

λ−z

is block diagonal as well. In order to establish (4.4), it is thus sufficient to prove that for each unit-normL-dimensional vectorb, it holds that

b^HR^m,m(z)(R^m,m(z))^Hb≥ δ²_z

16(η²+|z|²)² (4.6)

for some nice constantη(of course independent onmandb). For this, we remark that

b^HR^m,m(z)(R^m,m(z))^Hb≥b^HR^m,m(z)b²

We denoteξm the termξm(z) =b^HR^m,m(z)bwhich can be written as ξm(z) =

Z

R⁺

dµξm(λ) λ−z

where probability measureµξm is defined bydµξm(λ) =b^Hdν^m,m(λ)b. We claim that

|ξm(z)| ≥δz

Z

R⁺

b^Hdν^m,m(λ)b

|λ−z|² (4.7)

To justify this, we first remark that δz =|Im(z)| if Re(z) ≥0 and that δz =|z| if Re(z) ≤ 0. Next, we notice that |ξm(z)| ≥ |Im(ξm(z))| = |Im(z)|R

R⁺

dµ_ξm(λ)

|λ−z|²

whatever the sign of Re(z). Therefore, if Re(z)≥0, it holds that

|ξm(z)| ≥δz

Z

R⁺

dµξm(λ)

|λ−z|² If Re(z)≤0, Re(ξm(z)) =R

R⁺

λ−Re(z)

|λ−z|² dµξm(λ) verifies Re(ξm(z))≥ −Re(z)R

R⁺

dµ_ξm(λ)

|λ−z|² . Therefore, if Re(z)≤0,

|ξm(z)|²= (Im(ξm(z))²+(Re(ξm(z))²≥ |z|² Z

R⁺

dµξm(λ)

|λ−z|² 2

=δ_z² Z

R⁺

dµξm(λ)

|λ−z|² 2

Therefore, (4.7) holds. We now consider the family of probability measures {(b^H_Ndν^m,m_N (λ)bN)N≥1,m=1,...,M,kbN=1k}where we have mentioned the dependency ofbandν w.r.t.N. Using item (v) of Proposition 1.3 and hypothesis (1.8) , it is easily seen that

Z

R⁺

λb^H_Ndν^m,m_N (λ)bN =b^H_NΨ^m_L(IN)bN < C for some nice constantC. Therefore, it holds that

sup

N≥1,m=1,...,M,kbNk=1

Z

R⁺

λb^H_Ndν^m,m_N (λ)b_N <+∞

The family of probability measures is thus tight, and it exists a nice constant η such that

N≥1,m=1,...,M,inf kbNk=1b^H_Nν^m,m_N ([0, η])b_N >1/2 (4.8) We now use the obvious inequality

Z

R⁺

b^Hdν^m,m(λ)b

|λ−z|² ≥ Z η

0

b^Hdν^m,m(λ)b

|λ−z|² Ifλ∈[0, η], it is clear that|λ−z|²≤2(|z|²+η²), and that

Z η 0

b^Hdν^m,m(λ)b

|λ−z|² ≥ 1 4(|z|²+η²)

(4.7) eventually leads to

b^HR^m,m(z)(R^m,m(z))^Hb≥ |b^HR^m,m(z)b|²≥ δ_z² 16(|z|²+η²)² as expected.

In order to address the expectation of Q(z) and Qe(z) we will apply the integration by parts formula for the expectations of Gaussian functions, which is presented next.

Lemma 4.2. Letξ=ξ(W,W^∗)denote aC¹complex function such that both itself and its derivatives are polynomically bounded. Under the above assumptions, we can write

E W^m_i

1,j1ξ

= XL i2=1

XN j2=1

Eh W^m_i

1,j1 W^m_i

2,j2

∗i E

"

∂ξ

∂ W^m_i

2,j2

∗

#

whereW^m_i,j is the ((m−1)L+i, j)th entry ofW.

Proof. See [15, 10].

Consider the resolvent identity

zQ(z) =Q(z)WW^H−I_ML. (4.9)

Let w_k denote the kth column of matrix W, 1 ≤ k ≤ N. For an M L×M L matrix A, we recall that we denote as [A]^m1,m2 its (m1, m2)th block matrix (of sizeL×L) and as [A]^m_i1¹_,i^,m₂ ² the (i1, i2)th entry of its (m1, m2)th block. Applying the integration by parts formula in Lemma 4.2 and the identity in (3.3), we are able to write

E

Q(z)wkw^H_j ^m1,m2

i1,i2 = X

m3,i3

Eh

[Q(z)]^m_i1,i^1,m₃ ³W^m_i ³

3,k W^m_i ²

2,j

∗i

= XN

r=1

XL

i4=1

X

m3,i3

Eh W^m_i ³

3,k W^m_i ³

4,r

∗i E

"

∂[Q(z)]^m_i1¹_,i^,m₃ ³ W^m_i ²

2,j

∗

∂ W^m_i ³

4,r

∗

#

=− XN r=1

XM m3=1

XL i3=1

XL i4=1

rm3(k−r+i3−i4)

N Eh

Q(z)wrw^H_j m1,m2

i1,i2 [Q(z)]^m_i4,i^3,m₃ ³i +

XL i3=1

rm2(k−j+i3−i2)

N E[Q(z)]^m_i1,i^1,m₃ ².

Now, using the change of variablei=i4−i3 we can alternatively express E

Q(z)wkw^H_j m1,m2

i1,i2

=−L XN

r=1

XM

m3=1 LX−1

i=−L+1

rm3(k−r−i)

N Eh

Q(z)wrw^H_j ^m1,m2

i1,i2 τ(Q^m3,m3(z)) (i)i +

XL

i3=1

rm2(k−j+i3−i2)

N E[Q(z)]^m_i1¹_,i^,m₃ ²

where we recall that, for a given square matrixXof sizeR, the sequenceτ(X) (i) is defined in (2.3).

Using the definition of the operator Ψ^(m)_N and its averaged counterpart in (2.6), we may reexpress the above equation as

E

Q(z)wkw^H_j ^m1,m2

i1,i2 =−cNEh

Ψ (Q(z))W^TQ^T(z)e^m_i1¹ e^m_i ²

2

T

W^∗i

k,j (4.10) +

XL i3=1

rm2(k−j+i3−i2)

N Eh

(Q(z))^m_i1¹_,i^,m₃ ²i

where we recall that cN = ^ML_N . From (4.10) and using the definition of Ψ(·), we may generally write, for anyN×N deterministic matrixA

Eh

Q(z)WAW^Hi

=−cNEh

Q(z)WΨ^T(Q(z))AW^Hi +E

Q(z)Ψ A^T

. (4.11) Let us now consider the co-resolvent, namely Qe(z) = W^HW−zI_N−1

, together with the co-resolvent identity

zQ(z) =e Q(z)We ^HW−IN.

Observe that we can writeQ(z)We ^HW=W^HQ(z)Wand therefore Eh

Q(z)We ^HWi

j,k=E

W^HQ(z)W

j,k=Etr

Q(z)wkw^H_j .

Hence, using the expression for the expectation of the resolvent in (4.10), we can obtain

Eh

Q(z)We ^HW i

j,k=Etr

Q(z)wkw^H_j

=−cNEh

Q(z)We ^HWΨ^T(Q(z))i

j,k

+ XM m1=1

XL

i1=1

XL

i3=1

rm1(k−j+i3−i1)

N Eh

(Q(z))^m_i1¹_,i^,m₃ ¹i The second term can be further simplified by applying the change of variables i1=i+i3, namely

Eh

Q(z)We ^HWi

=−cNEh

Q(z)We ^HWΨ^T(Q(z))i

+cNΨ^T(EQ(z))

and therefore, by the resolvent’s identity, Eh

Q(z)e i

=−1

zI_N−cNEh

Q(z)Ψe ^T(Q(z))i .

Now, replacing Q(z) in the above equation by Q(z) = EQ(z) +Q^◦(z) (where X^◦=X−EX) we see that

Eh e Q(z)i

=Re(z) +zcNEh e

Q(z)Ψ^T(Q^◦(z))Re(z)i

whereR(z) is defined in (4.1). On the other hand, particularizing the equation ine (4.11) to the case A =R(z) and using the resolvent’s identity in (4.9), we alsoe obtain

E[Q(z)] =R(z)−zcNEh

Q(z)WΨ^T(Q^◦(z))R(z)We ^HR(z)i

whereR(z) is defined in (4.2). With this, we have arrived at the two fundamental equations, which are summarized in what follows:

E[Q(z)]−R(z) =∆(z), Eh Q(z)e i

−R(z) =e ∆(z)e where the error terms are defined as

∆(z) =−zcNEh

Q(z)WΨ^T(Q^◦(z))R(z)We ^HR(z)i

(4.12)

∆(z) =e zcNEh

Q(z)Ψe ^T(Q^◦(z))R(z)e i

. (4.13)

4.1. Control of the errors

We develop here a control on certain functionals of the error term ∆(z). In this section, we establish the following result.

Proposition 4.3. For each deterministic sequence ofM L×M Lmatrices(AN)N≥1

satisfyingsup_NkA_Nk< a <+∞, it holds that

1

M Ltr [AN∆(z)]

≤a C(z) L

M N (4.14)

where C(z) = P1(|z|)P2(1/δz) for some nice polynomials P1 and P2 (see Defini- tion 1.2) that do not depend on sequence (AN)N≥1. Moreover, if(b1,N)N≥1 and (b2,N)N≥1 are 2 sequences ofL dimensional vectors such thatsup_Nkbi,Nk< b <

+∞ for i = 1,2, and if ((dm,N)m=1,...,M)N≥1 are deterministic complex number verifying sup_N,m|dm,N|< d <+∞, then, it holds that

b^H_1,N 1 M

XM m=1

dm,N∆^m,m(z)

! b_2,N

≤d b²C(z)L^3/2

M N (4.15)

where C(z) is defined as above, where the nice polynomials P1 and P2 do not depend on(b1,N)N≥1,(b2,N)N≥1 and((dm,N)m=1,...,M)N≥1

We first establish (4.14), and denoteA_N,(bi,N)i=1,2 and (dm,N)m=1,...,M by A,(bi)i=1,2and (dm)m=1,...,M in order to simplify the notations. We denote byξ the term _ML¹ tr [A∆(z)], and expressξas

ξ= 1 M Ltr

Ψ^T(Q^◦)RWe ^HRAQW .

Using (2.7), we obtain immediately that ξ=

NX−1

n=−(N−1) LX−1

l=−(L−1)

τ^(M)(Q^◦(R(n−l)⊗IL)) (l)τ

RWe ^HRAQW (n).

Therefore,E(ξ) is equal to E(ξ) =

NX−1

n=−(N−1) LX−1

l=−(L−1)

Eh

τ^(M)(Q^◦(R(n−l)⊗IL)) (l)τ

RWe ^HRAQW◦

(n)i . Using the definition of operatorsτ andτ^(M), the Schwartz inequality, Lemma 3.1, and the inequalityJ^l_LJ^l(H)_L ≤I_L, we obtain that

Eh

τ^(M)(Q^◦(R(n−l)⊗I_L)) (l)τ e

RW^HRAQW◦

(n)i

≤C(z) 1 M N

1

M LTr(R(n−l)R^H(n−l)⊗I_L) 1/2

.

Therefore, it holds that

|E(ξ)| ≤ C(z) M N

NX−1

n=−(N−1) LX−1

l=−(L−1)

1 M

XM m=1

|rm(n−l)|²

!^1/2

and that

|E(ξ)| ≤C(z) L M N

X

n∈Z

1 M

XM m=1

|rm(n)|²

!^1/2 . Condition (1.7) thus implies (4.14).

In order to establish (4.15), we denote byηthe left hand side of (4.15), which can be written as

η= 1

Mtr ∆(D⊗b₂b^H₁)

whereDrepresents theM×M diagonal matrix with diagonal entriesd1, . . . , dM. Using the above calculations, we obtain thatE(η) can be expressed as

E(η) =L

NX−1

n=−(N−1) LX−1

l=−(L−1)

Eh

τ^(M)(Q^◦(R(n−l)⊗I_L)) (l)τ e

RW^HR(D⊗b₂b^H₁ )QW◦

(n)i .

We use again the Schwartz inequality to evaluateEh

τ^(M)(Q^◦(R(n−l)⊗I_L)) (l)τ e

RW^HR(D⊗b₂b^H₁ )QW◦

(n)i together with (3.2) forA=R(D⊗b₂b^H₁ ), and obtain that

E τ

e

RW^HR(D⊗b₂b^H₁ )QW◦

(n)

2

≤b²C(z) M N

1

M Ltr(DD^H⊗b₂b^H₂ ) The term _ML¹ tr(DD^H⊗b2b^H₂) can also be written as

1

M Ltr(DD^H⊗b₂b^H₂ ) = 1 L

1 M

XM m=1

|dm|²

! kb₂k²

and is thus bounded by ^b2_L^d2. Therefore, the Schwartz inequality leads to

Eh

τ^(M)(Q^◦(R(n−l)⊗I_L)) (l)τ e

RW^HR(D⊗b₂b^H₁ )QW◦

(n)i≤ b²d C(z) 1

M N√ L

1 M

XM m=1

|rm(n−l)|²

!^1/2

. (4.16) Using the same approach as above, we immediately obtain (4.15).

Corollary 4.4. It holds that

kΨ(∆)k ≤C(z)L^3/2

M N (4.17)

(4.17) follows immediately from kΨ(∆)k ≤ sup

ν∈[0,1]

1 M

XM m=1

S^m(ν)|a^H_L(ν)∆^m,ma_L(ν)|

and from the application of (4.15) to the caseb₁=b₂=a_L(ν) anddm=Sm(ν).

5. The deterministic equivalents.

As E(Q(z))−R(z) converges towards zero in some appropriate sense, (4.1) and (4.2) suggest that it is reasonable to expect that E(Q(z)) behaves as the first componentT(z) of the solution (T(z),T(z)) of the so-called canonical equatione

T(z) =−1 z

IML+ Ψ

Te^T(z)−1

(5.1) T(z) =e −1

z

I_N +cNΨ^T(T(z))−1

. (5.2)

In the following, we establish that the canonical equation has a unique solution.

More precisely: