Asymptotic theory of multiple-set linear canonical analysis

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Asymptotic theory of multiple-set linear canonical analysis

Guy Martial Nkiet

To cite this version:

Guy Martial Nkiet. Asymptotic theory of multiple-set linear canonical analysis. 2017. �hal-01511413�

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Asymptotic theory of multiple-set linear canonical analysis

Guy Martial Nkiet

Received: date / Revised: date

Abstract This paper deals with asymptotics for multiple-set linear canonical analysis (MSLCA). A definition of this analysis, that adapts the classical one to the context of Euclidean random variables, is given and properties of the related canonical coefficients are derived. Then, estimators of the MSLCA’s elements, based on empirical covariance operators, are proposed and asymp- totics for these estimators are obtained. More precisely, we prove their consis- tency and we obtain asymptotic normality for the estimator of the operator that gives MSLCA, and also for the estimator of the vector of canonical coeffi- cients. These results are then used to obtain a test for mutual non-correlation between the involved Euclidean random variables.

Keywords Multiple set canonical analysis · asymptotic study · non- correlation tests

1 Introduction

Multiple-set linear canonical analysis (MSLCA), also known as generalized canonical correlation analysis, has been extensively discussed in the literature, see Kettenring (1971), Gifi (1991), Gardner et al. (2006), Takane et al. (2008), Tenenhaus and Tenenhaus (2011), as well as the further references contained therein. It is a statistical method that generalizes linear canonical analysis (LCA) to the case where more than two sets of variables are considered, which is of a real interest since in applied statistical studies it is common to collect data from the observation of several sets of variables on a given population.

However, although this interest, several aspects under which LCA has been studied have never been addressed to MSLCA. For example, asymptotic the- ory for LCA and related applications have been tackled by several authors

Guy Martial Nkiet

Universit´e des Sciences et Techniques de Masuku, D´epartement de Math´ematiques et Infor- matique, BP 943 Franceville, Gabon. E-mail: gnkiet@hotmail.com

(e.g., Muirhead and Waternaux (1980), Anderson (1999), Pousse (1992), Fine (2000), Dauxois et al. (2004)). It would be natural to wonder how the obtained results extend to the case of MSLCA but, to the best of our knowledge, such an approach has never been tackled.

In this paper, we introduce an asymptotic theory for MSLCA. For doing that, we first define in Section 2 the notion of MSLCA for Euclidean random variables, that is random variables valued into Euclidean vector spaces. This analysis is defined from a maximization problem under specified constraints, and shown to be obtained from spectral analysis of a suitable operator. Prop- erties of the related eigenvalues, called canonical coefficients, are then given. In Section 3, we tackle the problem of estimating MSLCA. More precisely, estima- tors based on empirical covariance operators are introduced. Then, consistency of the obtained estimators is proved. Further, we derive the asymptotic distri- bution of the used estimator of the aforementioned operator, and also that of the estimator of the vector of canonical coefficients in the general case as well as in the case of elliptical distribution. Section 4 is devoted to the introduction of a test for mutual non-correlation between the random variables involved in MSLCA. The results obtained for asymptotic theory of MSLCA are then used in order to derive the asymptotic distribution of the used test statistic under null hypothesis.

2 Multiple-set canonical linear analysis of Euclidean random variables

For an integer K ≥2, let us consider random variables X1,· · ·, XK defined on a probability space (Ω,A, P) and valued into Euclidean vector spaces X1,· · ·,XK respectively. Denoting byEthe mathematical expectation related to P, we assume that, for any k ∈ {1,· · ·, K}, we have E(kXkk²k) < +∞ wherek · kk denotes the norm induced by the inner producth·,·ik ofXk, and, without loss of generality, thatE(Xk) = 0. Each vectorαin the vector space X :=X¹× · · · X^K will be writen as

α=



 α1

... αK



,

and we recall that X is an Euclidean vector space equipped with the inner producth·,·iX defined by:

∀α∈ X, ∀β∈ X, hα, βiX = XK k=1

hαk, βki^k.

We denote byk · kX the norm induced by this inner product. Considering the X-valued random variable

X =



 X1

... XK



,

we can give the following definition which adapts the classical definition of multiple-set canonical analysis (e.g., Gifi (1991), Gardner et al. (2006), Takane et al. (2008)) to the context of Euclidean random variables.

Definition 2.1. The multiple-set linear canonical analysis (MSLCA) of X is the search of a sequence α^(j)

1≤j≤q of vectors of E, where q = dim(X), satisfying:

α^(j)= arg max

α∈Cj

E < α, X >²_X

, (1)

where

C1= (

α∈ X/ XK k=1

var(< αk, Xk >k) = 1 )

, (2)

and, forj≥2: Cj=

(

α∈C1/ XK k=1

cov

< α^(r)_k , Xk>k, < αk, Xk>k

= 0, ∀r∈ {1,· · · , j−1} )

. (3)

Remark 2.1

1) The constraints sets given in (2) and (3) can be expressed by using covari- ance operators defined for (k, ℓ)∈ {1,· · · , K}²by:

Vkℓ=E(Xℓ⊗Xk) =V_ℓk^∗ andVk:=Vkk,

where⊗denotes the tensor product such that, for any (x, y),x⊗yis the linear map : h 7→< x, h > y, and T^∗ denotes the adjoint of T. Indeed, it is easily seen that, for (α, β)∈ X²,

var(hαk, Xkik) =E hαk, Xki²k

=E(hαk,(Xk⊗Xk)(αk)ik) =hαk, Vkαkik, and

cov(hαk, Xkik,hβℓ, Xℓiℓ) =E(hαk, Xkikhβℓ, Xℓiℓ) =E(hαk,(Xℓ⊗Xk)(βℓ)ik)

=hαk, Vkℓβℓik. Therefore,

C1= (

α∈ X/ XK k=1

< αk, Vkαk >k= 1 )

, (4)

and

Cj= (

α∈C1/ XK k=1

< α^(r)_k , Vkαk>k= 0, ∀r∈ {1,· · · , j−1} )

. (5) 2) For anyα∈C1, one has:

E < α, X >²_X

=E



 XK k=1

< αk, Xk>k

!2

= XK k=1

XK ℓ=1

E(< αk, Xk >k< αℓ, Xℓ>ℓ)

= XK k=1

E < αk, Xk >²_k +

XK k=1

XK ℓℓ=16=k

E(< αk, Xk >k< αℓ, Xℓ>ℓ)

= XK k=1

var(< αk, Xk>k) + XK k=1

XK

ℓ=1

ℓ6=k

< αk, Vkℓαℓ>k

= 1 + XK k=1

XK ℓℓ=16=k

< αk, Vkℓαℓ>k= 1 +ϕ(α),

where

ϕ(α) = XK k=1

XK ℓℓ=16=k

< αk, Vkℓαℓ>k. (6)

Then, the MSLCA ofX is obtained by minimizingϕ(α) under the constraints expressed in (4) and (5).

Fork∈ {1,· · · , K}, the covariance operatorVk is a self-adjoint non-negative operator. From now on, we assume that it is invertible. Letτkbe the canonical projection defined as

τk : α∈ X 7→αk ∈ X^k; its adjointτ_k^∗ ofτk is the map given by:

τ_k^∗ : t∈ Xk7→( 0,· · ·,0

| {z }

k−1 times

, t,0,· · ·,0)^T ∈ X,

where we denote bya^T the transposed ofa. Now, let us consider the operators ofL(X) given by:

Φ= XK k=1

τ_k^∗Vkτk and Ψ = XK k=1

XK ℓℓ=16=k

τ_k^∗Vkℓτℓ.

From the fact thatτkτ_ℓ^∗=δkℓIk, whereδkℓ is the usual Kronecker symbol and Ikis the identity operator ofXk, it is easily seen thatΦis also an invertible self- adjoint and non-negative operator, withΦ⁻¹=PK

k=1τ_k^∗V_k⁻¹τk and Φ^−1/2 =

PK

k=1τ_k^∗V_k^−1/2τk. The following theorem shows how to obtain a MSLCA ofX. It just repeats a known result (e.g., Gifi (1991), Takane et al. (2008)) within the framework used for this paper.

Theorem 2.1. Letting

β⁽¹⁾,· · · , β^(q) be an orthonormal basis ofX such that β^(j) is an eigenvector of the operatorT =Φ^−1/2Ψ Φ^−1/2 associated with thej-th largest eigenvalueρj of T. Then, the sequence α^(j)

1≤j≤q given by:

α^(j)=Φ^−1/2β^(j)=

V₁^−1/2β₁^(j),· · · , V_K^−1/2β_K^(j) ,

consists of solutions of (1) under the constraints (2) and (3), and we have:

ρj=< β^(j), T β^(j)>E=ϕ(α^(j)).

Proof.Puttingβk=V_k^1/2αk andβ^(r)=V_k^1/2α^(r)_k , we have:

ϕ(α) = XK k=1

XK ℓℓ=16=k

< V_k^−1/2βk, VkℓV_ℓ^−1/2βℓ>k

= XK k=1

XK

ℓ=1

ℓ6=k

< βk, V_k^−1/2VkℓV_ℓ^−1/2βℓ>k=:ψ(β), (7)

where

β=



 β1

... βK



∈ X.

SinceVk=V_k^1/2V_k^1/2, havingα∈Cj is equivalent to havingβ ∈C_j^′, where:

C₁^′ = (

β ∈ X/ XK k=1

kβkk²k= 1 )

=

β∈ X/kβk²_X = 1 , (8) and forj≥2:

C_j^′ = (

β ∈C1/ XK k=1

< β_k^(r), βk >k= 0, ∀r∈ {1,· · ·, j−1} )

=n

β ∈C1/ < β^(r), β >_X= 0, ∀r∈ {1,· · ·, j−1}o

. (9)

Further, for anyβ ∈ X: Ψ Φ^−1/2β=

XK k=1

XK ℓℓ=16=k

XK j=1

τ_k^∗Vkℓτℓτ_j^∗V_j^−1/2τjβ = XK k=1

XK ℓℓ=16=k

XK j=1

δℓjτ_k^∗VkℓV_j^−1/2τjβ

= XK k=1

XK ℓℓ=16=k

τ_k^∗VkℓV_ℓ^−1/2τℓβ,

and

Φ^−1/2Ψ Φ^−1/2β = XK k=1

XK

ℓ=1

ℓ6=k

XK j=1

τ_j^∗V_j^−1/2τjτ_k^∗VkℓV_ℓ^−1/2τℓβ

= XK k=1

XK ℓℓ=16=k

XK j=1

δjkτ_j^∗V_j^−1/2VkℓV_ℓ^−1/2τℓβ

= XK k=1

XK

ℓ=1

ℓ6=k

τ_k^∗V_k^−1/2VkℓV_ℓ^−1/2τℓβ

= XK k=1

XK ℓℓ=16=k

τ_k^∗V_k^−1/2VkℓV_ℓ^−1/2βℓ.

Thus,

< β, Φ^−1/2Ψ Φ^−1/2β >_X = XK k=1

XK ℓℓ=16=k

< β, τ_k^∗V_k^−1/2VkℓV_ℓ^−1/2βℓ>_X

= XK k=1

XK

ℓ=1

ℓ6=k

< τkβ, V_k^−1/2VkℓV_ℓ^−1/2βℓ>k

= XK k=1

XK ℓℓ=16=k

< βk, V_k^−1/2VkℓV_ℓ^−1/2βℓ>k=ψ(β),

where ψ is defined in (7). Then, the MSLCA optimization problem reduces to the maximization of < β, Φ^−1/2Ψ Φ^−1/2β >_X under the constraints (8) and (9). Since T = Φ^−1/2Ψ Φ^−1/2 is a self-adjoint operator, this is a well known maximization problem for which a solution is obtained from the spectral

analysis ofT as stated in the theorem.

Definition 2.2.Theρj’s are termed the canonical coefficients. Theα^(j)’s are termed vectors of canonical directions.

The following theorem gives some properties of the canonical coefficients.

Theorem 2.2.

(i)∀j ∈ {1,· · ·, q},−1≤ρj≤K(K−1).

(ii)∀j ∈ {1,· · · , q},ρj= 0⇔ ∀(k, ℓ)∈ {1,· · ·, K}², k6=ℓ, Vkℓ = 0.

Proof.

(i) First, using (6), we have for anyj∈ {1,· · ·, q}, ρj =ϕ(α^(j)) =E

< α^(j), X >²_X

−1≥ −1.

On the other hand, we have:

ρj=ϕ(α^(j)) = XK k=1

XK

ℓ=1

ℓ6=k

E

< α^(j)_k , Xk >k< α^(j)_ℓ , Xℓ>ℓ

≤ XK k=1

XK ℓℓ=16=k

r E

< α^(j)_k , Xk>²_kr E

< α^(j)_ℓ , Xℓ>²_ℓ .

Since, for anyk∈ {1,· · ·, K}, one has:

E

< α^(j)_k , Xk>²_k

=var

< α^(j)_k , Xk >k

≤ XK ℓ=1

var

< α^(j)_ℓ , Xℓ>ℓ

= 1,

it follows that:

ρj≤ XK k=1

XK ℓℓ=16=k

1 =K(K−1).

(ii) Since theρj’s are the eigenvalues ofT, we have:

∀j∈ {1,· · · , q}, ρj= 0⇔T = 0⇔Ψ = 0⇔ ∀(k, ℓ)∈ {1,· · ·, K}², k6=ℓ, Vkℓ= 0.

Remark 2.2.

1) When K = 2, one hasΦ =τ₁^∗V1τ1+τ₂^∗V2τ2 and Ψ =τ₁^∗V12τ2+τ₂^∗V21τ1. Then it is easy to check thatT =τ₁^∗Sτ2+τ₂^∗S^∗τ1, whereS=V₁^−1/2V12V₂^−1/2. Let xbe an eigenvector of T associated with an eigenvalueρ 6= 0. We have T x=ρx, that is equivalent to having:

τ₁^∗(Sτ2x−ρτ1x) =−τ₂^∗(S^∗τ1x−ρτ2x).

This implies:

Sτ2x=ρ τ1x S^∗τ1x=ρ τ2x and, puttingx1=τ1xandx2=τ2x, we obtain

x2=ρ⁻¹S^∗x1 and Rx1=ρ²x1, (10) where

R=SS^∗=V₁^−1/2V12V₂⁻¹V21V₂^−1/2.

Conversely, if (10) holds then, puttingx=τ₁^∗x1+τ₂^∗x2, we have:

T x=τ₁^∗Sτ2x+τ₂^∗S^∗τ1x=τ₁^∗Sx2+τ₂^∗S^∗x1=ρ⁻¹τ₁^∗SS^∗x1+ρ τ₂^∗x2

=ρ⁻¹τ₁^∗Rx1+ρ τ₂^∗x2=ρ(τ₁^∗x1+τ₂^∗x2) =ρ x.

Moreover, since

kx2k2=ρ⁻¹kS^∗x1k1=ρ⁻¹p

< S^∗x1, S^∗x1>2=ρ⁻¹p

< SS^∗x1, x1>1=kx1k1

and

kxk²_X =kx1k²1+kx2k²2

it follows that

kx1k¹=kx2k²= 1

√2kxkX.

2) The preceding remark shows the equivalence between MSLCA and linear canonical analysis (LCA) when K = 2. Recall that LCA of X1 and X2 is obtained from the spectral analysis ofR(see, e.g., Dauxois and Pouse (1975), Pousse (1992), Fine (2000)). More precisely,

β^(j), ρj 1≤j≤q is defined as in Theorem 2.1 if, and only if, n

u^(j)₁ , u^(j)₂ ρ²_jo

1≤j≤q, where u^(j)_ℓ = ^√1₂τℓβ^(j) (ℓ ∈ {1,2}), is a LCA ofX1andX2.

3 Estimation and asymptotic theory

In this section, we deal with estimation of MSLCA. For k = 1,· · ·, K, let {X_k⁽ⁱ⁾}1≤i≤n be an i.i.d. sample ofXk. We use empirical covariance operators for defining estimators of MSLCA elements. Then, consistency and asymptotic normality are obtained for the resulting estimators of the vectors of canonical directions and the canonical coefficients.

3.1 Estimation and almost sure convergence

For (k, ℓ) ∈ {1,· · ·, K}², let us consider the sample means and covariance operators:

Xk·n= 1 n

Xn i=1

X_k⁽ⁱ⁾, Vbkℓ·n= 1 n

Xn i=1

X_ℓ⁽ⁱ⁾−Xℓ·n

⊗

X_k⁽ⁱ⁾−Xk·n

, Vbk·n:=Vbkk·n, and the random operators valued intoL(X) defined as

Φbn= XK k=1

τ_k^∗Vbk·nτk and Ψbn= XK k=1

XK

ℓ=1

ℓ6=k

τ_k^∗Vbkℓ·nτℓ. Then, we estimateT by

Tbn=Φb⁻_n^1/2ΨbnΦb⁻_n^1/2.

Considering the eigenvaluesbρ1·n≥ρb2·n· · · ≥ρbq·n ofTbn, andn

βbn⁽¹⁾,· · ·,βbn^(q)

o an orthonormal basis ofXsuch thatβb^(j)n is an eigenvector ofTbnassociated with b

ρj·n. Then, we estimate ρj byρbj·n, and β^(j) byβbn^(j). The following theorem establishes strong consistency for these estimators.

Theorem 3.1. For any integerj∈ {1,· · ·, q}: (i)ρbj·n converge almost surely, asn→+∞, toρj.

(ii)sign(hβb^(j)n , β^(j)iX)βb^(j)n converges almost surely, asn→+∞, toβ^(j)inX. Proof. From obvious applications of the strong law of large numbers, it is easily seen that Tbn converges almost surely inL(X), asn→+∞toT. Then using Lemma 1 in Ferr´e and Yao (2003), we obtain the inequality|bρj·n−ρj| ≤ kTbn−Tkfrom what (i) is deduced. Clearly, eachβ^(j)⊗β^(j)and is a projector onto an eigenspace. Thefore, using Proposition 3 in Dossou-Gbete and Pousse (1991), we deduce thatβbn^(j)⊗βbn^(j)converges almost surely inL(X) toβ^(j)⊗β^(j), as n → +∞. Using again Lemma 1 in Ferr´e and Yao (2003), we obtain the inequality

sign(hβb_n^(j), β^(j)iX)βb_n^(j)−β^(j)

X ≤2√

2 bβ_n^(j)⊗βb^(j)_n −β^(j)⊗β^(j)

from what we deduce (ii).

3.2 Asymptotic distribution

In this section, we assume that, for k ∈ {1,· · ·, K}, we have E kXkk⁴k

<

+∞ and Vk = Ik, where Ik denotes the identity operator of Xk. We first derive an asymptotic distribution forTbn, then we obtain these of the canonical coefficients.

Theorem 3.2. √ n

Tbn−T

converges in distribution, as n → +∞, to a random variable U having a normal distribution in L(X), with mean 0 and covariance operatorΓ equal to that of the random operator:

Z= XK k=1

XK ℓℓ=16=k

−1

2(τ_k^∗(Xk⊗Xk)Vkℓτℓ+τ_ℓ^∗Vℓk(Xk⊗Xk)τk) +τ_k^∗(Xℓ⊗Xk)τℓ.

Proof. Under the above assumptions,

Φ= XK k=1

τ_k^∗Vkτk = XK k=1

τ_k^∗τk=I_X,

whereI_X is the indentity operator ofX, and

√n

Tbn−T

=√ n

Φb⁻_n^1/2ΨbnΦb⁻_n^1/2−Ψ

=√ n

Φb⁻_n^1/2−I_X

ΨbnΦb⁻_n^1/2+√ n

Ψbn−Ψ

Φb⁻_n^1/2+Ψ√

n(Φb⁻_n^1/2−I_X)

=−Φb⁻_n¹√

n(Φbn−I_X) Φb⁻_n^1/2+I_X−1

ΨbnΦb⁻_n^1/2+√ n

Ψbn−Ψ Φb⁻_n^1/2

−ΨΦb⁻_n¹√

n(Φbn−I_X) Φb⁻_n^1/2+I_X−1

. (11)

Clearly,

Vkℓ=E(τℓ(X)⊗τk(X)) =τkV τ_ℓ^∗, (12) whereV =E(X⊗X). Moreover, putting

X⁽ⁱ⁾=





 X₁⁽ⁱ⁾

... X_K⁽ⁱ⁾





,

we have

Vbkℓ·n = 1 n

Xn i=1

X_ℓ⁽ⁱ⁾⊗X_k⁽ⁱ⁾−Xℓ·n⊗Xk·n

= 1 n

Xn i=1

τℓ(X⁽ⁱ⁾)⊗τk(X⁽ⁱ⁾)−τℓ(Xn)⊗τk(Xn)

=τkVbnτ_ℓ^∗, (13)

whereXn=n⁻¹Pn

i=1X⁽ⁱ⁾ and Vbn = 1

n Xn i=1

X⁽ⁱ⁾⊗X⁽ⁱ⁾−Xn⊗Xn. (14) Therefore, using (12) and (13), we obtain

√n

Ψbn−Ψ

= XK k=1

XK ℓℓ=16=k

τ_k^∗τkHbnτ_ℓ^∗τℓ=f(Hbn), (15)

whereHbn=√

n(Vbn−V) andf is the operator defined as f : A∈ L(X)7→

XK k=1

XK

ℓ=1

ℓ6=k

τ_k^∗τkAτ_ℓ^∗τℓ∈ L(X).

Further, sinceI_X =PK

k=1τ_k^∗τk, we obtain

√n(Φbn−I_X) = XK k=1

τ_k^∗τkHbnτ_k^∗τk=g(Hbn), (16)

where g is the operator g : A ∈ L(X) 7→ PK

k=1τ_k^∗τkAτ_k^∗τk ∈ L(X). Then, using (11), (15) and (16), we obtain√

n

Tbn−T

=ϕbn(Hbn), whereϕbn is the random operator fromL(X) to itself defined by

b

ϕn(A) =−(Φb⁻_n^1/2+I_X)⁻¹g(A)Φb⁻_n¹ΨbnΦb⁻_n^1/2+f(A)Φb⁻_n^1/2−Ψ(Φb⁻_n^1/2+I_X)⁻¹g(A)Φb⁻_n¹. Considering the operator

ϕ : A∈ L(X)7→ −1

2g(A)Ψ+f(A)−1

2Ψ g(A)∈ L(X),

and denoting byk·k∞(resp.k·k∞∞) the norm ofL(X) (resp.L(L(X))) defined

bykAk∞= sup_{x∈X −{0}}kAxkX/kxkX(resp.khk∞∞= sup_{B∈L(X)−{0}}kh(B)k∞/kBk∞) for anyA(resp.h) inL(X) (resp.L(L(X))), we have

kϕbn(Hbn)−ϕ(Hbn)k∞= −

(Φb⁻_n^1/2+I_X)⁻¹−1 2I_X

g(Hbn)bΦ⁻_n¹ΨbnΦb⁻_n^1/2

− 1

2g(Hbn)

Φb⁻_n¹ΨbnΦb⁻_n^1/2−Ψ

+f(Hbn)(bΦ⁻_n^1/2−I_X)

− Ψ

(Φb⁻_n^1/2+I_X)⁻¹−1 2I_X

g(Hbn)Φb⁻_n¹−1

2Ψ(Φb⁻_n¹−I_X)

∞

≤ k(Φb⁻_n^1/2+I_X)⁻¹−1

2I_Xk∞kg(Hbn)k∞kΦb⁻_n¹ΨbnΦb⁻_n^1/2k∞. + 1

2kg(Hbn)k∞kΦb⁻_n¹ΨbnΦb⁻_n^1/2−Ψk∞+kf(Hbn)k∞kΦb⁻_n^1/2−I_Xk∞

+kΨk∞k(Φb⁻_n^1/2+I_X)⁻¹−1

2I_Xk∞kg(Hbn)k∞kΦb⁻_n¹k∞

+ 1

2kΨk∞kg(Hbn)k∞kΦb⁻_n¹−I_Xk∞

≤

k(Φb⁻_n^1/2+I_X)⁻¹−1

2I_Xk∞kgk∞∞kΦb⁻_n¹ΨbnΦb⁻_n^1/2k∞

+ 1

2kgk∞∞kΦb⁻_n¹ΨbnΦb⁻_n^1/2−Ψk∞+kfk∞∞kΦb⁻_n^1/2−I_Xk∞

+kΨk∞k(Φb⁻_n^1/2+I_X)⁻¹−1

2I_Xk∞kgk∞∞kΦb⁻_n¹k∞

+ 1

2kΨk∞kgk∞∞kΦb⁻_n¹−I_Xk∞

kHbnk∞. (17) Using the strong law of large numbers, it is easy to verify that, for any (k, ℓ)∈

{1,· · ·, K}²withk6=ℓ,Vbkℓ·n (resp.Vbk·n) converge almost surely toVkℓ (resp.

Vbk), as n → +∞. Consequently, Φbn (resp. Ψbn) converge almost surely to Φ= I_X (resp.Ψ), as n→ +∞. This implies the almost sure convergence of (Φb⁻n^1/2+I_X)⁻¹ (resp.Φb⁻_n¹ΨbnΦb⁻n^1/2; resp.Φb⁻_n¹; resp. Φb⁻n^1/2) to ¹₂I_X (resp.Ψ; resp.I_X; resp.I_X), asn→+∞. Furthermore, denoting byk · k the norm of L(X) defined bykAk=p

tr(AA^∗) and using the properties (a⊗b)(c⊗d) =<

a, d > c⊗band tr(a⊗b) =< a, b >of the tensor product (see Dauxois et al.

(1994)), we have:

E kX⊗Xk²

=E(tr((X⊗X)(X⊗X)) =E kXk⁴_X

=E



 XK k=1

kXkk²k

!2



= XK k=1

E(kXkk⁴k) + XK k=1

XK ℓℓ=16=k

E(kXkk²kkXℓk²ℓ)

≤ XK k=1

E(kXkk⁴k) + XK k=1

XK ℓℓ=16=k

q

E(kXkk⁴k) q

E(kXℓk⁴ℓ)<+∞.

Then, the central limit theorem can be used. It gives the convergence in distri- bution, asn→+∞, of√

n n⁻¹P

i=1X⁽ⁱ⁾⊗X⁽ⁱ⁾−V

to an random variable H having the normal distribution inL(X) with mean equal to 0 and a covari- ance operator equal to that ofX⊗X. Since, by the central limit theorem again,

√nXn converges in distribution, asn→+∞, to an random variable having a normal distribution inX with mean equal to 0 and a covariance operator equal toV, we deduce from the equality√n Xn⊗Xn

=n^−1/2 √nXn

⊗ √nXn

that√n Xn⊗Xn

converges in probability to 0, asn→+∞. Therefore, from (14) and Slutsky’s theorem, we deduce thatHbn converges in distribution, as n→+∞toH. Then, from (17), we conclude thatϕbn(Hbn)−ϕ(Hbn) converges in probability to 0, as n → +∞. Then, using again Slutsky’s theorem, we deduce that ϕbn(Hbn) and ϕ(Hbn) both converge in distribution to the same distribution. Sinceϕ is a linear map (and is, therefore, continuous), this dis- tribution just is that of the random variableU = ϕ(H), that is the normal distribution in L(X) with mean 0 and covariance operator equal to that of Z=ϕ(X⊗X). Clearly,

g(X⊗X) = XK k=1

τ_k^∗τk(X⊗X)τ_k^∗τk = XK k=1

τ_k^∗((τk(X))⊗(τk(X)))τk= XK k=1

τ_k^∗(Xk⊗Xk)τk,

and

f(X⊗X) = XK k=1

XK ℓℓ=16=k

τ_k^∗τk(X⊗X)τ_ℓ^∗τℓ= XK k=1

XK ℓℓ=16=k

τ_k^∗(Xℓ⊗Xk)τℓ.

Then, sinceτkτ_j^∗=δkjIk, it follows

g(X⊗X)Ψ = XK k=1

XK j=1

XK ℓℓ=16=j

τ_k^∗(Xk⊗Xk)τkτ_j^∗Vjℓτℓ= XK k=1

XK ℓℓ=16=k

τ_k^∗(Xk⊗Xk)Vkℓτℓ

and

Ψ g(X⊗X) = XK k=1

XK ℓℓ=16=k

XK j=1

τ_k^∗Vkℓτℓτ_j^∗(Xj⊗Xj)τj= XK k=1

XK ℓℓ=16=k

τ_k^∗Vkℓ(Xℓ⊗Xℓ)τℓ

= XK k=1

XK

ℓ=1

ℓ6=k

τ_ℓ^∗Vℓk(Xk⊗Xk)τk.

Thus, Z=

XK k=1

XK ℓℓ=16=k

−1

2(τ_k^∗(Xk⊗Xk)Vkℓτℓ+τ_ℓ^∗Vℓk(Xk⊗Xk)τk) +τ_k^∗(Xℓ⊗Xk)τℓ. Using the preceding theorem and results in Eaton and Tyler (1991,1994), we can now give asymptotic distributions for the canonical coefficients. We denote by ρ^′_j

1≤j≤r (with r ∈ N^∗) the sequence of distinct eigienvalues of T in decreasing order, that is ρ^′₁ > · · · > ρ^′_r. Putting m0 = 0, denoting by mj

the multiplicity ofρ^′_j and putting νj =Pj−1

k=0mk for any j ∈ {1,· · ·r}, it is clear that for anyi∈ {νj−1+ 1,· · ·, νj} one hasρi=ρ^′_j. Further, considering the eigenspace Ej = ker(T −ρ^′_jI), we have the following decomposition in orthogonal direct sum:X =E1⊕ · · · ⊕Er. We denote by Πj the orthogonal projector fromX ontoEj, and by∆ the continuous map which associates to each self-adjoint operatorAthe vector∆(A) of its eigenvalues in nonincreasing order. For j ∈ {1,· · ·r}, we consider mj-dimensional vector given by υj = ρ^′_jJ_mj, where J_q denotes the q-dimensional vector with elements all equal to 1, and theR^mj- valued random vector:

ˆ υ_jⁿ=



 b ρν_j−1+1·n

... b ρνj·n



.

Then, putting

Λbn=



 ˆ υ₁ⁿ

... ˆ υ_rⁿ



 and Λ=



 υ1

... υr



,

we have:

Theorem 3.3. √ n

Λbn−Λ

converges in distribution, asn→+∞, to the R^p-valued random vector

ζ=





∆(Π1W Π1) ...

∆(ΠrW Πr)



, (18)

where W is a random variable having a normal distribution in L(X), with mean0and covariance operatorΘ given by:

Θ= X

1≤m,r,s,t≤p

C(m, r, s, t) (em⊗er)e⊗(es⊗et) with

C(m, r, s, t) = XK k=1

XK j=1

XK

ℓ=1

ℓ6=k

XK qq=16=j

γ_kℓjq^m,r,s,t+γ_kℓjq^m,r,t,s+γ_kℓjq^r,m,s,t+γ_kℓjq^r,m,t,s

−θ^m,r,s,t_kℓjq −θ^r,m,s,t_kℓjq −θ^s,t,m,r_kℓjq −θ_kℓjq^t,s,m,r+λ^m,r,s,t_kℓjq , γ_kℓjq^a,b,c,d= 1

4E

< Xk, τkβ^(a)>k< Xk, Vkℓτℓβ^(b)>k< Xj, τjβ^(c)>j< Xj, Vjqτqβ^(d)>j

,

θ^a,b,c,d_kℓjq = 1 2E

< Xk, τkβ^(a)>k< Xk, Vkℓτℓβ^(b)>k< Xj, τjβ^(c)>j< Xq, τqβ^(d)>q

and γ_kℓjq^a,b,c,d=E

< Xk, τkβ^(a)>k< Xℓ, τℓβ^(b)>ℓ< Xj, τjβ^(c)>j< Xq, τqβ^(d)>q

. Proof. Since∆(Tbn) =Λbn and ∆(T) = Λ, we deduce from Theorem 3.2 and the Theorem 2.1 of Eaton and Tyler (1994) that √

n

Λbn−Λ

converges in distribution, as n → +∞, to the random variable given in (18) with W = P^∗U P, whereP =Pp

ℓ=1eℓ⊗β^(ℓ). Clearly,W has a normal distribution with mean 0 and covariance operatorΘequal to that ofP^∗ZP. In order to give an explicit expression ofΘ, let us first note that:

P^∗ZP = XK k=1

XK

ℓ=1

ℓ6=k

−1

2(P^∗τ_k^∗(Xk⊗Xk)VkℓτℓP+P^∗τ_ℓ^∗Vℓk(Xk⊗Xk)τkP) +P^∗τ_k^∗(Xℓ⊗Xk)τℓP

= XK k=1

XK ℓℓ=16=k

−1

2((P^∗τ_ℓ^∗VℓkXk)⊗(P^∗τ_k^∗Xk) + (P^∗τ_k^∗Xk)⊗(P^∗τ_ℓ^∗VℓkXk)) +(P^∗τ_ℓ^∗Xℓ)⊗(P^∗τ_k^∗Xk).

Since

P^∗τ_ℓ^∗VℓkXk = Xp m=1

β^(m)⊗em

!

τ_ℓ^∗VℓkXk= Xp m=1

< β^(m), τ_ℓ^∗VℓkXk>_X em

= Xp m=1

< τℓβ^(m), VℓkXk>ℓem

and, similarly,P^∗τ_k^∗Xk =Pp

m=1< τkβ^(m), Xk>kem, it follows:

P^∗ZP = Xp m=1

Xp r=1



 XK k=1

XK

ℓ=1

ℓ6=k

− 1

2(< τℓβ^(m), VℓkXk >ℓ< τkβ^(r), Xk>k

+< τℓβ^(r), VℓkXk >ℓ< τkβ^(m), Xk>k) + < τℓβ^(m), Xℓ>ℓ< τkβ^(r), Xk >k

i em⊗er.

From:

E

< τℓβ^(m), VℓkXk>ℓ< τkβ^(r), Xk >k

=E

<(Xk⊗Xk)(τkβ^(r)), Vkℓτℓβ^(m)>k

=<E(Xk⊗Xk)(τkβ^(r)), Vkℓτℓβ^(m)>k

=< Vkτkβ^(r), Vkℓτℓβ^(m)>k

=< τkβ^(r), Vkℓτℓβ^(m)>k, E

< τℓβ^(r), VℓkXk >ℓ< τkβ^(m), Xk>k

=< τkβ^(m), Vkℓτℓβ^(r)>k

and E

< τℓβ^(m), Xℓ>ℓ< τkβ^(r), Xk>k

=E

<(Xℓ⊗Xk)(τℓβ^(m)), τkβ^(r)>k

=<E(Xℓ⊗Xk)(τℓβ^(m)), τkβ^(r)>k

=< Vkℓτℓβ^(m), τkβ^(r)>k, we deduce thatE(P^∗ZP) = 0. Thus,

Θ=E (P^∗ZP)e⊗(P^∗ZP)

= X

1≤m,r,s,t≤p

C(m, r, s, t) (em⊗er)⊗e(es⊗et),

where

C(m, r, s, t) = XK k=1

XK j=1

XK

ℓ=1

ℓ6=k

XK qq=16=j

E Y_kℓ^m,rY_jq^s,q

with Y_kℓ^m,r= −1

2

< τℓβ^(m), VℓkXk>ℓ< τkβ^(r), Xk >k+< τℓβ^(r), VℓkXk >ℓ< τkβ^(m), Xk>k

+< τℓβ^(m), Xℓ>ℓ< τkβ^(r), Xk >k. Further calculations give

E Y_kℓ^m,rY_jq^s,q

= γ_kℓjq^m,r,s,t+γ_kℓjq^m,r,t,s+γ_kℓjq^r,m,s,t+γ_kℓjq^r,m,t,s

−θ_kℓjq^m,r,s,t−θ_kℓjq^r,m,s,t−θ_kℓjq^s,t,m,r−θ^t,s,m,r_kℓjq +λ^m,r,s,t_kℓjq .

WhenT has simple eigenvalues, that isρ1> ρ2>· · ·> ρq, the preceding theorem has a simpler statement. We have:

Corollary 3.1. When the eigenvalues of T are simple, √n

Λbn−Λ con- verges in distribution, as n → +∞, to a random variable having a normal distribution inR^p with mean0 and covariance matrixΣ= (σij)_1≤i,j≤p with:

σij = X

1≤m,r,s,t≤p

β_m⁽ⁱ⁾β_r⁽ⁱ⁾β^(j)_s β_t^(j)C(m, r, s, t).

Proof. In this case, m1 = · · ·mp = 1 and, for any j ∈ {1,· · ·, p}, Πj = β^(j)⊗β^(j). Thus

ΠjW Πj= ((β^(j)⊗β^(j))W(β^(j)⊗β^(j)) = (β^(j)⊗β^(j))(β^(j)⊗(W β^(j)))

=< β^(j), W β^(j)>_X β^(j)⊗β^(j),

and, therefore,∆(ΠjW Πj) =< β^(j), W β^(j)>_X. Then,ζis a linear function of W and, consequently, it has a normal distribution with mean 0 and covariance matrix Σ = (σij)_1≤i,j≤p with σij = E < β⁽ⁱ⁾, W β⁽ⁱ⁾>_X< β^(j), W β^(j)>_X

. Denoting by <·,· >the inner product of operators defined by < A, B >=

tr (AB^∗), we have:

< W, β^(j)⊗β^(j)>= tr

W(β^(j)⊗β^(j))

= tr

β^(j)⊗(W β^(j))

=< β^(j), W β^(j)>_X, it follows that

σij =E

< β⁽ⁱ⁾, W β⁽ⁱ⁾>_X< β^(j), W β^(j)>_X

=E

< W, β⁽ⁱ⁾⊗β⁽ⁱ⁾>< W, β^(j)⊗β^(j)>

=E

<(W⊗eW)(β⁽ⁱ⁾⊗β⁽ⁱ⁾), β^(j)⊗β^(j)>

=<E(W⊗eW)(β⁽ⁱ⁾⊗β⁽ⁱ⁾), β^(j)⊗β^(j)>

=< Θ(β⁽ⁱ⁾⊗β⁽ⁱ⁾), β^(j)⊗β^(j)>

= X

1≤m,r,s,t≤p

C(m, r, s, t)< (em⊗er)⊗e(es⊗et)

(β⁽ⁱ⁾⊗β⁽ⁱ⁾), β^(j)⊗β^(j)>

= X

1≤m,r,s,t≤p

C(m, r, s, t)< em⊗er, β⁽ⁱ⁾⊗β⁽ⁱ⁾>< es⊗et, β^(j)⊗β^(j)> .

Then, the required result is obtained from

< em⊗er, β⁽ⁱ⁾⊗β⁽ⁱ⁾>= tr

(em⊗er)(β⁽ⁱ⁾⊗β⁽ⁱ⁾>)

= tr

< em, β⁽ⁱ⁾>_X β⁽ⁱ⁾⊗er

=< em, β⁽ⁱ⁾>_X< er, β⁽ⁱ⁾_X >

=β⁽ⁱ⁾_mβ_r⁽ⁱ⁾

and< es⊗et, β^(j)⊗β^(j)>=βs^(j)β_t^(j).