Table Of ContentLimiting eigenvectors of outliers for Spiked
7 Information-Plus-Noise type matrices
1
0
2 M. Capitaine∗
n
a
J
7
Abstract
2
We consider an Information-Plus-Noise type matrix where the Infor-
]
mation matrix is aspiked matrix. Whensome eigenvalues of therandom
R
matrix separate from the bulk, we study how the corresponding eigen-
P
vectors project onto those of the spikes. Note that, in an Appendix, we
.
h presentalternativeversionsoftheearlierresultsof[3](“noeigenvalueout-
t sidethesupportofthedeterministicequivalentmeasure”)and[11](“exact
a
m separation phenomenon”) where we remove some technical assumptions
that were difficult tohandle.
[
1
1 Introduction
v
9
6 In this paper, we consider the so-called Information-Plus-Noise type model
0
8 X
N
M =Σ Σ where Σ =σ +A ,
0 N N ∗N N √N N
.
1
0 defined as follows.
7
1 n=n(N), n N, cN =n/N N + c ]0;1].
• ≤ → → ∞ ∈
:
v σ ]0;+ [.
i • ∈ ∞
X
X = [X ] where X ,i N,j N is an infinite set of
N ij 1 i n;1 j N ij
ar • complex rand≤om≤ va≤ri≤ables such{that ∈(Xij),∈(Xi}j),i N,j N are
{ℜ ℑ ∈ ∈ }
independent centered random variables with variance 1/2 and satisfy
1. There exists K > 0 and a random variable Z with finite fourth mo-
ment for which there exists x > 0 and an integer number n > 0
0 0
such that, for any x > x and any integer numbers n ,n > n , we
0 1 2 0
have
1
P(X >x) KP(Z >x). (1)
ij
n n | | ≤ | |
1 2
i≤nX1,j≤n2
∗CNRS, Institut de Math´ematiques de Toulouse, F-31062 Toulouse Cedex 09. E-mail:
[email protected]
1
2.
sup E(X 3)<+ . (2)
ij
(i,j) N2 | | ∞
∈
Let ν be a compactly supported probability measure on R whose support
•
hasafinitenumberofconnectedcomponents. LetΘ= θ ;...;θ where
1 J
{ }
θ > ... > θ 0 are J fixed real numbers independent of N which
1 J
≥
are outside the support of ν. Let k ,...,k be fixed integer numbers
1 J
independent of N and r = J k . Let β (N) 0, r+1 j n, be
j=1 j j ≥ ≤ ≤
such that 1 n δ weakly converges to ν and
n j=r+1 βj(N) P
P max dist(β (N),supp(ν)) 0 (3)
j
r+1 j n N−→
≤ ≤ →∞
where supp(ν) denotes the support of ν.
Let α (N),j =1,...,J, be real nonnegative numbers such that
j
lim α (N)=θ .
j j
N +
→ ∞
Let A be a n N deterministic matrix such that, for each j =1,...,J,
N
×
α (N)isaneigenvalueofA A withmultiplicityk ,andtheothereigen-
j N ∗N j
values of A A are the β (N), r+1 j n. Note that the empirical
N ∗N j ≤ ≤
spectral measure of A A weakly converges to ν.
N ∗N
Remark 1.1. Note that assumption such as (1) appears in [14]. It obviously
holds if the X ’s are identically distributed with finite fourth moment.
ij
For any Hermitian n n matrix Y, denote by spect(Y) its spectrum, by
×
λ (Y) ... λ (Y)
1 n
≥ ≥
the ordered eigenvalues of Y and by µ the empirical spectral measure of Y:
Y
n
1
µ := δ .
Y n λi(Y)
i=1
X
For a probability measure τ on R, denote by g its Stieltjes transform defined
τ
for z C R by
∈ \
dτ(x)
g (z)= .
τ
R z x
Z −
When the X ’s are identically distributed, Dozier and Silverstein established
ij
in[15] thatalmostsurely the empiricalspectralmeasureµ of M converges
MN N
weaklytowardsanonrandomdistributionµ whichis characterizedinterms
σ,ν,c
of its Stieljes transform which satisfies the following equation: for any z C+,
∈
1
g (z)= dν(t). (4)
µσ,ν,c (1 σ2cg (z))z t σ2(1 c)
Z − µσ,ν,c − 1−σ2cgµσ,ν,c(z) − −
This result of convergence was extended to independent but non identically
distributedrandomvariablesbyXie in[32]. (Notethat, in[19],the authorsin-
2
vestigatedthecasewhereσ isreplacedbyaboundedsequenceofrealnumbers.)
In [11], the author carries on with the study of the support of the limiting
spectralmeasurepreviouslyinvestigatedin[16]andlaterin[29,25]andobtains
that there is a one-to-one relationship between the complement of the limiting
supportandsomesubsetinthecomplementofthesupportofν whichisdefined
in (6) below.
Proposition 1.2. Define differentiable functions ω and Φ on respec-
σ,ν,c σ,ν,c
tively R supp(µ ) and R supp(ν) by setting
σ,ν,c
\ \
R supp(µ ) R
ωσ,ν,c : x \ x(1 σσ,ν2,ccg→ (x))2 σ2(1 c)(1 σ2cg (x)) (5)
7→ − µσ,ν,c − − − µσ,ν,c
and
R supp(ν) R
Φσ,ν,c : x \ x(1+c→σ2g (x))2+σ2(1 c)(1+cσ2g (x)) .
ν ν
7→ −
Set
1
:= x R supp(ν),Φ′ (x)>0,g (x)> . (6)
Eσ,ν,c ∈ \ σ,ν,c ν −σ2c
(cid:26) (cid:27)
ω is an increasing analytic diffeomorphism with positive derivative from R
σ,ν,c
\
supp(µ ) to , with inverse Φ .
σ,ν,c σ,ν,c σ,ν,c
E
Moreover, extending previous results in [25] and [8] involving the Gaussian
caseandfinite rank perturbations,[11]establishes a one-to-onecorrespondance
between the θ ’s that belong to the set (counting multiplicity) and the
i σ,ν,c
E
outliers in the spectrum of M . More precisely, setting
N
1
′
Θ = θ Θ,Φ (θ)>0,g (θ)> , (7)
σ,ν,c ∈ σ,ν,c ν −σ2c
(cid:26) (cid:27)
and
= supp (µ ) Φ (θ),θ Θ , (8)
σ,ν,c σ,ν,c σ,ν,c
S ∪{ ∈ }
we have the following results.
Theorem 1.3. [11] For any ǫ>0,
P[for all large N,spect(M ) x R,dist(x, ) ǫ ]=1.
N
⊂{ ∈ S ≤ }
Theorem 1.4. [11] Let θ be in Θ and denote by n +1,...,n +k
j σ,ν,c j 1 j 1 j
− −
the descending ranks of α (N) among the eigenvalues of A A . Then the
j N ∗N
k eigenvalues (λ (M ), 1 i k ) converge almost surely outside the
j nj−1+i N ≤ ≤ j
support of µ towards ρ :=Φ (θ ). Moreover, these eigenvalues asymp-
σ,ν,c θj σ,ν,c j
totically separate from the rest of the spectrum since (with the conventions that
λ (M ) = + and λ (M ) = ) there exists δ > 0 such that almost
0 N N+1 N 0
∞ −∞
surely for all large N,
λ (M )>ρ +δ and λ (M )<ρ δ . (9)
nj−1 N θj 0 nj−1+kj+1 N θj − 0
3
Remark 1.5. Note that Theorems 1.3 and 1.4 were established in [11] for A
N
as (14) below and with 0 instead of but they hold true as stated above
S ∪{ } S
and in the more general framework of this paper. Indeed, these extensions can
be obtained sticking to the proof of the corresponding results in [11] but using
the new versions of [3] and of the exact separation phenomenon of [11] which
are presented in the Appendix A of the present paper.
The aim of this paper is to study how the eigenvectors corresponding to the
outliers of M project onto those corresponding to the spikes θ ’s. Note that
N i
there are some pionneering results investigating the eigenvectorscorresponding
totheoutliersoffiniterankperturbationsofclassicalrandommatricialmodels:
[28] in the real Gaussian sample covariance matrix setting, and [7, 8] dealing
with finite rank additive or multiplicative perturbations of unitarily invariant
matrices. For a general perturbation, dealing with sample covariance matrices,
S.P´ech´eandO.Ledoit[23]introduceda toolto study the averagebehaviourof
theeigenvectorsbutitseemsthatthisdidnotallowthemtofocusontheeigen-
vectorsassociatedwiththeeigenvaluesthatseparatefromthebulk. Itturnsout
that further studies [10, 5] point out that the angle between the eigenvectors
of the outliers of the deformed model and the eigenvectors associated to the
corresponding original spikes is determined by Biane’s subordination function.
For the model investigated in this paper, such a free interpretation holds but
we choose not to develop this free probabilistic point of view in this paper and
we refer the reader to the paper [13]. Here is the main result of the paper.
Theorem 1.6. Let θ be in Θ (defined in (7)) and denote by n +
j σ,ν,c j 1
−
1,...,n +k thedescendingranksofα (N)amongtheeigenvalues ofA A .
j 1 j j N ∗N
−
Let ξ(j) be a normalized eigenvector of M relative to one of the eigenvalues
N
(λ (M ), 1 q k ). Denote by the Euclidean norm on Cn. Then,
nj−1+q N ≤ ≤ j k·k2
almost surely
2
(i) lim P ξ(j) =τ(θ )
N + Ker (αj(N)IN−ANA∗N) 2 j
→ ∞(cid:13) (cid:13)
(cid:13) (cid:13)
where (cid:13) (cid:13)
1 σ2cg (ρ ) Φ (θ )
τ(θ )= − µσ,ν,c θj = ′σ,ν,c j (10)
j ω (ρ ) 1+σ2cg (θ )
σ′,ν,c θj ν j
(ii) for any θ in Θ θ ,
i σ,ν,c j
\{ }
lim P ξ(j) =0.
N + Ker (αi(N)IN−ANA∗N) 2
→ ∞(cid:13) (cid:13)
(cid:13) (cid:13)
The sketch of the proof o(cid:13)f Theorem 1.6 follows the(cid:13)analysis of [10] as ex-
plained in Section 2. In Section 3, we prove a universal result allowing to
reduce the study to estimating expectations of Gaussian resolvent entries car-
riedonSection4. InSection5,weexplainhowtodeduceTheorem1.6fromthe
previousSections. InanAppendixA,wepresentalternativeversionsontheone
handofthe resultin[3]aboutthe lackofeigenvaluesoutsidethe supportofthe
4
deterministic equivalent measure, and, on the other hand, of the result in [11]
aboutthe exactseparationphenomenon. These new versionsdealwith random
variables whose imaginaryand realparts are independent but removethe tech-
nicalassumptions((1.10)and“b >0”inTheorem1.1in[3]and“ω (b)>0”
1 σ,ν,c
in Theorem 1.2 in [11]). This allows us to claim that Theorem 1.4 holds in our
context(seeRemark1.5). Finally,wepresent,inanAppendixB,sometechnical
lemmas that are used throughout the paper.
2 Sketch of the proof
Throughoutthe paper, for any m p matrix B, (m,p) N2, we will denote by
B the largest singular value of ×B, and by B 2 = T∈r(BB∗) 21 its Hilbert-
k k k k { }
Schmidt norm.
The proof of Theorem 1.6 follows the analysis in two steps of [10].
Step A. First, we shall prove that, for any orthonormal system (ξ , ,ξ )
1 ··· kj
of eigenvectors associated to the k eigenvalues λ (M ), 1 q k , the
j nj−1+q N ≤ ≤ j
following convergence holds almost surely: l=1,...,J,
∀
kj 2 k δ (1 σ2cg (ρ ))
Xp=1(cid:13)Pker(αl(N)IN−ANA∗N)ξp(cid:13)2 →N→+∞ j jl ω−σ′,ν,c(ρµθσj,)ν,c θj . (11)
(cid:13) (cid:13)
Note tha(cid:13)t for any smooth funct(cid:13)ions h and f on R, if v ,...,v are eigenvec-
1 n
tors associated to λ (A A ),...,λ (A A ) and w ,...,w are eigenvectors
1 N ∗N n N ∗N 1 n
associated to λ (M ),...,λ (M ), one can easily check that
1 N n N
n
Tr[h(M )f(A A )]= h(λ (M ))f(λ (A A )) v ,w 2. (12)
N N ∗N p N m N ∗N |h m pi|
m,p=1
X
Thus,sinceα (N)ononehandandthek eigenvaluesofM in(ρ ε,ρ +ε)
l j N θj− θj
(forǫsmallenough)ontheotherhand,asymptoticallyseparatefromtherestof
thespectrumofrespectivelyA A andM ,afitchoiceofhandf willallowthe
N ∗N N
2
studyofthe restrictivesum kp=j 1 Pker(αl(N)IN−ANA∗N)ξp 2. Thereforeproving
(11)isreducedtothestudyofthea(cid:13)symptoticbehaviourof(cid:13)Tr[h(M )f(A A )]
P (cid:13) (cid:13) N N ∗N
for some functions f and h respec(cid:13)tively concentrated on(cid:13)a neighborhood of θ
l
and ρ .
θj
Step B: In the second, and final, step, we shall use a perturbation argument
identical to the one used in [10] to reduce the problem to the case of a spike
with multiplicity one, case that follows trivially from Step A.
Step B closely follows the lines of [10] whereas Step A requires substantial
work. We first reduce the investigations to the mean Gaussian case by proving
the following.
5
Proposition 2.1. Let X as defined in Section 1. Let =[ ]
N N ij 1 i n,1 j N
G G ≤≤ ≤ ≤
be a n N random matrix with i.i.d. standard complex normal entries. Let h
be a fun×ction in (R,R) with compact support, and Γ be a n n Hermitian
∞ N
C ×
matrix such that
sup Γ < and suprank(Γ )< . (13)
N N
k k ∞ ∞
n,N n,N
Then almost surely,
Tr h σXN +A σXN +A ∗ Γ
√N N √N N N
(cid:16) (cid:16)(cid:16) (cid:17)(cid:16) (cid:17) (cid:17) (cid:17)
E Tr h σ GN +A σ GN +A ∗ Γ 0.
N N N N +
− √N √N → → ∞
(cid:18) (cid:20) (cid:18)(cid:18) (cid:19)(cid:18) (cid:19) (cid:19) (cid:21)(cid:19)
TheasymptoticbehaviourofE Tr h σ√GNN +AN σ√GNN +AN ∗ f(ANA∗N)
can be deduced, by using the b(cid:16)i-unhita(cid:16)r(cid:16)ily invariance(cid:17)(cid:16)of the distrib(cid:17)ut(cid:17)ion of N, i(cid:17)
G
from the following Proposition2.2 and Lemma 5.9.
Proposition 2.2. Let =[ ] be a n N random matrix with
N ij 1 i n,1 j N
G G ≤≤ ≤ ≤ ×
i.i.d. complex standard normal entries. Assume that A is such that
N
d (N) (0)
1
(0)
AN = ... (0) (14)
(0)
d (N) (0)
n
wheren=n(N),n N,c =n/N c ]0;1],fori=1,...,n,d (N)
N N + i
C, sup max ≤d (N) < + →and→1 ∞n ∈δ weakly converges to∈a
compacNtly supip=o1r,.t.e.,dn|priobabi|lity m∞easure νnoniR=1wh|dein(NN)|2goes toinfinity. Define
for all z C R, P
∈ \
1
N N ∗ −
GGN(z)= zI− σ√GN +AN σ√GN +AN .
(cid:18) (cid:18) (cid:19)(cid:18) (cid:19) (cid:19)
Define for any q =1,...,n,
γ (N)=(A A ) = d (N)2. (15)
q N ∗N qq | q |
There is a polynomial P with nonnegative coefficients, a sequence (u ) of
N N
nonnegative real numbers converging to zero when N goes to infinity and some
nonnegative real number l, such that for any (p,q) in 1,...,n 2, for all z
C R, { } ∈
\
1 σ2cg (z)
E GGN(z) pq = ω − (z)µσ,γν,c(N)δpq+∆p,q,N(z), (16)
σ,ν,c q
(cid:16)(cid:0) (cid:1) (cid:17) −
with
∆ (z) (1+ z )lP( z 1)u .
p,q,N − N
| |≤ | | |ℑ |
6
3 Proof of Proposition 2.1
In the following,we will denote by o (1) any deterministic sequence ofpositive
C
realnumbersdepending onthe parameterC andconvergingforeachfixedC to
zerowhenN goestoinfinity. TheaimofthissectionistoproveProposition2.1.
For any n N matrix B, let us introduce the (N +n) (N +n) matrix
× ×
0 B+A
MN+n(B)=(cid:18) Bn∗×+n A∗N 0NN×N (cid:19).
Set for any z C R,
∈ \
1
G˜(z)= zI σXN − .
N+n N+n
−M √N
(cid:18) (cid:18) (cid:19)(cid:19)
Define for any C > 0, XC = (XC) , where for any 1 i n,1
ij 1≤i≤n;1≤j≤N ≤ ≤ ≤
j N,
≤ YC
XC = ij ,
ij
E YC 2
| ij|
q
with (cid:0) (cid:1)
YC =X 1I E X 1I .
ij ij |Xij|≤C − ij |Xij|≤C
Let = [ ij]1 i n,1 j N be a n N ra(cid:0)ndom matrix(cid:1) with i.i.d. standard
G G ≤≤ ≤ ≤ ×
complex normal entries, independent from X , and define for any α>0,
N
XC +α
Xα,C = G.
√1+α2
Define for any z C R,
∈ \
G˜α,C(z)= zI σXα,C −1.
N+n N+n
−M √N
(cid:18) (cid:18) (cid:19)(cid:19)
Denote by U(n+N) the set of unitary (n+N) (n+N) matrices. We first
×
establish the following approximation result.
Lemma 3.1. There exist some positive deterministic functions u and v on
[0,+ [ such that lim u(C) = 0 and lim v(α) = 0, and a polynomial
C + α 0
∞ → ∞ →
P with nonnegative coefficients such that for any α,C, we have that
almost surely, for all large N,
•
sup sup sup z 2 (U∗G˜α,C(z)U)ij (U∗G˜(z)U)ij
U U(n+N)(i,j) 1,...,n+N 2z C R|ℑ | −
∈ ∈{ } ∈ \ (cid:12) (cid:12)
(cid:12) (cid:12)
u(C)+(cid:12) v(α), (cid:12) (17)
≤
7
for all large N,
• 1
sup sup sup E (U G˜α,C(z)U) (U G˜(z)U)
U∈U(n+N)(i,j)∈{1,...,n+N}2z∈C\RP(|ℑz|−1)(cid:12) (cid:16) ∗ ij − ∗ ij(cid:17)(cid:12)
(cid:12) (cid:12)
u(C)+v(α)(cid:12)+o (1). (18)(cid:12)
C
≤
Proof. Set
θ∗ = sup E(Xij 3)<+ .
(i,j) N2 | | ∞
∈
We have
E X YC 2 = E X 21I
| ij − ij| | ij| |Xij|>C
(cid:0) (cid:1) (cid:0)E eX 1I (cid:1) 2 E mX 1I 2
− ℜ ij |Xij|>C − ℑ ij |Xij|>C
so that (cid:8) (cid:0) (cid:1)(cid:9) (cid:8) (cid:0) (cid:1)(cid:9)
θ
sup E X YC 2 ∗.
(i,j) N2 | ij − ij| ≤ C
∈ (cid:0) (cid:1)
Note also that
1 E YC 2
XC YC =XC − | ij| ,
ij − ij ij1+E (cid:0)YC 2 1(cid:1)/2
| ij|
with (cid:0) (cid:1)
1 E YC 2 = E X 21I + E eX 1I 2
− | ij| | ij| |Xij|>C ℜ ij |Xij|>C
(cid:0) (cid:1) +(cid:0)E mX 1I (cid:1) (cid:8) (cid:0)2 (cid:1)(cid:9)
ℑ ij |Xij|>C
≤ 2E(cid:8)|X(cid:0)ij|21I|Xij|>C (cid:1)(cid:9)
so that (cid:0) (cid:1)
2θ
sup 1 E YC 2 ∗.
(i,j) N2| − | ij| |≤ C
∈ (cid:0) (cid:1)
Let us assume that C >4θ . Then,
∗
1/2
2θ
sup E XC YC 2 ∗, and sup E XC YC 3 < .
((i,j) N2 | ij − ij| ) ≤ C (i,j) N2 | ij − ij| ∞
∈ ∈
(cid:0) (cid:1) (cid:0) (cid:1)
Itisstraightfowardtosee,usinglemma5.8,thatforanyunitary(n+N) (n+N)
×
matrix U,
(U G˜α,C(z)U) (U G˜(z)U)
∗ ij ∗ ij
−
(cid:12) (cid:12)
(cid:12) σ X (cid:12)Xα,C
(cid:12) N (cid:12)
−
≤ z 2 √N
|ℑ | (cid:13) (cid:13)
σ (cid:13)(cid:13) XN YC (cid:13)(cid:13) XC YC
(cid:13) − (cid:13)+ −
≤ z 2 √N √N
|ℑ | (cid:26)(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13)(cid:13) 1 (cid:13)(cid:13)XC(cid:13)(cid:13) (cid:13)(cid:13)
+ 1 (cid:13) (cid:13) (cid:13) +α G(cid:13) . (19)
− √1+α2 √N √N
(cid:18) (cid:19)(cid:13) (cid:13) (cid:13) (cid:13)(cid:27)
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
8
From Bai-Yin’s theorem (Theorem 5.8 in [2]) , we have
G =2+o(1).
√N
(cid:13) (cid:13)
(cid:13) (cid:13)
(cid:13) (cid:13) 0 B
Applying Remark 5.4 to the ((cid:13)n+N(cid:13)) (n+N) matrix B˜ = n×n
× B∗ 0N N
(cid:18) × (cid:19)
for B X YC,XC YC,XC (see alsoAppendix B of [14]), we have that
N
∈{ − − }
almost surely
XC XC YC 4√2θ
limsup 2√2, limsup − ∗,
N→+∞(cid:13)√N(cid:13)≤ N→+∞(cid:13) √N (cid:13)≤ C
(cid:13) (cid:13) (cid:13) (cid:13)
and (cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
X YC θ
limsup N − 2√2 ∗.
N→+∞(cid:13) √N (cid:13)≤ rC
(cid:13) (cid:13)
Then, (17) readily follows. (cid:13) (cid:13)
(cid:13) (cid:13)
Let us introduce
XC X YC θ XC YC 8θ
Ω = G 4, 4, N − 4 ∗, − ∗ .
N,C
((cid:13)√N(cid:13)≤ (cid:13)√N(cid:13)≤ (cid:13) √N (cid:13)≤ rC (cid:13) √N (cid:13)≤ C )
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
Using (19)(cid:13), we h(cid:13)ave (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
E (U G˜α,C(z)U) (U G˜(z)U)
∗ ij ∗ ij
−
(cid:12) (cid:16) (cid:17)(cid:12)
(cid:12) (cid:12)
(cid:12) 4σ θ∗ + 2(cid:12)θ∗ +α+ 1 1
≤ |ℑz|2 "rC C (cid:18) − √1+α2(cid:19)#
2
+ P(Ωc ).
z N,C
|ℑ |
Thus (18) follows.
Now, Lemma 5.9, Lemma 3.1 and Lemma 5.10 readily yields the following
approximationlemma.
Lemma 3.2. Let h be in (R,R) with compact support and Γ˜ be a (n+
∞ N
C
N) (n+N) Hermitian matrix such that such that
×
sup Γ˜ < and suprank(Γ˜ )< . (20)
N N
k k ∞ ∞
n,N n,N
Then, there exist some deterministic functions u and v on [0,+ [ such that
∞
lim u(C) = 0 and lim v(α) = 0, such that for all C > 0, α > 0, we
C + α 0
→ ∞ →
have almost surely for all large N,
Xα,C X
Tr h ( Γ˜ Tr h ( N Γ˜ a(1) ,
MN+n √N N − MN+n √N N ≤ C,α
(cid:12) (cid:20) (cid:18) (cid:18) (cid:19)(cid:19) (cid:21) (cid:20) (cid:18) (cid:18) (cid:19)(cid:19) (cid:21)(cid:12)
(cid:12) (cid:12) (21)
(cid:12) (cid:12)
(cid:12) (cid:12)
9
and for all large N,
Xα,C X
ETr h ( Γ˜ ETr h ( N Γ˜ a(2) ,
MN+n √N N − MN+n √N N ≤ C,α,N
(cid:12) (cid:20) (cid:18) (cid:18) (cid:19)(cid:19) (cid:21) (cid:20) (cid:18) (cid:18) (cid:19)(cid:19) (cid:21)(cid:12)
(cid:12) (cid:12) (22)
(cid:12) (cid:12)
w(cid:12)here (cid:12)
a(1) =u(C)+v(α), a(2) =u(C)+v(α)+o (1).
C,α C,α,N C
Note that the distributions of the independent random variables (Xα,C),
ℜ ij
(Xα,C) are all a convolution of a centred Gaussian distribution with some
ℑ ij
variancev ,withsomelawwithboundedsupportinaballofsomeradiusR ;
α C,α
thus, according to Lemma 5.11, they satisfy a Poincar´e inequality with some
common constant C (C,α) and therefore so does their product (see the Ap-
PI
pendixB).AnimportantconsequenceofthePoincar´einequalityisthefollowing
concentration result.
Lemma 3.3. Lemma 4.4.3 and Exercise 4.4.5 in [1] or Chapter 3 in [24].
There exists K >0 and K >0 such that for any probability measure P on RM
1 2
which satisfies a Poincar´e inequality with constant C , and for any Lipschitz
PI
function F on RM with Lipschitz constant F , we have
Lip
| |
ǫ
ǫ>0, P(F EP(F) >ǫ) K1exp .
∀ | − | ≤ −K √C F
(cid:18) 2 PI| |Lip(cid:19)
In order to apply Lemma 3.3, we need the following preliminary lemmas.
Lemma 3.4. (see Lemma 8.2 [10]) Let f be a real C -Lipschitz function on
R. Then its extension on the N N Hermitian matricLes is C -Lipschitz with
× L
respect to the Hilbert-Schmidt norm.
Lemma 3.5. Let Γ˜ be a (n+N) (n+N) matrix and h be a real Lipschitz
N
function on R. For any n N matr×ix B,
×
( B(i,j), B(i,j)) Tr h(( (B))Γ˜
ℜ ℑ 1 i n,1 j N 7→ MN+n N
≤≤ ≤ ≤
n o h i
is Lipschitz with constant bounded by √2 Γ˜ h .
N Lip
2k k
(cid:13) (cid:13)
Proof. Tr h( N+p(B))Γ˜N Tr h( N(cid:13)(cid:13)+p((cid:13)(cid:13)B′))Γ˜N
M − M
(cid:12) h i h i(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Γ˜ h( (B)) h( (B′))
N N+p N+p
≤ 2 M − M 2
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13)Γ˜ (cid:13) (cid:13)h (B) (B′)(cid:13) . (23)
≤ (cid:13) N(cid:13)2k(cid:13) kLip MN+p −MN+p (cid:13) 2
(cid:13) (cid:13) (cid:13) (cid:13)
where we used Lemm(cid:13)a 3.4(cid:13)in the la(cid:13)st line. Now, (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
2 2
′ ′
(B) (B ) =2 B B . (24)
N+p N+p
M −M 2 − 2
(cid:13) (cid:13) (cid:13) (cid:13)
Lemma 3.5 readil(cid:13)y follows from (23) and (2(cid:13)4). (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
10
