Asymptotic theory of multiple-set linear canonical analysis

TL;DR

This paper develops the asymptotic theory for multiple-set linear canonical analysis (MSLCA), including estimators, their properties, and a test for mutual non-correlation among Euclidean random variables.

Contribution

It introduces a new asymptotic framework for MSLCA, including consistent and asymptotically normal estimators based on empirical covariance operators.

Findings

Proved consistency of MSLCA estimators

Established asymptotic normality of estimators

Developed a statistical test for mutual non-correlation

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 asymptotics for these estimators are obtained. More precisely, we prove their consistency and we obtain asymptotic normality for the estimator of the operator that gives MSLCA, and also for the estimator of the vector of canonical coefficients. These results are then used to obtain a test for mutual non-correlation between the involved Euclidean random variables.

Equations

$$\alpha=\left(\begin{array}[c]{c}\alpha_{1}\\ \vdots\\ \alpha_{K}\end{array}\right),$$

$$\forall \alpha \in \mathcal{X},\forall \beta \in \mathcal{X},⟨\alpha ,\beta {⟩}_{\mathcal{X}}=\sum _{k=1}^K⟨{\alpha }_k,{\beta }_k{⟩}_k.$$

$$X=\left(\begin{array}[c]{c}X_{1}\\ \vdots\\ X_{K}\end{array}\right),$$

$${\alpha }^{(j)}=\arg \underset{\alpha \in C_j}{\max }\mathbb{E}\left(<\alpha ,X{>}_{\mathcal{X}}^2\right),$$

$$C_1=\left\{\alpha \in \mathcal{X}/\sum _{k=1}^{K}var(<{\alpha }_k,X_k{>}_k)=1\right\},$$

$$C_j=\left\{\alpha \in C_1/\sum _{k=1}^{K}cov\left(<{\alpha }_k^{(r)},X_k{>}_k,<{\alpha }_k,X_k{>}_k\right)=0,\forall r\in \{1,\cdots ,j-1\}\right\}.$$

$$V_{k\ell }=\mathbb{E}\left(X_{\ell }\otimes X_k\right)=V_{\ell k}^{\ast }\text{ and }{V}_k:=V_{kk},$$

$$var\left(⟨{\alpha }_k,X_k{⟩}_k\right)=\mathbb{E}\left(⟨{\alpha }_k,X_k{⟩}_k^2\right)=\mathbb{E}\left(⟨{\alpha }_k,(X_k\otimes X_k)({\alpha }_k){⟩}_k\right)=⟨{\alpha }_k,V_k{\alpha }_k{⟩}_k,$$

$$cov\left(⟨{\alpha }_k,X_k{⟩}_k,⟨{\beta }_{\ell },X_{\ell }{⟩}_{\ell }\right)$$

$$C_1=\left\{\alpha \in \mathcal{X}/\sum _{k=1}^K<{\alpha }_k,V_k{\alpha }_k{>}_k=1\right\},$$

$$C_j=\left\{\alpha \in C_1/\sum _{k=1}^K<{\alpha }_k^{(r)},V_k{\alpha }_k{>}_k=0,\forall r\in \{1,\cdots ,j-1\}\right\}.$$

$$\mathbb{E}\left(<\alpha ,X{>}_{\mathcal{X}}^2\right)$$

$$\phi (\alpha )=\sum _{k=1}^K\sum _{\stackrel{\ell =1}{\ell \ne k}}^K<{\alpha }_k,V_{k\ell }{\alpha }_{\ell }{>}_k.$$

$${\tau }_k:\alpha \in \mathcal{X}\mapsto {\alpha }_k\in {\mathcal{X}}_k;$$

$${\tau }_k^{\ast }:t\in {\mathcal{X}}_k\mapsto (\underset{k-1\text{ times}}{\underbrace{0,\cdots ,0}},t,0,\cdots ,0{)}^T\in \mathcal{X},$$

$$\Phi =\sum _{k=1}^K{\tau }_k^{\ast }{V}_k{\tau }_k\text{ and }\Psi =\sum _{k=1}^K\sum _{\stackrel{\ell =1}{\ell \ne k}}^K{\tau }_k^{\ast }{V}_{k\ell }{\tau }_{\ell }.$$

$${\alpha }^{(j)}={\Phi }^{-1/2}{\beta }^{(j)}=\left(V_1^{-1/2}{\beta }_1^{(j)},\cdots ,V_K^{-1/2}{\beta }_K^{(j)}\right),$$

$$\phi (\alpha )$$

$$\beta=\left(\begin{array}[]{c}\beta_{1}\\ \vdots\\ \beta_{K}\end{array}\right)\in\mathcal{X}.$$

$$C_1^{\prime }$$

$$C_j^{\prime }$$

$$\Psi {\Phi }^{-1/2}\beta$$

$${\Phi }^{-1/2}\Psi {\Phi }^{-1/2}\beta$$

$$<\beta ,{\Phi }^{-1/2}\Psi {\Phi }^{-1/2}\beta {>}_{\mathcal{X}}$$

$${\rho }_j=\phi ({\alpha }^{(j)})=\mathbb{E}\left(<{\alpha }^{(j)},X{>}_{\mathcal{X}}^2\right)-1\ge -1.$$

$${\rho }_j=\phi ({\alpha }^{(j)})$$

$$\mathbb{E}\left(<{\alpha }_k^{(j)},X_k{>}_k^2\right)=var\left(<{\alpha }_k^{(j)},X_k{>}_k\right)\le \sum _{\ell =1}^{K}var\left(<{\alpha }_{\ell }^{(j)},X_{\ell }{>}_{\ell }\right)=1,$$

$${\rho }_j\le \sum _{k=1}^K\sum _{\stackrel{\ell =1}{\ell \ne k}}^{K}1=K(K-1).$$

$$\forall j\in \{1,\cdots ,q\},{\rho }_j=0\iff T=0\iff \Psi =0\iff \forall (k,\ell )\in \{1,\cdots ,K{\}}^2,k\ne \ell ,V_{k\ell }=0.$$

$${\tau }_1^{\ast }\left(S{\tau }_{2}x-\rho {\tau }_{1}x\right)=-{\tau }_2^{\ast }\left(S^{\ast }{\tau }_{1}x-\rho {\tau }_{2}x\right).$$

Full text

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 theory for LCA and related applications have been tackled by several authors (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. Properties of the related eigenvalues, called canonical coefficients, are then given. In Section 3, we tackle the problem of estimating MSLCA. More precisely, estimators based on empirical covariance operators are introduced. Then, consistency of the obtained estimators is proved. Further, we derive the asymptotic distribution 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 $K\geq 2$, let us consider random variables $X_{1},\cdots,X_{K}$ defined on a probability space $(\Omega,\mathcal{A},P)$ and valued into Euclidean vector spaces $\mathcal{X}_{1},\cdots,\mathcal{X}_{K}$ respectively. Denoting by $\mathbb{E}$ the mathematical expectation related to $P$, we assume that, for any $k\in\{1,\cdots,K\}$, we have $\mathbb{E}(\|X_{k}\|_{k}^{2})<+\infty$ where $\|\cdot\|_{k}$ denotes the norm induced by the inner product $\langle\cdot,\cdot\rangle_{k}$ of $\mathcal{X}_{k}$, and, without loss of generality, that $\mathbb{E}(X_{k})=0$. Each vector $\alpha$ in the vector space $\mathcal{X}:=\mathcal{X}_{1}\times\cdots\mathcal{X}_{K}$ will be writen as

[TABLE]

and we recall that $\mathcal{X}$ is an Euclidean vector space equipped with the inner product $\langle\cdot,\cdot\rangle_{\mathcal{X}}$ defined by:

[TABLE]

We denote by $\|\cdot\|_{\mathcal{X}}$ the norm induced by this inner product. Considering the $\mathcal{X}$-valued random variable

[TABLE]

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 $\left(\alpha^{(j)}\right)_{1\leq j\leq q}$ of vectors of $E$, where $q=\dim(\mathcal{X})$, satisfying:

[TABLE]

where

[TABLE]

and, for $j\geq 2$ :

[TABLE]

Remark 2.1

1. The constraints sets given in (2) and (3) can be expressed by using covariance operators defined for $(k,\ell)\in\{1,\cdots,K\}^{2}$ by:

[TABLE]

where $\otimes$ denotes the tensor product such that, for any $(x,y)$, $x\otimes y$ is the linear map : $h\mapsto<x,h>y$, and $T^{\ast}$ denotes the adjoint of $T$. Indeed, it is easily seen that, for $(\alpha,\beta)\in\mathcal{X}^{2}$,

[TABLE]

and

[TABLE]

Therefore,

[TABLE]

and

[TABLE]

2. For any $\alpha\in C_{1}$, one has:

[TABLE]

where

[TABLE]

Then, the MSLCA of $X$ is obtained by minimizing $\varphi(\alpha)$ under the constraints expressed in (4) and (5).

For $k\in\{1,\cdots,K\}$, the covariance operator $V_{k}$ is a self-adjoint non-negative operator. From now on, we assume that it is invertible. Let $\tau_{k}$ be the canonical projection defined as

[TABLE]

its adjoint $\tau_{k}^{\ast}$ of $\tau_{k}$ is the map given by:

[TABLE]

where we denote by $a^{T}$ the transposed of $a$. Now, let us consider the operators of $\mathcal{L}(\mathcal{X})$ given by:

[TABLE]

From the fact that $\tau_{k}\tau^{\ast}_{\ell}=\delta_{k\ell}I_{k}$, where $\delta_{k\ell}$ is the usual Kronecker symbol and $I_{k}$ is the identity operator of $\mathcal{X}_{k}$, it is easily seen that $\Phi$ is also an invertible self-adjoint and non-negative operator, with $\Phi^{-1}=\sum_{k=1}^{K}\tau_{k}^{\ast}V_{k}^{-1}\tau_{k}$ and $\Phi^{-1/2}=\sum_{k=1}^{K}\tau_{k}^{\ast}V_{k}^{-1/2}\tau_{k}$. The following theorem shows how to obtain a MSLCA of $X$. 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 $\left\{\beta^{(1)},\cdots,\beta^{(q)}\right\}$ be an orthonormal basis of $\mathcal{X}$ such that $\beta^{(j)}$ is an eigenvector of the operator $T=\Phi^{-1/2}\Psi\Phi^{-1/2}$ associated with the $j$-th largest eigenvalue $\rho_{j}$ of $T$. Then, the sequence $\left(\alpha^{(j)}\right)_{1\leq j\leq q}$ given by:

[TABLE]

consists of solutions of (1) under the constraints (2) and (3), and we have: $\rho_{j}=<\beta^{(j)},T\beta^{(j)}>_{E}=\varphi(\alpha^{(j)})$.

Proof. Putting $\beta_{k}=V_{k}^{1/2}\alpha_{k}$ and $\beta^{(r)}=V_{k}^{1/2}\alpha^{(r)}_{k}$, we have:

[TABLE]

where

[TABLE]

Since $V_{k}=V_{k}^{1/2}V_{k}^{1/2}$, having $\alpha\in C_{j}$ is equivalent to having $\beta\in C_{j}^{\prime}$, where:

[TABLE]

and for $j\geq 2$:

[TABLE]

Further, for any $\beta\in\mathcal{X}$:

[TABLE]

and

[TABLE]

Thus,

[TABLE]

where $\psi$ is defined in (7). Then, the MSLCA optimization problem reduces to the maximization of $<\beta,\Phi^{-1/2}\Psi\Phi^{-1/2}\beta>_{\mathcal{X}}$ under the constraints (8) and (9). Since $T=\Phi^{-1/2}\Psi\Phi^{-1/2}$ is a self-adjoint operator, this is a well known maximization problem for which a solution is obtained from the spectral analysis of $T$ as stated in the theorem. $\Box$

**Definition 2.2. **The $\rho_{j}$’s are termed the canonical coefficients. The $\alpha^{(j)}$’s are termed vectors of canonical directions.

The following theorem gives some properties of the canonical coefficients.

**Theorem 2.2. **

(i) $\forall j\in\{1,\cdots,q\}$, $-1\leq\rho_{j}\leq K(K-1).$

(ii) $\forall j\in\{1,\cdots,q\}$, $\rho_{j}=0\Leftrightarrow\forall(k,\ell)\in\{1,\cdots,K\}^{2},\,k\neq\ell,\,V_{k\ell}=0.$

*Proof. *

(i) First, using (6), we have for any $j\in\{1,\cdots,q\}$,

[TABLE]

On the other hand, we have:

[TABLE]

Since, for any $k\in\{1,\cdots,K\}$, one has:

[TABLE]

it follows that:

[TABLE]

(ii) Since the $\rho_{j}$’s are the eigenvalues of $T$, we have:

[TABLE]

$\Box$

**Remark 2.2. **

1. When $K=2$, one has $\Phi=\tau_{1}^{\ast}V_{1}\tau_{1}+\tau_{2}^{\ast}V_{2}\tau_{2}$ and $\Psi=\tau_{1}^{\ast}V_{12}\tau_{2}+\tau_{2}^{\ast}V_{21}\tau_{1}$. Then it is easy to check that $T=\tau_{1}^{\ast}S\tau_{2}+\tau_{2}^{\ast}S^{\ast}\tau_{1}$, where $S=V_{1}^{-1/2}V_{12}V_{2}^{-1/2}$. Let $x$ be an eigenvector of $T$ associated with an eigenvalue $\rho\neq 0$. We have $Tx=\rho x$, that is equivalent to having:

[TABLE]

This implies:

[TABLE]

and, putting $x_{1}=\tau_{1}x$ and $x_{2}=\tau_{2}x$, we obtain

[TABLE]

where

[TABLE]

Conversely, if (10) holds then, putting $x=\tau_{1}^{\ast}x_{1}+\tau_{2}^{\ast}x_{2}$, we have:

[TABLE]

Moreover, since

[TABLE]

and

[TABLE]

it follows that

[TABLE]

2. The preceding remark shows the equivalence between MSLCA and linear canonical analysis (LCA) when $K=2$. Recall that LCA of $X_{1}$ and $X_{2}$ is obtained from the spectral analysis of $R$ (see, e.g., Dauxois and Pouse (1975), Pousse (1992), Fine (2000)). More precisely, $\left\{\beta^{(j)},\rho_{j}\right\}_{1\leq j\leq q}$ is defined as in Theorem 2.1 if, and only if, $\left\{u_{1}^{(j)},u_{2}^{(j)}\rho_{j}^{2}\right\}_{1\leq j\leq q}$, where $u_{\ell}^{(j)}=\frac{1}{\sqrt{2}}\tau_{\ell}\beta^{(j)}$ ($\ell\in\{1,2\}$), is a LCA of $X_{1}$ and $X_{2}$.

3 Estimation and asymptotic theory

In this section, we deal with estimation of MSLCA. For $k=1,\cdots,K$, let $\{X_{k}^{(i)}\}_{1\leq i\leq n}$ be an i.i.d. sample of $X_{k}$. 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 $(k,\ell)\in\{1,\cdots,K\}^{2}$, let us consider the sample means and covariance operators:

[TABLE]

and the random operators valued into $\mathcal{L}(\mathcal{X})$ defined as

[TABLE]

Then, we estimate $T$ by

[TABLE]

Considering the eigenvalues $\widehat{\rho}_{1\cdot n}\geq\widehat{\rho}_{2\cdot n}\cdots\geq\widehat{\rho}_{q\cdot n}$ of $\widehat{T}_{n}$, and $\left\{\widehat{\beta}^{(1)}_{n},\cdots,\widehat{\beta}^{(q)}_{n}\right\}$ an orthonormal basis of $\mathcal{X}$ such that $\widehat{\beta}^{(j)}_{n}$ is an eigenvector of $\widehat{T}_{n}$ associated with $\widehat{\rho}_{j\cdot n}$. Then, we estimate $\rho_{j}$ by $\widehat{\rho}_{j\cdot n}$, and $\beta^{(j)}$ by $\widehat{\beta}^{(j)}_{n}$. The following theorem establishes strong consistency for these estimators.

**Theorem 3.1. ** For any integer $j\in\{1,\cdots,q\}$ :

(i) $\widehat{\rho}_{j\cdot n}$ converge almost surely, as $n\rightarrow+\infty$, to $\rho_{j}$.

(ii) $sign(\langle\widehat{\beta}^{(j)}_{n},\beta^{(j)}\rangle_{\mathcal{X}})\,\widehat{\beta}^{(j)}_{n}$ converges almost surely, as $n\rightarrow+\infty$, to $\beta^{(j)}$ in $\mathcal{X}$.

Proof. From obvious applications of the strong law of large numbers, it is easily seen that $\widehat{T}_{n}$ converges almost surely in $\mathcal{L}(\mathcal{X})$, as $n\rightarrow+\infty$ to $T$. Then using Lemma 1 in Ferré and Yao (2003), we obtain the inequality $|\widehat{\rho}_{j\cdot n}-\rho_{j}|\leq\|\widehat{T}_{n}-T\|$ from what $(i)$ is deduced. Clearly, each $\beta^{(j)}\otimes\beta^{(j)}$ and is a projector onto an eigenspace. Thefore, using Proposition 3 in Dossou-Gbete and Pousse (1991), we deduce that $\widehat{\beta}^{(j)}_{n}\otimes\widehat{\beta}^{(j)}_{n}$ converges almost surely in ${\mathcal{L}(\mathcal{X})}$ to $\beta^{(j)}\otimes\beta^{(j)}$, as $n\rightarrow+\infty$. Using again Lemma 1 in Ferré and Yao (2003), we obtain the inequality

[TABLE]

from what we deduce (ii). $\Box$

3.2 Asymptotic distribution

In this section, we assume that, for $k\in\{1,\cdots,K\}$, we have $\mathbb{E}\left(\|X_{k}\|_{k}^{4}\right)<+\infty$ and $V_{k}=I_{k}$, where $I_{k}$ denotes the identity operator of $\mathcal{X}_{k}$. We first derive an asymptotic distribution for $\widehat{T}_{n}$, then we obtain these of the canonical coefficients.

**Theorem 3.2. ** $\sqrt{n}\left(\widehat{T}_{n}-T\right)$ converges in distribution, as $n\rightarrow+\infty$, to a random variable $U$ having a normal distribution in $\mathcal{L}(\mathcal{X})$, with mean [math] and covariance operator $\Gamma$ equal to that of the random operator:

[TABLE]

*Proof. * Under the above assumptions,

[TABLE]

where $I_{\mathcal{X}}$ is the indentity operator of $\mathcal{X}$, and

[TABLE]

Clearly,

[TABLE]

where $V=\mathbb{E}(X\otimes X)$. Moreover, putting

[TABLE]

we have

[TABLE]

where $\overline{X}_{n}=n^{-1}\sum_{i=1}^{n}X^{(i)}$ and

[TABLE]

Therefore, using (12) and (13), we obtain

[TABLE]

where $\widehat{H}_{n}=\sqrt{n}(\widehat{V}_{n}-V)$ and $f$ is the operator defined as

[TABLE]

Further, since $I_{\mathcal{X}}=\sum_{k=1}^{K}\tau_{k}^{\ast}\tau_{k}$, we obtain

[TABLE]

where $g$ is the operator $g\,:\,A\in\mathcal{L}(\mathcal{X})\mapsto\sum_{k=1}^{K}\tau_{k}^{\ast}\tau_{k}A\tau_{k}^{\ast}\tau_{k}\in\mathcal{L}(\mathcal{X})$. Then, using (11), (15) and (16), we obtain $\sqrt{n}\left(\widehat{T}_{n}-T\right)=\widehat{\varphi}_{n}(\widehat{H}_{n})$, where $\widehat{\varphi}_{n}$ is the random operator from $\mathcal{L}(\mathcal{X})$ to itself defined by

[TABLE]

Considering the operator

[TABLE]

and denoting by $\|\cdot\|_{\infty}$ (resp. $\|\cdot\|_{\infty\infty}$) the norm of $\mathcal{L}(\mathcal{X})$ (resp. $\mathcal{L}(\mathcal{L}(\mathcal{X}))$) defined by $\|A\|_{\infty}=\sup_{x\in\mathcal{X}-\{0\}}\|Ax\|_{\mathcal{X}}/\|x\|_{\mathcal{X}}$ (resp. $\|h\|_{\infty\infty}=\sup_{B\in\mathcal{L}(\mathcal{X})-\{0\}}\|h(B)\|_{\infty}/\|B\|_{\infty}$) for any $A$ (resp. $h$) in $\mathcal{L}(\mathcal{X})$ (resp. $\mathcal{L}(\mathcal{L}(\mathcal{X}))$), we have

[TABLE]

Using the strong law of large numbers, it is easy to verify that, for any $(k,\ell)\in\{1,\cdots,K\}^{2}$ with $k\neq\ell$, $\widehat{V}_{k\ell\cdot n}$ (resp. $\widehat{V}_{k\cdot n}$) converge almost surely to $V_{k\ell}$ (resp. $\widehat{V}_{k}$), as $n\rightarrow+\infty$. Consequently, $\widehat{\Phi}_{n}$ (resp. $\widehat{\Psi}_{n}$) converge almost surely to $\Phi=I_{\mathcal{X}}$ (resp. $\Psi$), as $n\rightarrow+\infty$. This implies the almost sure convergence of $(\widehat{\Phi}_{n}^{-1/2}+I_{\mathcal{X}})^{-1}$ (resp. $\widehat{\Phi}_{n}^{-1}\widehat{\Psi}_{n}\widehat{\Phi}_{n}^{-1/2}$; resp. $\widehat{\Phi}_{n}^{-1}$; resp. $\widehat{\Phi}_{n}^{-1/2}$) to $\frac{1}{2}I_{\mathcal{X}}$ (resp. $\Psi$; resp. $I_{\mathcal{X}}$; resp. $I_{\mathcal{X}}$), as $n\rightarrow+\infty$. Furthermore, denoting by $\|\cdot\|$ the norm of $\mathcal{L}(\mathcal{X})$ defined by $\|A\|=\sqrt{\textrm{tr}(AA^{\ast})}$ and using the properties $(a\otimes b)(c\otimes d)=<a,d>c\otimes b$ and tr$(a\otimes b)=<a,b>$ of the tensor product (see Dauxois et al. (1994)), we have:

[TABLE]

Then, the central limit theorem can be used. It gives the convergence in distribution, as $n\rightarrow+\infty$, of $\sqrt{n}\left(n^{-1}\sum_{i=1}X^{(i)}\otimes X^{(i)}-V\right)$ to an random variable $H$ having the normal distribution in $\mathcal{L}(\mathcal{X})$ with mean equal to [math] and a covariance operator equal to that of $X\otimes X$. Since, by the central limit theorem again, $\sqrt{n}\overline{X}_{n}$ converges in distribution, as $n\rightarrow+\infty$, to an random variable having a normal distribution in $\mathcal{X}$ with mean equal to [math] and a covariance operator equal to $V$, we deduce from the equality $\sqrt{n}\left(\overline{X}_{n}\otimes\overline{X}_{n}\right)=n^{-1/2}\left(\sqrt{n}\overline{X}_{n}\right)\otimes\left(\sqrt{n}\overline{X}_{n}\right)$ that $\sqrt{n}\left(\overline{X}_{n}\otimes\overline{X}_{n}\right)$ converges in probability to [math], as $n\rightarrow+\infty$. Therefore, from (14) and Slutsky’s theorem, we deduce that $\widehat{H}_{n}$ converges in distribution, as $n\rightarrow+\infty$ to $H$. Then, from (17), we conclude that $\widehat{\varphi}_{n}(\widehat{H}_{n})-\varphi(\widehat{H}_{n})$ converges in probability to [math], as $n\rightarrow+\infty$. Then, using again Slutsky’s theorem, we deduce that $\widehat{\varphi}_{n}(\widehat{H}_{n})$ and $\varphi(\widehat{H}_{n})$ both converge in distribution to the same distribution. Since $\varphi$ is a linear map (and is, therefore, continuous), this distribution just is that of the random variable $U=\varphi(H)$, that is the normal distribution in $\mathcal{L}(\mathcal{X})$ with mean [math] and covariance operator equal to that of $Z=\varphi(X\otimes X)$. Clearly,

[TABLE]

and

[TABLE]

Then, since $\tau_{k}\tau_{j}^{\ast}=\delta_{kj}I_{k}$, it follows

[TABLE]

and

[TABLE]

Thus,

[TABLE]

$\Box$

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 $\left(\rho^{\prime}_{j}\right)_{1\leq j\leq r}$ (with $r\in\mathbb{N}^{\ast}$) the sequence of distinct eigienvalues of $T$ in decreasing order, that is $\rho^{\prime}_{1}>\cdots>\rho^{\prime}_{r}$. Putting $m_{0}=0$, denoting by $m_{j}$ the multiplicity of $\rho^{\prime}_{j}$ and putting $\nu_{j}=\sum_{k=0}^{j-1}m_{k}$ for any $j\in\{1,\cdots r\}$, it is clear that for any $i\in\{\nu_{j-1}+1,\cdots,\nu_{j}\}$ one has $\rho_{i}=\rho^{\prime}_{j}$. Further, considering the eigenspace $E_{j}=\ker(T-\rho^{\prime}_{j}I)$, we have the following decomposition in orthogonal direct sum: $\mathcal{X}=E_{1}\oplus\cdots\oplus E_{r}$. We denote by $\Pi_{j}$ the orthogonal projector from $\mathcal{X}$ onto $E_{j}$, and by $\Delta$ the continuous map which associates to each self-adjoint operator $A$ the vector $\Delta(A)$ of its eigenvalues in nonincreasing order. For $j\in\{1,\cdots r\}$, we consider $m_{j}$-dimensional vector given by $\upsilon_{j}=\rho^{\prime}_{j}\mathbb{J}_{m_{j}}$, where $\mathbb{J}_{q}$ denotes the $q$-dimensional vector with elements all equal to $1$, and the $\mathbb{R}^{m_{j}}$- valued random vector:

[TABLE]

Then, putting

[TABLE]

we have:

**Theorem 3.3. ** $\sqrt{n}\left(\widehat{\Lambda}_{n}-\Lambda\right)$ converges in distribution, as $n\rightarrow+\infty$, to the $\mathbb{R}^{p}$-valued random vector

[TABLE]

where $W$ is a random variable having a normal distribution in $\mathcal{L}(\mathcal{X})$, with mean [math] and covariance operator $\Theta$ given by:

[TABLE]

with

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

*Proof. * Since $\Delta(\widehat{T}_{n})=\widehat{\Lambda}_{n}$ and $\Delta(T)=\Lambda$, we deduce from Theorem 3.2 and the Theorem 2.1 of Eaton and Tyler (1994) that $\sqrt{n}\left(\widehat{\Lambda}_{n}-\Lambda\right)$ converges in distribution, as $n\rightarrow+\infty$, to the random variable given in (21) with $W=P^{\ast}UP$, where $P=\sum_{\ell=1}^{p}e_{\ell}\otimes\beta^{(\ell)}$. Clearly, $W$ has a normal distribution with mean [math] and covariance operator $\Theta$ equal to that of $P^{\ast}ZP$. In order to give an explicit expression of $\Theta$, let us first note that:

[TABLE]

Since

[TABLE]

and, similarly, $P^{\ast}\tau_{k}^{\ast}X_{k}=\sum_{m=1}^{p}<\tau_{k}\beta^{(m)},X_{k}>_{k}e_{m}$, it follows:

[TABLE]

From:

[TABLE]

[TABLE]

and

[TABLE]

we deduce that $\mathbb{E}(P^{\ast}ZP)=0$. Thus,

[TABLE]

where

[TABLE]

with

[TABLE]

Further calculations give

[TABLE]

$\Box$

When $T$ has simple eigenvalues, that is $\rho_{1}>\rho_{2}>\cdots>\rho_{q}$, the preceding theorem has a simpler statement. We have:

**Corollary 3.1. ** When the eigenvalues of $T$ are simple, $\sqrt{n}\left(\widehat{\Lambda}_{n}-\Lambda\right)$ converges in distribution, as $n\rightarrow+\infty$, to a random variable having a normal distribution in $\mathbb{R}^{p}$ with mean [math] and covariance matrix $\Sigma=\left(\sigma_{ij}\right)_{1\leq i,j\leq p}$ with:

[TABLE]

*Proof. *In this case, $m_{1}=\cdots m_{p}=1$ and, for any $j\in\{1,\cdots,p\}$, $\Pi_{j}=\beta^{(j)}\otimes\beta^{(j)}$. Thus

[TABLE]

and, therefore, $\Delta(\Pi_{j}W\Pi_{j})=<\beta^{(j)},W\beta^{(j)}>_{\mathcal{X}}$. Then, $\zeta$ is a linear function of $W$ and, consequently, it has a normal distribution with mean [math] and covariance matrix $\Sigma=\left(\sigma_{ij}\right)_{1\leq i,j\leq p}$ with $\sigma_{ij}=\mathbb{E}\left(<\beta^{(i)},W\beta^{(i)}>_{\mathcal{X}}<\beta^{(j)},W\beta^{(j)}>_{\mathcal{X}}\right)$. Denoting by $<\cdot,\cdot>$ the inner product of operators defined by $<A,B>=\textrm{tr }(AB^{\ast})$, we have:

[TABLE]

it follows that

[TABLE]

Then, the required result is obtained from

[TABLE]

and $<e_{s}\otimes e_{t},\beta^{(j)}\otimes\beta^{(j)}>=\beta^{(j)}_{s}\beta^{(j)}_{t}$. $\Box$

4 Testing for mutual non-correlation

In this section, we consider the problem of testing for mutual non-correlation between $X_{1},X_{2},\cdots,X_{K}$, that is testing for non-correlation between any pair $(X_{k},X_{\ell})$ for $(k,\ell)\in\{1,\cdots,K\}^{2}$. The null hypothesis is

[TABLE]

and the alternative is given by:

[TABLE]

that is testing for non-correlation between any pair $(X_{k},X_{\ell})$ of variables. We will first introduce a test statistic for this problem, then we will derive its asymptotic distribution under the null hypothesis, in the general case and in case $X$ has an elliptical distribution, by using the results of the preceding section.

4.1 A test statistic

For $(k,\ell)\in\{1,\cdots,K\}^{2}$, denoting by $\pi_{k\ell}$ the operator

[TABLE]

we have for $k\neq\ell$:

[TABLE]

Therefore, $\mathcal{H}_{0}$ is equivalent to having $\sum_{k=2}^{K}\sum_{\ell=1}^{k-1}\textrm{tr}\left(\pi_{k\ell}(T)\pi_{k\ell}(T)^{\ast}\right)=0$. This leads us to take as test statistic the random variable $\widehat{S}_{n}$ given by:

[TABLE]

4.2 Asymptotic distribution under null hypothesis

For $k\in\{1,\cdots,K\}$, we denote by $\{e_{j}^{(k)};\,j=1,\cdots,p_{k}\}$ an orthonormal basis of $\mathcal{X}_{k}$, where $p_{k}=\dim(\mathcal{X}_{k})$, and we consider the matrix:

[TABLE]

where $\Gamma_{k\ell,rs}$ is the $p_{k}p_{\ell}\times p_{r}p_{s}$ matrix given by

[TABLE]

with

[TABLE]

Then, putting $d=\sum_{k=1}^{K}\sum_{\ell=1}^{k-1}p_{k}p_{\ell}$, we have:

Theorem 4.1. Under $\mathcal{H}_{0}$, the sequence $n\widehat{S}_{n}$ converges in distribution, as $n\rightarrow+\infty$, to $\mathcal{Q}=\mathbb{W}^{T}\mathbb{W}$, where $\mathbb{W}$ is a random variable having a centered normal distribution in $\mathbb{R}^{d}$ with covariance matrix $\Gamma$.

*Proof. * First, note that under $\mathcal{H}_{0}$ we have $\Psi=0$ and, therefore, $T=0$. Then, the asymptotic distribution of $\widehat{S}_{n}$ under $\mathcal{H}_{0}$ can be obtained Theorem 3.2 which shows that $\sqrt{n}\widehat{T}_{n}$ converges in distribution, as $n\rightarrow+\infty$, to a r.v. $U$ having a normal distribution in $\mathcal{L}(\mathcal{X})$ with mean [math] and covariance operator equal to that of $Z$. Then, since the map $A\in\mathcal{L}(\mathcal{X})\mapsto\sum_{k=2}^{K}\sum_{\ell=1}^{k-1}\textrm{tr}\left(\pi_{k\ell}(A)\pi_{k\ell}(A)^{\ast}\right)\in\mathbb{R}$ is continuous, we deduce that $n\widehat{S}_{n}$ converges in distribution, as $n\rightarrow+\infty$, to $\mathcal{Q}=\sum_{k=2}^{K}\sum_{\ell=1}^{k-1}\textrm{tr}\left(W_{k\ell}W_{k\ell}^{\ast}\right)$, where $W_{k\ell}=\pi_{k\ell}(U)$. We can write $W_{k\ell}=\sum_{i=1}^{p_{k}}\sum_{j=1}^{p_{\ell}}W_{k\ell}^{ij}\,e_{j}^{(\ell)}\otimes e_{i}^{(k)}$ where $W_{k\ell}^{ij}=<W_{k\ell},e_{j}^{(\ell)}\otimes e_{i}^{(k)}>$, and, using the equalities $(u\otimes v)(w\otimes z)=<u,z>w\otimes v$ and $\textrm{tr}(u\otimes v)=<u,v>$ (cf. Dauxois et al. (1994)), we have:

[TABLE]

where $\delta$ denotes the usual Kronecker symbol and

[TABLE]

Hence, $\mathcal{Q}=\mathbb{W}^{T}\mathbb{W}$, where $\mathbb{W}=\left(\mathbb{W}_{2,1}^{T},\mathbb{W}_{3,1}^{T},\mathbb{W}_{3,2}^{T},\cdots,\mathbb{W}_{K,1}^{T},\cdots,\mathbb{W}_{K,K-1}^{T}\right)^{T}$. Since $\mathbb{W}$ is a linear function of $U$, it is also normally distributed with mean [math] and covariance matrix $\Gamma$ defined in (22) and (23) with $\gamma^{k\ell,rs}_{ijpq}=\mathbb{E}\left(W_{k\ell}^{ij}W_{rs}^{pq}\right)$. Further,

[TABLE]

where $\widetilde{\otimes}$ denotes the tensor product related to the inner product of operators $<A,B>=$tr $(AB^{\ast})$. Recalling that $U$ has the same covariance operator than $\varphi(X\otimes X)$, that is $\mathbb{E}(U\widetilde{\otimes}U)=\mathbb{E}\left((\varphi(X\otimes X))\widetilde{\otimes}(\varphi(X\otimes X))\right)$, we obtain

[TABLE]

Thus, $\mathbb{E}\left(\mathbb{W}_{k\ell}\mathbb{W}_{rq}^{T}\right)=$, $\mathbb{E}\left(<X_{\ell}\otimes X_{k},e_{j}^{(\ell)}\otimes e_{i}^{(k)}><X_{s}\otimes X_{r},e_{q}^{(s)}\otimes e_{p}^{(r)}>\right)$

[TABLE]

$\Box$

4.3 Case of elliptical distribution

Recall that a random variable valued into an Euclidean space $E$ is said to have an elliptical distribution $\mathcal{E}(\mu,\Sigma,h)$ if its characteristic function is of the form:

[TABLE]

In this subsection, we assume that $X$ has an elliptical distribution $\mathcal{E}(0,V,h)$. Then, the previous theorem has a more explicit formulation stated below.

**Theorem 4.2.**When $X$ has the elliptical distribution $\mathcal{E}(0,V,h)$ then, under $\mathcal{H}_{0}$, the sequence $n\widehat{S}_{n}$ converges in distribution, as $n\rightarrow+\infty$, to $4h^{\prime\prime}(0)\chi^{2}_{d}$. *Proof. * The $\mathbb{R}^{4}$- valued random vector

[TABLE]

can obviously been writen as $S=L(X)$, where $L$ in a linear map from $\mathcal{X}$ to $\mathbb{R}^{4}$. Then, $S$ has the elliptical distribution $E(0,LVL^{\ast},h)$ and, therefore,

[TABLE]

Since, under $\mathcal{H}_{0}$, we have for any $(m,l)\in\{1,\cdots,K\}^{2}$, $V_{ml}=\delta_{ml}V_{l}=\delta_{ml}I_{l}$, where $\delta$ denotes the usual Kronecker symbol, we finally obtain:

[TABLE]

If $k=r$, $\ell=s$ with $k\in\{2,\cdots,K\}$ and $\ell\in\{1,\cdots,k-1\}$, since $\ell\neq k$ and, equivalently, $s\neq r$, it follows that

[TABLE]

Hence,

[TABLE]

where $\mathbb{I}_{m}$ denotes the $m$-dimensional identity matrix. If $(k,\ell)\neq(r,s)$ with $(k,r)\in\{2,\cdots,K\}^{2}$ and $(\ell,s)\in\{1,\cdots,k-1\}\times\{1,\cdots,r-1\}$, suppose that $\gamma^{k\ell,rs}_{ijpq}\neq 0$. Then, since $\ell\neq k$, $s\neq r$ and $(k,\ell)\neq(r,s)$, we deduce from (24) that we necessarily have $s=k$ and $\ell=r$. Thus $k<r$ and, since $r=\ell$, we obtain the inequality $k<\ell$. That is not possible since $\ell\in\{1,\cdots,k-1\}$. So, we deduce that $\gamma^{k\ell,rs}_{ijpq}=0$. Using (22), (23) and (25) we can conclude that $\Gamma=4h^{\prime\prime}(0)\,\mathbb{I}_{d}$. Then, we deduce from Theorem 4.1 that $\mathcal{Q}=4h^{\prime\prime}(0)\mathcal{Q}^{\prime}$, where $\mathcal{Q}^{\prime}$ is a random variable with distribution equal to $\chi^{2}_{d}$. $\Box$