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The empirical distribution of the eigenvalues of a Gram matrix with a given variance profile
W. Hachem
a, P. Loubaton
b,c, J. Najim
d,∗aSupélec (Ecole Supérieure d’Electricité), Plateau de Moulon, 3, rue Joliot-Curie, 91192 Gif Sur Yvette cedex, France bIGM LabInfo, UMR 8049, Institut Gaspard Monge, Université de Marne La Vallée, France
c5, Bd Descartes, Champs sur Marne, 77454 Marne La Vallée cedex 2, France dCNRS, Télécom Paris, 46, rue Barrault, 75013 Paris, France
Received 3 November 2004; received in revised form 12 September 2005; accepted 7 October 2005 Available online 27 March 2006
Abstract
Consider aN×nrandom matrixYn=(Yijn)where the entries are given byYijn= σ (i/N,j/n)√
n Xijn, theXnij being centered i.i.d.
andσ:[0,1]2→Rbeing a function whose square is continuous and called a variance profile. Consider now a deterministicN×n matrixΛn=(Λnij)whose off-diagonal entries are zero. Denote byΣnthe non-centered matrixYn+Λnand byN∧n=min(N, n).
Then under the assumption that limn→∞Nn =c >0 and 1
N∧n N∧n
i=1 δ( i
N∧n,(Λnii)2) −→
n→∞H (dx,dλ),
whereHis a probability measure, it is proved that the empirical distribution of the eigenvalues ofΣnΣnTconverges almost surely to a non-random probability measure. This measure is characterized in terms of its Stieltjes transform, which is obtained with the help of an auxiliary system of equations. This kind of results is of interest in the field of wireless communication.
©2006 Elsevier Masson SAS. All rights reserved.
Résumé
SoitYn=(Yijn)une matriceN×ndont les entrées sont données parYijn=σ (i/N,j/n)√
n Xnij, lesXnijétant des variables aléatoires centrées, i.i.d. et oùσ:[0,1]2→Rest une fonction de carré continu qu’on appelera profil de variance. Considérons une matrice déterministeΛn=(Λnij)de dimensionsN×ndont les éléments non diagonaux sont nuls. AppelonsΣnla matrice non centrée définie parΣn=Yn+Λnet notonsN∧n=min(N, n). Sous les hypothèses que limn→∞Nn =c >0 et que
1 N∧n
N∧n
i=1
δ(Ni∧n,(Λnii)2) −→
n→∞H (dx,dλ),
oùHest une probabilité, on démontre que la mesure empirique des valeurs propres deΣnΣnTconverge presque sûrement vers une mesure de probabilité déterministe. Cette mesure est caractérisée par sa transformée de Stieltjes, qui s’obtient à l’aide d’un système auxiliaire d’équations. Ce type de résultats présente un intérêt dans le domaine des communications numériques sans fil.
©2006 Elsevier Masson SAS. All rights reserved.
* Corresponding author.
E-mail addresses:walid.hachem@supelec.fr (W. Hachem), loubaton@univ-mlv.fr (P. Loubaton), najim@tsi.enst.fr (J. Najim).
0246-0203/$ – see front matter ©2006 Elsevier Masson SAS. All rights reserved.
doi:10.1016/j.anihpb.2005.10.001
MSC:primary 15A52; secondary 15A18, 60F15
Keywords:Random matrix; Empirical distribution of the eigenvalues; Stieltjes transform
1. Introduction
Consider aN×nrandom matrixYn=(Yijn)where the entries are given by
Yijn=σ (i/N, j/n)
√n Xijn, (1.1)
whereσ:[0,1] × [0,1] → [0,∞)is a function whose square is continuous called a variance profile and the random variablesXnij are real, centered, independent and identically distributed (i.i.d.) with finite 4+ moment. Consider a real deterministicN×nmatrixΛn=(Λnij)whose off-diagonal entries are zero and letΣn be the matrix Σn= Yn+Λn. This model has two interesting features: The random variables are independent but not i.i.d. since the variance may vary andΛn, the centering perturbation ofYn, though (pseudo) diagonal can be of full rank. The purpose of this article is to study the convergence of the empirical distribution of the eigenvalues of the Gram random matrix ΣnΣnT(ΣnTbeing the transpose ofΣn) whenn→ +∞andN→ +∞in such a way that Nn →c, 0< c <+∞.
The asymptotics of the spectrum ofN×N Gram random matricesZnZnT have been widely studied in the case whereZn is centered (see Marˇcenko and Pastur [15], Yin [23], Silverstein et al. [17,18], Girko [7,8], Khorunzhy et al. [13], Boutet de Monvel et al. [3], etc.). For an overview on asymptotic spectral properties of random matrices, see Bai [1]. The case of a Gram matrixZnZTn whereZnis non-centered has comparatively received less attention. Let us mention Girko ([8], Chapter 7) where a general study is carried out for the matrixZn=(Wn+An)whereWn has a given variance profile andAn is deterministic. In [8], it is proved that the entries of the resolvent(ZnZnT−zI )−1 have the same asymptotic behaviour as the entries of a certain deterministic holomorphicN ×N matrix valued functionTn(z). This matrix-valued function is characterized by a non-linear system of(n+N )coupled functional equations (see also [10]). Using different methods, Dozier and Silverstein [6] study the eigenvalue asymptotics of the matrix(Rn+Xn)(Rn+Xn)Tin the case where the matricesXn andRnare independent random matrices,Xnhas i.i.d. entries and the empirical distribution ofRnRnTconverges to a non-random distribution. It is proved there that the eigenvalue distribution of(Rn+Xn)(Rn+Xn)Tconverges almost surely towards a deterministic distribution whose Stieltjes transform is uniquely defined by a certain functional equation.
As in [6], the model studied in this article, i.e.Σn=Yn+Λn, is a particular case of the general case studied in ([8], Chapter 7, equationK7) for which there exists a limiting distribution for the empirical distribution of the eigenvalues.
Since the centering termΛn is pseudo-diagonal, the proof of the convergence of the empirical distribution of the eigenvalues is based on a direct analysis of the diagonal entries of the resolvent(ΣnΣnT−zI )−1. This analysis leads in a natural way to the equations characterizing the Stieltjes transform of the limiting probability distribution of the eigenvalues.
In the Wigner case with a variance profile, that is when matrixYn and the variance profile are symmetric (such matrices are also called band matrices), the limiting behaviour of the empirical distribution of the eigenvalues has been studied by Shlyakhtenko [16] in the Gaussian case (see Section 3.4 for more details).
Recently, many of these results have been applied to the field of Signal Processing and Communication Systems and some new ones have been developed for that purpose (Silverstein and Combettes [19], Tse et al. [20,21], Debbah et al. [5], Li et al. [14], etc.). The issue addressed in this paper is mainly motivated by the performance analysis of multiple-input multiple-output (MIMO) digital communication systems. In MIMO systems withntransmit antennas andN receive antennas, one can model the communication channel by aN×nmatrixHn=(Hijn)where the entries Hijn represent the complex gain between transmit antennaiand receive antennaj. The statisticsCn=n1log det(In+ HnHn/σ2)(where Hn is the hermitian adjoint and σ2represents the variance of an additive noise corrupting the received signals) is a popular performance analysis index since it has been shown in information theory thatCn is
the mutual information, that is the maximum number of bits per channel use and per antenna that can be transmitted reliably in a MIMO system with channel matrixHn. Since
Cn=1 n
N k=1
log
1+μk σ2
,
where(μk)1kN are the eigenvalues ofHnHn, the empirical distribution of the eigenvalues ofHnHn gives direct information onCn(see Tulino and Verdu [22] for an exhaustive review of recent results). For wireless systems, matrix Hnis often modelled as a zero-mean Gaussian random matrix and several articles have recently been devoted to the study of the impact of the channel statistics (via the eigenvalues ofHnHn) on the probability distribution ofCn(Chuah et al. [4], Goldsmith et al. [9], see also [22] and the references therein). Of particular interest is also the channel matrix Hn=FN(Yn+Λn)FnT whereFk=(Fpqk )1p,qk is the Fourier matrix (i.e.Fpqk =k−1/2exp(2iπ(p−1)(qk −1))) and the matrixYnis given by (1.1) (see [22], p. 139 for more details). The matricesHnandΣnhaving the same singular values, we will focus on the study of the empirical distribution of the singular values ofΣn. Moreover, we will focus on matrices with real entries since the complex case is a straightforward extension.
In the sequel, we will study simultaneously quantities (Stieltjes kernels) related to the Stieltjes transforms ofΣnΣnT andΣnTΣn. Even if the Stieltjes transforms ofΣnΣnT andΣnTΣn are related in an obvious way, the corresponding Stieltjes kernels are not, as we shall see. We will prove that ifN/n−→
n→∞c >0 (since we study at the same timeΣnΣnT andΣnTΣn, we assume without loss of generality thatc1) and if there exists a probability measureH on[0,1] ×R with compact supportHsuch that
1 N
N i=1
δ(i/N,(Λn ii)2)
−→D
n→∞H (dx,dλ),
whereDstands for convergence in distribution, then almost surely, the empirical distribution of the eigenvalues of the random matrixΣnΣnT(resp.ΣnTΣn) converges in distribution to a deterministic probability distributionP(resp.P).
The probability distributionsPandPare characterized in terms of their Stieltjes transform f (z)=
R+
P(dx)
x−z and f (z)˜ =
R+
P(dx)
x−z, Im(z)=0 as follows. Consider the following system of equations
gdπz=
g(u, λ)
−z(1+
σ2(u,·)dπ˜z)+λ/(1+c
σ2(·, cu)dπz)H (du,dλ),
gdπ˜z=c
g(cu, λ)
−z(1+c
σ2(·, cu)dπz)+λ/(1+
σ2(u,·)dπ˜z)H (du,dλ) +(1−c)
1 c
g(u,0)
−z(1+c
σ2(·, u)dπz)du,
where the unknown parameters are the complex measuresπzandπ˜zand whereg:H→Ris a continuous test func- tion. Then, this system admits a unique pair of solutions (πz(dx,dλ),π˜z(dx,dλ)). In particular, πz is absolutely continuous with respect toH whileπ˜z is not (see Section 2 for more details). The Stieltjes transformsf andf˜are then given by
f (z)=
[0,1]×R
πz(dx,dλ) and f (z)˜ =
[0,1]×R
˜
πz(dx,dλ).
The article is organized as follows. In Section 2, the notations and the assumptions are introduced and the main result (Theorem 2.3) is stated. Section 3 is devoted to corollaries and remarks. Section 4 is devoted to the proof of Theorem 2.3.
2. Convergence of the Stieltjes transform 2.1. Notations and assumptions
LetN=N (n)be a sequence of integers such that
nlim→∞
N (n) n =c.
Consider aN×nrandom matrixYnwhere the entries are given by Yijn=σ (i/N, j/n)
√n Xijn, whereXnijandσ are defined below.
Assumption A-1.The random variables(Xijn; 1iN, 1jn, n1)are real, independent and identically distributed. They are centered withE(Xnij)2=1 and satisfy:
∃ >0, EXijn4+<∞, whereEdenotes the expectation.
Remark 2.1.Using truncation arguments à la Bai and Silverstein [2,17,18], one may improve Assumption (A-1).
Assumption A-2.The real functionσ:[0,1] × [0,1] →Ris such that σ2is continuous. Therefore there exists a non-negative constantσmaxsuch that
∀(x, y)∈ [0,1]2, 0σ2(x, y)σmax2 <∞. (2.1)
Remark 2.2.The functionσcan vanish on portions of the domain[0,1]2.
Denote by var(Z)the variance of the random variableZ. Since var(Yijn)=σ2(i/N, j/n)/n, the functionσ will be called a variance profile. Denote byδz0(dz)the Dirac measure at pointz0. The indicator function ofAis denoted by 1A(x). Denote byCb(X)(resp.Cb(X;C))the set of real (resp. complex) continuous and bounded functions over the topological setX and byf∞=supx∈X|f (x)|, the supremum norm. IfX is compact, we simply writeC(X) (resp.C(X;C))instead ofCb(X)(resp.Cb(X;C)). We denote by−→D the convergence in distribution for probability measures and by−→w the weak convergence for bounded complex measures.
Consider a real deterministicN×nmatrixΛn=(Λnij)whose off-diagonal entries are zero. We often writeΛij instead ofΛnij. We introduce theN×nmatrixΣn=Yn+Λn.
For every matrixA, we denote byATits transpose, by Tr(A)its trace (ifAis square) and byFAAT, the empirical distribution function of the eigenvalues ofAAT. Denote by diag(ai;1ik) the k×k diagonal matrix whose diagonal entries are theai’s. Since we will study at the same time the limiting spectrum of the matricesΣnΣnT and ΣnTΣn, we can assume without loss of generality thatc1. We also assume for simplicity thatNn.
We assume that:
Assumption A-3.There exists a probability measureH (du,dλ)over the set[0,1] ×Rwith compact supportHsuch that
Hn(du,dλ)≡ 1 N
N i=1
δ(i/N,(Λn
ii)2)(du,dλ)−→D
n→∞H (du,dλ). (2.2)
Remark 2.3 (The probability measure H). Assumption (A-3) accounts for the presence of probability measure H (du,dλ)in forthcoming Eqs. (2.6) and (2.7). IfΛ2ii=f (ni), then (A-3) is automatically fulfilled withH (du,dλ)= δf (x)(du)dx. This is in particular the case in Boutet de Monvel et al. [3], Schlyakhtenko [16], Hachem et al. ([11], Theorem 4.2).
Remark 2.4 (The complex case). Assumptions (A-1), (A-2) and (A-3) must be slightly modified in the complex setting. Related modifications are stated in Section 3.3.
When dealing with vectors, the norm · will denote the Euclidean norm. In the case of matrices, the norm · will refer to the spectral norm.
Remark 2.5(Boundedness of theΛii’s).Due to (A-3), we can assume without loss of generality that theΛnii’s are bounded fornlarge enough. In fact, suppose not, then by (A-3), N1 Ni=1δΛ2
ii→HΛ(dλ)whose support is compact and, say, included in[0, K]. Then Portmanteau’s theorem yieldsN1 N1 1[0, K+δ](Λ2ii)→1 thus
#{Λii, Λ2ii∈ [0, K/ +δ]}
N =1− 1
N N
1
1[0, K+δ] Λ2ii
n−→→∞0. (2.3)
Denote by Λˇn=(Λˇnij)the matrix whose off-diagonal entries are zero and set Λˇnii =Λnii1{(Λn
ii)2K+δ}. Then it is straightforward to check that N1 Ni=1δ(i/N,Λˇ2
ii)(du,dλ)→H (du,dλ). Moreover, ifΣˇn=Yn+ ˇΛnthen FΣ ΣT−FΣˇΣˇT
∞
(a) rank(Σˇ −Σ ) N
(b)#{Λii, Λ2ii∈ [/ 0, K+δ]}
N
−→(c) n→∞0,
where (a) follows from Lemma 3.5 in [23] (see also [18], Section 2), (b) follows from the fact that for a rectangular matrixA, rank(A)the number of non-zero entries ofAand (c) follows from (2.3). Therefore,FΣ ΣT converges iff FΣˇΣˇT converges. In this case they share the same limit. Remark 2.5 is proved.
Remark 2.6(Compacity of the support ofHn).Due to Remark 2.5, we will assume in the sequel that for alln, the support of N1 δ(i/N,Λ2
ii)is included in a compact setK⊂ [0,1] ×R.
LetC+= {z∈C, Im(z) >0}andC∇= {z∈C+, |Re(z)|Im(z)}. 2.2. Stieltjes transforms and Stieltjes kernels
Letνbe a bounded non-negative measure overR. Its Stieltjes transformf is defined by:
f (z)=
R
ν(dλ)
λ−z, z∈C+.
We list below the main properties of the Stieltjes transforms that will be needed in the sequel.
Proposition 2.1.The following properties hold true:
(1) Letf be the Stieltjes transform ofν, then – the functionf is analytic overC+, – the functionf satisfies:|f (z)|Im(z)ν(R), – ifz∈C+thenf (z)∈C+,
– ifν(−∞,0)=0thenz∈C+implieszf (z)∈C+.
(2) Conversely, letf be a function analytic over C+ such that f (z)∈C+ if z∈C+ and |f (z)||Im(z)| bounded onC+. Then,f is the Stieltjes transform of a bounded positive measureμandμ(R)is given by
μ(R)= lim
y→+∞−iyf (iy).
If moreoverzf (z)∈C+forz∈C+then,μ(R−)=0.
(3) LetPnandPbe probability measures overRand denote byfnandf their Stieltjes transforms. Then ∀z∈C+, fn(z)−→
n→∞f (z)
⇒ Pn −→D n→∞P.
LetAbe ann×pmatrix and letInbe then×nidentity. The resolvent ofAATis defined by Q(z)=
AAT−zIn−1
= qij(z)
1i,jn, z∈C\R. The following properties are straightforward.
Proposition 2.2.LetQ(z)be the resolvent ofAAT, then:
(1) For allz∈C+,Q(z)(Im(z))−1. Similarly,|qij(z)|(Im(z))−1.
(2) The functionhn(z)=1nTrQ(z)is the Stieltjes transform of the empirical distribution probability associated to the eigenvalues ofAAT. Since these eigenvalues are non-negative,zhn(z)∈C+forz∈C+.
(3) Letξbe an×1vector, thenIm(zξ Q(z)ξT)∈C+forz∈C+.
Denote byMC(X)the set of complex measures over the topological setX. In the sequel, we will call Stieltjes kernel every application
μ:C+→MC(X)
either denotedμ(z,dx)orμz(dx)and satisfying:
(1) ∀g∈Cb(X),
gdμzis analytic overC+, (2) ∀z∈C+,∀g∈Cb(X),
gdμz
g∞ Im(z),
(3) ∀z∈C+,∀g∈Cb(X)andg0 then Im(
gdμz)0, (4) ∀z∈C+,∀g∈Cb(X)andg0 then Im(z
gdμz)0.
Let us introduce the following resolvents:
Qn(z)=
ΣnΣnT−zIN−1
= qij(z)
1i,jN, z∈C+, Qn(z)=
ΣnTΣn−zIn−1
=
˜ qij(z)
1i,jn, z∈C+,
and the following empirical measures defined forz∈C+by (recall thatNn) Lnz(du,dλ)= 1
N N i=1
qii(z)δ(i/N,Λ2
ii)(du,dλ), (2.4)
L˜nz(du,dλ)=1 n
N i=1
˜
qii(z)δ(i/n,Λ2
ii)(du,dλ)+ 1
n n i=N+1
˜
qii(z) δi/n(du)⊗δ0(dλ)
1{N <n}, (2.5)
where⊗denotes the product of measures. Sinceqii(z)(resp.q˜ii(z)) is analytic overC+, satisfies|qii(z)|(Im(z))−1 and min(Im(qii(z)),Im(zqii(z))) >0,Ln(resp.L˜n) is a Stieltjes kernel. Recall that due to Remark 2.6,Ln andL˜n have supports included in the compact setK.
Remark 2.7(on the limiting support ofLn).Consider a converging subsequence ofLnz, then its limiting support is necessarily included inH.
Remark 2.8(on the limiting support ofL˜n).Denote byHcthe image of the probability measureH under the appli- cation(u, λ)→(cu, λ), byHcits support, byRthe support of the measure 1[c,1](u)du⊗δ0(dλ). LetH=Hc∪R.
Notice thatHis obviously compact. Consider a converging subsequence ofL˜nz, then its limiting support is necessarily included inH.
2.3. Convergence of the empirical measuresLnz andL˜nz
Theorem 2.3.Assume that Assumptions(A-1),(A-2)and(A-3)hold and consider the following system of equations
gdπz=
g(u, λ)
−z(1+
σ2(u, t )π (z,˜ dt,dζ ))+λ/(1+c
σ2(t, cu)π(z,dt,dζ ))H (du,dλ), (2.6)
gdπ˜z=c
g(cu, λ)
−z(1+c
σ2(t, cu)π(z,dt,dζ ))+λ/(1+
σ2(u, t )π (z,˜ dt,dζ ))H (du,dλ) +(1−c)
1 c
g(u,0)
−z(1+c
σ2(t, u)π(z,dt,dζ ))du, (2.7)
where(2.6)and(2.7)hold for everyg∈C(H). Then,
(1) this system admits a unique couple of solutions(π(z,dt,dλ),π (z,˜ dt,dλ))among the set of Stieltjes kernels for which the support of measureπzis included inHand the support of measureπ˜zis included inH,
(2) the functionsf (z)=
dπzandf (z)˜ =
dπ˜zare the Stieltjes transforms of probability measures, (3) the following convergences hold true:
a.s.∀z∈C+, Lnz −→w
n→∞πz, a.s.∀z∈C+, L˜nz −→w
n→∞π˜z,
whereLnandL˜nare defined by(2.4)and(2.5).
Remark 2.9(on the absolute continuity ofπzandπ˜z).Due to (2.6), the complex measureπzis absolutely continuous with respect toH. However, it is clear from (2.7) thatπ˜zhas an absolutely continuous part with respect toHc(recall that Hc is the image ofH under(u, λ)→(cu, λ)) and an absolutely continuous part with respect to 1[c,1](u)du⊗ δ0(dλ)(which is in general singular with respect toHc). Therefore, it is much more convenient to work with Stieltjes kernelsπ andπ˜ rather than with measure densities indexed byz.
Proof of Theorem 2.3 is postponed to Section 4.
Corollary 2.4.Assume that(A-1),(A-2)and(A-3)hold and denote byπ and π˜ the two Stieltjes kernels solutions of the coupled equations (2.6) and (2.7). Then the empirical distribution of the eigenvalues of the matrix ΣnΣnT converges almost surely to a non-random probability measurePwhose Stieltjes transformf (z)=
R+ P(dx)
x−z is given by:
f (z)=
H
πz(dx,dλ).
Similarly, the empirical distribution of the eigenvalues of the matrixΣnTΣnconverges almost surely to a non-random probability measurePwhose Stieltjes transformf (z)˜ is given by:
f (z)˜ =
H˜
˜
πz(dx,dλ).
Proof of Corollary 2.4. The Stieltjes transform ofΣnΣnTis equal to N1 Ni=1qii(z)=
dLnz. By Theorem 2.3(3), a.s.∀z∈C+,
dLnz −→
n→∞
dπz. (2.8)
Since
dπz is the Stieltjes transform of a probability measurePby Theorem 2.3(2), Eq. (2.8) implies thatFΣnΣnT converges weakly toP. One can similarly prove thatFΣnTΣnconverges weakly to a probability measureP. 2
3. Further results and remarks
In this section, we present two corollaries of Theorem 2.3. We will discuss the case whereΛn=0 and the case where the variance profile σ (x, y)is constant. These results are already well-known [3,6–8]. We also show how Assumptions (A-1), (A-2) and (A-3) must be modified in the complex case. Finally, we give some comments on Shlyakhtenko’s result [16].
3.1. The centered case
Corollary 3.1.Assume that(A-1)and (A-2)hold. Then the empirical distribution of the eigenvalues of the matrix YnYnTconverges a.s. to a non-random probability measurePwhose Stieltjes transformf is given by
f (z)=
[0,1]
πz(dx),
whereπzis the unique Stieltjes kernel with support included in[0,1]and satisfying
∀g∈C [0,1]
,
gdπz= 1 0
g(u)
−z+1
0σ2(u, t )/(1+c1
0σ2(x, t )πz(dx))dtdu. (3.1) Remark 3.1(Absolute continuity of the Stieltjes kernel).In this case,πzis absolutely continuous with respect to du, i.e. πz(du)=k(z, u)du. Furthermore, one can prove that z→k(z, u) is analytic and u→k(z, u) is continuous.
Eq. (3.1) becomes
∀u∈ [0,1], ∀z∈C+, k(u, z)= 1
−z+1
0σ2(u, t )/(1+c1
0σ2(x, t )k(x, z)dx)dt
. (3.2)
Eq. (3.2) appears (up to notational differences) in [7] and in [3] in the setting of Gram matrices based on Gaussian fields. Links between Gram matrices with a variance profile and Gram matrices based on Gaussian fields are studied in [11].
Proof. Assumption (A-3) is satisfied withΛnii=0 andH (du,dλ)=du⊗δ0(λ)where dudenotes Lebesgue measure on[0,1]. Therefore Theorem 2.3 yields the existence of kernelsπzandπ˜zsatisfying (2.6) and (2.7). It is straightfor- ward to check that in this caseπzandπ˜zdo not depend on variableλ. Therefore (2.6) and (2.7) become:
gdπz=
g(u)
−z(1+
σ2(u, t )π (z,˜ dt ))du (3.3)
and
gdπ˜z=c
[0,1]
g(cu)
−z(1+c
σ2(t, cu)π(z,dt ))du+(1−c)
[c,1]
g(u)
−z(1+c
σ2(t, u)π(z,dt ))du
=
[0,1]
g(u)
−z(1+c
σ2(t, u)π(z,dt ))du, (3.4)
whereg∈C([0,1]). Replacing
σ2(u, t )π (z,˜ dt )in (3.3) by the expression given by (3.4), one gets the following equation satisfied byπz(du):
gdπz=
g(u)
−z+
σ2(u, t )/(1+c
σ2(s, t )π(z,ds))dtdu. 2
3.2. The non-centered case with i.i.d. entries
Corollary 3.2.Assume that(A-1)and (A-2)hold whereσ (x, y)=σ is a constant function. Assume moreover that
1 N
N i=1δΛ2
ii→HΛ(dλ)weakly, whereHΛhas a compact support. Then the empirical distribution of the eigenvalues of the matrixΣnΣnTconverges a.s. to a non-random probability measurePwhose Stieltjes transform is given by
f (z)=
HΛ(dλ)
−z(1+cσ2f (z))+(1−c)σ2+λ/(1+cσ2f (z)). (3.5) Remark 3.2.Eq. (3.5) appears in [6] in the case whereΣn=σ Zn+Rn whereZn andRn are assumed to be inde- pendent,Zijn =Xij/√
n, theXij being i.i.d. and the empirical distribution of the eigenvalues ofRnRnTconverging to a given probability distribution. SinceRn is not assumed to be diagonal in [6], the results in [6] do not follow from Corollary 3.2.
Proof. We first sort theΛ2ii’s. Denote byONtheN×Npermutation matrix such thatONdiag(Λii; 1iN )ONT= diag(Λ˜ii; 1iN )whereΛ˜211· · ·Λ˜2N N. ThenONΣnΣnTONT andΣnΣnT have the same eigenvalues. Denote byΛ˜nthe pseudo-diagonalN×nmatrix whose diagonal entries are theΛ˜iiand byOˇnthe block matrix:
Oˇn=
ON 0 0 In−N
. Then
ONΣnOˇnT=ON(Yn+Λn)OˇnT=ONYnOˇnT+ ˜Λn.
In particularONYnOˇnTremains a matrix with i.i.d. entries. Therefore, we can assume without loss of generality that:
Λ211· · ·Λ2N N.
In this case, one can prove that the empirical distribution N1 Ni=1δi/N,Λ2
ii converges. In fact, denote byFΛ(y)= HΛ([0, y])and consider the function
F (x, y)=x∧FΛ(y), (x, y)∈ [0,1] ×R+,
where∧denotes the infimum. Assume thatFΛis continuous aty, then:
1 N
N i=1
δi/N,Λ2 ii
[0, x] × [0, y]
= #{i, i/Nx andΛ2iiy} N
= card(An∩Bn)
N whereAn= {i, i/Nx}andBn=
i, Λ2iiy
(a)= card(An)
N ∧card(Bn)
N −→
n→∞x∧FΛ(y),
where (a) follows from the fact thatAn∩Bnis either equal toAnifAn⊂Bnor toBnifBn⊂Andue to the fact that theΛ2ii are sorted. The probability measureH associated to the cumulative distribution functionF readily satisfies (A-3). Theorem 2.3 yields the existence of kernelsπzandπ˜zsatisfying (2.6) and (2.7). It is straightforward to check that in this caseπzandπ˜zdo not depend on variableu. Eq. (2.6) becomes
gdπz=
g(u, λ)
−z(1+σ2
˜
π (z,dt,dζ ))+λ/(1+cσ2
π(z,dt,dζ ))H (du,dλ).
Letg(u, λ)=1 and denote byf =
dπ, then (2.6) becomes f (z)=
1
−z(1+σ2f (z))˜ +λ/(1+cσ2f (z))HΛ(dλ). (3.6)
Denote by f˜n(z)=1
n n
1
˜
qii(z)=1 nTr
ΣnTΣn−zI−1
.
Sincefn(z)=N1 Tr(ΣnΣnT−zI )−1andf˜n(z)=1nTr(ΣnTΣn−zI )−1(recall thatNn), we havef˜n(z)=Nnfn(z)+ (1−Nn)(−1z). This yieldsf (z)˜ =cf (z)−1−zc. Replacingf (z)˜ in (3.6) by this expression, we get (3.5). 2 3.3. Statement of the results in the complex case
In the complex setting, Assumptions (A-1)–(A-3) must be modified in the following way:
Assumption A-1.The random variables(Xnij; 1iN, 1j n, n1)are complex, independent and identi- cally distributed. They are centered withE|Xijn|2=1 and satisfy:
∃ >0, EXijn4+<∞.
Assumption A-2.The complex functionσ:[0,1] × [0,1] →Cis such that|σ|2is continuous and therefore there exists a non-negative constantσmaxsuch that
∀(x, y)∈ [0,1]2, 0σ (x, y)2σmax2 <∞. (3.7)
IfΛnis a complex deterministicN×nmatrix whose off-diagonal entries are zero, assume that:
Assumption A-3.There exists a probability measureH (du,dλ)over the set[0,1] ×Rwith compact supportHsuch that
1 N
N i=1
δ(i/N,|Λn
ii|2)(du,dλ)−→D
n→∞H (du,dλ). (3.8)
In Eqs. (2.6) and (2.7), one must replaceσby its module|σ|. The statements of Theorem 2.3 and Corollary 2.4 are not modified.
3.4. Further remarks
The problem of studying the convergence of the empirical distribution ofΣnΣnTcould have been addressed differ- ently. In particular, one could have more relied on Shlyakhtenko’s ideas [16]. We give some details in this section. We also take this opportunity to thank the referee whose remarks led to this section.
An extension of Shlyakhtenko’s results. Using the concept of freeness with amalgamation [16], Shlyakhtenko de- scribes the spectral distribution of then×nmatrixMn=Λn+An whereΛn is a diagonal matrix whose entries are approximately samples of a bounded functionf on a regular grid, i.e.Λn is close to diag(f (ni); 1in)and Anis a Wigner matrix with a variance profile:Anij=σ (i/n,j/n)√
n Xij, theXij’s being i.i.d. (apart from the symmetry constraint) standard Gaussian random variables andσ being symmetric, i.e.σ (x, y)=σ (y, x). If instead ofMn, one can describe the limiting spectral distribution of
Mn=
0 Σn ΣnT 0
=
0 Yn+Λn YnT+ΛTn 0
,
then one can also describe the limiting spectral distribution ofΣnΣnTsince Mn2=
ΣnΣnT 0 0 ΣnTΣn
as noticed by Shlyakhtenko. It is not clear however how to adapt the concept of freeness with amalgamation to the study ofMnsince the deterministic elements ofMnare not diagonal entries any more.
Apart from this, two issues remain with this approach: The extension of the result to the non-Gaussian case and to the case where the diagonal deterministic entries are not samples of a bounded function (which is a case covered by Assumption (A-3)).
The direct study of a Wigner matrix with a variance profile. Another approach is based on Shlyakhtenko’s re- mark together with the technique developed in this paper: Since it is sufficient to study the spectral measure of Mn in order to get the result for the spectral measure ofMn2, one can directly study the diagonal elements of the resolvent (Mn−zIn)−1. This approach yields to results expressed in terms of the limiting Stieltjes transform of
1
nTr(Mn−zIn)−1. 4. Proof of Theorem 2.3
We first give an outline of the proof.
4.1. Outline of the proof, more notations
The proof is carried out following four steps:
(1) Recall the definitions of the supportsHandH(cf. Assumption (A-3) and Remark 2.8). We first prove that the system of Eqs. (2.6) and (2.7) admits at most a unique couple of solutions(π(z,dt,dλ),π (z,˜ dt,dλ)) among the set of Stieltjes kernels for which the support of measureπz is included inHand the support of measureπ˜z is included inH(Section 4.2). We also prove that if such solutions exist, then necessarily,f (z)=
dπz and f (z)˜ =
dπ˜zare Stieltjes transforms of probability measures.
(2) We prove that for each subsequenceM(n)ofnthere exists a subsequenceMsub=Msub(n)such that
∀z∈C+, LMzsub −→w
n→∞μz and L˜Mzsub −→w
n→∞μ˜z, (4.1)
whereμzandμ˜zare complex measures, a priori depending onω∈Ω (ifΩ denotes the underlying probability space), with support respectively included inHandH(Section 4.3). Note that the convergence of the subse- quences stands everywhere (for everyω∈Ω).
(3) We then prove thatz→μz andz→ ˜μz are Stieltjes kernels (Section 4.4). As in the previous step, this result holds everywhere.
(4) We finally consider a countable collection C of z∈C+ with a limit point. The asymptotic behaviour of the diagonal entryqii(z)of the resolvent(ΣnΣnT−zI )−1is established almost surely (Lemma 4.1) overC(one can check that the set of probability one related to this almost sure property does not depend on any subsequence ofLn orL˜n). We prove that the measuresμzandμ˜zsatisfy Eqs. (2.6) and (2.7) almost surely for allz∈C. Analyticity arguments yield then that almost surely,∀z∈C+,μzandμ˜zsatisfy Eqs. (2.6) and (2.7). Otherwise stated:
– Almost surely, the system (2.6), (2.7) admits a solution(μz,μ˜z),
– This solution being unique, there exists a couple of deterministic Stieltjes kernels(πz,π˜z)such that almost surely(μz,μ˜z)=(πz,π˜z).
– Finally since almost surely the limit is the same for every subsequence in (4.1), the convergence holds for the whole sequence:
Lnz −→w
n→∞πz and L˜nz −→w
n→∞π˜z. This will conclude the proof.
We introduce some notations. Denote by D(π˜z, πz)(u, λ)= −z
1+
σ2(u, t )π (z,˜ dt,dζ )
+ λ
1+c
σ2(t, cu)π(z,dt,dζ ),
