Zhidong Bai

Jian-Feng Yao

CENTRAL LIMIT THEOREMS FOR EIGENVALUES IN A SPIKED POPULATION MODEL

Keywords: Sample covariance matrices, Spiked population model, Central limit theorems, Largest eigenvalue, Extreme eigenvalues, Random sesquilinear forms, Random quadratic forms

In a spiked population model, the population covariance matrix has all its eigenvalues equal to unit except for a few fixed eigenvalues (spikes). This model is proposed by Johnstone to cope with empirical findings on various data sets. The question is to quantify the effect of the perturbation caused by the spike eigenvalues. A recent work by Baik and Silverstein establishes the almost sure limits of the extreme sample eigenvalues associated to the spike eigenvalues when the population and the sample sizes become large. This paper establishes the limiting distributions of these extreme sample eigenvalues. As another important result of the paper, we provide a central limit theorem on random sesquilinear forms.

Titre et résumé en francais.

Un théorème limite central pour les valeurs propres empiriques d'un modèle de variances hétérogènes Résumé. Dans un modèle de variances hétérogènes, les valeurs propres de la matrice de covariance des variables sont toutes unitaires sauf un faible nombre d'entre elles. Ce modèle a été introduit par Johnstone comme une explication possible de la structure des valeurs propres de la matrice de covariance empirique constatée sur plusieurs ensembles de données réelles. Une question importante est de quantifier la perturbation causée par ces valeurs propres non unitaires. Un travail récent de Baik and Silverstein établit la limite presque sure des valeurs propres empiriques extrêmes lorsque le nombre de variables tend vers l'infini proportionnellement à la taille de l'échantillon. Ce travail établit un théorème limite central pour ces valeurs propres empiriques extrêmes. Il est basé sur un autre nouveau théorème limite central pour les formes sesquilinéaires aléatoires.

Introduction.

It is well-known that the empirical spectral distribution (E.S.D) of a large sample covariance matrix converges to the family of Marčenko-Pastur laws under fairly general condition on the sample variables [START_REF] Marčenko | Distribution of eigenvalues for some sets of random matrices[END_REF][START_REF] Bai | Methodologies in spectral analysis of large dimensional random matrices. a review[END_REF]. On the other hand, the study of the largest or smallest eigenvalues is more complex. In a variety of situations, the almost sure limits of these extreme eigenvalues are proved to coincide with the boundaries of the support of the limiting distribution. As an example, when the sample vectors have independent coordinates and unit variances and assuming that the ratio p/n of the population size p over the sample size n tends to a positive limit y ∈ (0, 1), then the limiting distribution is the classical Marčenko-Pastur law F y (dx)

F y (dx) = 1 2πxy (x -a y )(b y -x)dx, a y ≤ x ≤ b y , (1.1)
where a y = (1 -√ y) 2 , and b y = (1 + √ y) 2 . Moreover, the smallest and the largest eigenvalue converge almost surely to the boundary a y and b y , respectively.

Recent empirical data analysis from fields like wireless communication engineering, speech recognition or gene expression experiments suggest that frequently, some extreme eigenvalues of sample covariance matrices are well-separated from the rest. For instance, see Figures 1 and2 in Johnstone [START_REF] Johnstone | On the distribution of the largest eigenvalue in principal components analysis[END_REF] which display the sample eigenvalues of the functional data consisting of a speech dataset of 162 instances of a phoneme "dcl" spoken by males calculated at 256 points. As a way for possible explanation of this phenomenon, this author proposes a spiked population model where all eigenvalues of the population covariance matrix are equal to one except a fixed and relatively small number among them (spikes). Clearly, a spiked population model can be considered as a small perturbation of the so-called null case where all the eigenvalues of the population covariance matrix are unit. It then raises the question how such a small perturbation affects the limits of the extreme eigenvalues of the sample covariance matrix as compared to the null case.

The behavior of the largest eigenvalue in case of complex Gaussian variables has been recently studied in Baik et al. [START_REF] Baik | Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices[END_REF]. These authors prove a transition phenomenon: the weak limit as well as the scaling of the largest eigenvalue is different according to the largest spike eigenvalue is larger, equal or less than the critical value 1 + √ y. In Baik and Silverstein [START_REF] Baik | Eigenvalues of large sample covariance matrices of spiked population models[END_REF], the authors consider the spiked population model with general random variables: complex or real and not necessarily Gaussian.

For the almost sure limits of the extreme sample eigenvalues, they also find that these limits depend on the critical values 1 + √ y and 1 -√ y from above and below, respectively. For example, if there are M eigenvalues in the population covariance matrix larger than 1 + √ y, then the M largest eigenvalues from the sample covariance matrix will have their (almost surely) limits above the right edge b y of of the limiting Marčhenko-Pastur law. Analogous results are also proposed for the case y > 1 and y = 1.

An important question here is to find the limiting distributions of these extreme eigenvalues. As mentioned above, the results are proposed in [START_REF] Baik | Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices[END_REF] for the largest eigenvalue and the Gaussian complex case. In this perspective, assuming that the population vector is real Gaussian with a diagonal covariance matrix and that the M spike eigenvalues are all simple, Paul [START_REF] Paul | Asymptotics of the leading sample eigenvalues for a spiked covariance model[END_REF] found that each of the M largest sample eigenvalues has a Gaussian limiting distribution.

In this paper, we follow the general set-up of [START_REF] Baik | Eigenvalues of large sample covariance matrices of spiked population models[END_REF]. Assuming y ∈ (0, 1) and general population variables, we will establish central limit theorems for the largest as well as for the smallest sample eigenvalues associated to spike eigenvalues outside the interval [1-√ y, 1+ √ y]. Furthermore, we prove that the limiting distribution of such sample extreme eigenvalues is Gaussian only if the corresponding spike population eigenvalue is simple. Otherwise, if a spiked eigenvalue is multiple, say of index k, then there will be k packed-consecutive sample eigenvalues λ n,1 , . . . , λ n,k which converge jointly to the distribution of a k × k symmetric (or Hermitian) Gaussian random matrix. Consequently in this case, the limiting distribution of a single λ n,j is generally non Gaussian.

The main tools of our analysis are borrowed from the random matrix theory on one hand. For general background of this theory, we refer to the book Mehta [START_REF] Mehta | Random Matrices[END_REF] and a modern review by Bai [START_REF] Bai | Methodologies in spectral analysis of large dimensional random matrices. a review[END_REF]. On the other hand, we introduce in this paper another important tool, namely a CLT for random sequilinear forms which should have its own interests. This CLT, independent from the rest of the paper, is presented in the last section 7.

The remaining sections of the paper are organized as follows. First in Section 2, we introduce the spiked population model and recall known results on the almost sure limits of extreme sample eigenvalues. The main result of the paper, namely a general CLT for extreme sample eigenvalues, Theorem 3.1, is then introduced in Section 3. To provide a better account of this CLT, Section 4 develops in details several meaningful examples. Several set of numerical computations are also conducted to give concrete illustration of the main result. In particular, we recover a CLT given in [START_REF] Paul | Asymptotics of the leading sample eigenvalues for a spiked covariance model[END_REF] as a special instance. In Section 5, we discuss some extensions of these results to the case where spiked eigenvalues are inside the gaps located in the center of the spectrum of the population covariance matrix. Finally, Section 6 collects the proofs of the presented results based on a CLT for random sequilinear forms which is itself introduced and proved in Section 7.

Spiked population model and convergence of extreme eigenvalues.

We consider a zero-mean, complex-valued random vector x = (ξ T , η T ) T where ξ = (ξ(1), . . . , ξ(M )) T , η = (η(1), . . . , η(p)) T are independent, of dimension M and p respectively. Moreover, we assume that E[ x 4 ] < ∞ and the coordinates of η are independent and identically distributed with unit variance. The population covariance matrix of the vector x is therefore

V = cov(x) = Σ 0 0 I p .
We consider the following spiked population model by assuming that Σ has K non null and non unit eigenvalues α 1 , . . . , α K with respective multiplicity n 1 , . . . , n K (n

1 + • • •+ n K = M ).
Therefore, the eigenvalues of the population covariance matrix V are unit except the (α j ), called spike eigenvalues.

Let x i = (ξ T i , η T i ) T be n copies i.i.d. of x.
The sample covariance matrix is

S n = 1 n n i=1 x i x * i
which can be rewritten as

S n = S 11 S 12 S 21 S 22 = X 1 X * 1 X 1 X * 2 X 2 X * 1 X 2 X * 2 = 1 n ξ i ξ * i ξ i η * i η i ξ * i η i η * i , (2.2)
with

X 1 = 1 √ n (ξ 1 , • • • , ξ n ) M ×n = 1 √ n ξ 1:n , X 2 = 1 √ n (η 1 , • • • , η n ) p×n = 1 √ n η 1:n .
It is assumed in the sequel that M is fixed, and p and n are related so that when n → ∞, p/n → y ∈ (0, 1). The E.S.D of S n , as well as the one of S 22 , converges to the Marčenko-Pastur distribution F y (dx) given in (1.1). As explained in Introduction, a central question is to quantify the effect caused by the small number of spiked eigenvalues on the asymptotic of the extreme sample eigenvalues.

As a first general answer to this question, Baik and Silverstein [START_REF] Baik | Eigenvalues of large sample covariance matrices of spiked population models[END_REF] completely determines the almost sure limits of largest and smallest sample eigenvalues. More precisely, assume that among the M eigenvalues of Σ, there are exactly M b greater than 1 + √ y and M a smaller than 1 -√ y:

α 1 > • • • > α M b > 1 + √ y , α M < • • • < α M -M b +1 < 1 - √ y , (2.3)
and 1 -√ y ≤ α k ≤ 1 + √ y for the other α k 's. Moreover, for α = 1, we define the

function λ = φ(α) = α + yα α -1 . (2.4)
As y < 1, we have p ≤ n for large n. Let

λ n,1 ≥ λ n,2 ≥ • • • ≥ λ n,p
be the eigenvalues of the sample covariance matrix S n . Let

s i = n 1 + • • • + n i for 1 ≤ i ≤ M b and t j = n M + • • • + n j for 1 ≤ j ≤ M a (by convention s 0 = t 0 = 0).
Therefore, Baik and Silverstein [START_REF] Baik | Eigenvalues of large sample covariance matrices of spiked population models[END_REF] proves that for each k ∈ {1, . . . , M b } and

s k-1 < j ≤ s k (largest eigenvalues) or k ∈ {1, . . . , M a } and p -t k < j ≤ p -t k-1 (smallest eigenvalues), λ n,j → φ(α k ) = α k + yα k α k -1 , almost surely. (2.5)
In other words, if a spike eigenvalue α k lies outside the interval [1 -√ y, 1 + √ y] and has multiplicity n k , then φ(α k ) is the limit of n k packed sample eigenvalue {λ n,j , j ∈ J k }. Here we have denoted by J k the corresponding set of indexes:

J k = {s k-1 + 1, . . . , s k } for α k > 1 + √ y and J k = {p -t k + 1, . . . , p -t k-1 } for α k < 1 - √ y.
= |λI -S n | = |λI -S 22 | |λI -K n (λ)| , (3.6)
where

K n (λ) = S 11 + S 12 (λI -S 22 ) -1 S 21 . (3.7)
As when n → ∞, with probability 1, the limit λ n,j → φ(α k ) / ∈ [a y , b y ] and the eigenvalues of S 22 go inside the interval [a y , b y ], the probability of the event

Q n Q n = {λ n,j / ∈ [a y , b y ]} ∩ {spectrum of S 22 ⊂ [a y , b y ]}
tends to 1. Conditional on this event, the (λ n,j )'s then solve the determinant equation

|λI -K n (λ)| = 0. (3.8)
Therefore without loss of generality, we can assume that λ n,j / ∈ [a y , b y ] and they are solutions of this equation.

Furthermore, let

A n = (a ij ) = A n (λ) = X * 2 (λI -X 2 X * 2 ) -1 X 2 , λ / ∈ [a y , b y ]. (3.9)
Lemma 6.1 detailed in Section 6.1 establishes the convergence of several statistics of the matrix A n . In particular, n -1 trA n , n -1 trA n A * n and n -1 n i=1 a 2 ii converges in probability to ym 1 (λ), ym 2 (λ) and (y[1 + m 1 (λ)]/{λ -y[1 + m 1 (λ)]}) 2 , respectively. Here, the m j (λ) are some specific transforms of the Marçenko-Pastur law F y (see Section 6.1 for more details).

Therefore, the random form K n in (3.7) can be decomposed as follows

K n (λ) = S 11 + X 1 A n X * 1 = 1 n ξ 1:n (I + A n )ξ * 1:n = 1 n {ξ 1:n (I + A n )ξ * 1:n -Σtr(I + A n )} + 1 n Σtr(I + A n ) = 1 √ n R n + [1 + ym 1 (λ)] Σ + o P ( 1 √ n ), (3.10)
with

R n = R n (λ) = 1 √ n {ξ 1:n (I + A n )ξ * 1:n -Σtr(I + A n )} . (3.11)
In the last derivation, we have used the fact 1 n tr(I

+ A n ) = 1 + ym 1 (λ) + o P ( 1 √ n ),
which follows from a CLT for tr(A n ) [see 4].

3.2.

Limit distribution of the random matrices {R n (λ)} . The next step is to find the limit distribution of the sequence of random matrices {R n (λ)}. The situation is different for the real and complex cases. Define the constants

θ = 1 + 2ym 1 (λ) + ym 2 (λ) , (3.12
)

ω = 1 + 2ym 1 (λ) + y[1 + m 1 (λ)] λ -y[1 + m 1 (λ)] 2 .
(3.13) Proposition 3.1 Limiting distribution of R n (λ): real variables case. Assume that the variables ξ and η are real-valued. Then, the random matrix R n converges weakly to a symmetric random matrix R = (R ij ) with zero-mean Gaussian entries having the following covariance function: for

1 ≤ i ≤ j ≤ M , cov(R ij , R i ′ j ′ ) = ω E ξ(i)ξ(j)ξ(i ′ )ξ(j ′ ) -Σ ij Σ i ′ j ′ +(θ -ω) E[ξ(i)ξ(j ′ )] E[ξ(i ′ )ξ(j)] +(θ -ω) E[ξ(i)ξ(i ′ )] E[ξ(j)ξ(j ′ )] . (3.14)
Note that in particular, the following formula holds for the variances

var(R ij ) = θ(Σ ii Σ jj + Σ 2 ij ) + ω E[ξ 2 (i)ξ 2 (j)] -2Σ 2 ij -Σ ii Σ jj . (3.15)
In case of a diagonal element R ii , this expression simplifies to

var(R ii ) = [2θ + β i ω]Σ 2 ii , with β i = E[ξ(i) 4 ] Σ 2 ii -3. (3.16) If moreover, ξ(i) is Gaussian, β i = 0.
Remark. If the coordinates {ξ(i)} of ξ are independent, then the limiting covariance matrix in (3.14) is diagonal: the limiting Gaussian matrix is made with independent entries. Their variances simplify to (3.16) and var(R ij ) = θΣ ii Σ jj , i < j.

(3.17) Proposition 3.2 Limiting distribution of R n (λ): complex variables case. Assume the general case with complex-valued variables ξ and η and that the following limit exists

m 4 (λ) = lim n 1 n tr A n A T n , λ / ∈ [a y , b y ]. (3.18)
Then, the random matrix R n converges weakly to a zero-mean Hermitian random matrix R = (R ij ). Moreover, the joint distribution of the real and imaginary parts of the upper-triangular bloc

{R ij , 1 ≤ i ≤ j ≤ M } is a 2K-dimensional Gaussian vector with covariance matrix Γ = Γ 11 Γ 12 Γ 21 Γ 22 (3.19)
where

Γ 11 = 1 4 3 j=1 {2ℜ(B j ) + B ja + B jb } , Γ 22 = 1 4 3 j=1 {-2ℜ(B j ) + B ja + B jb } , Γ 12 = 1 2 3 j=1 ℑ(B j ) ,
and for

1 ≤ i ≤ j ≤ M and 1 ≤ i ′ ≤ j ′ ≤ M , B 1 (ij, i ′ j ′ ) = ω E[ξ i ξ j ξ i ′ ξ j ′ ] -Σ ij Σ i ′ j ′ , B 2 (ij, i ′ j ′ ) = (θ -ω)Σ ij ′ Σ i ′ j , B 3 (ij, i ′ j ′ ) = (τ -ω) E[ξ i ξ i ′ ] E[ξ j ξ j ′ ] , B 1a (ij, i ′ j ′ ) = ω E[|ξ i ξ i ′ | 2 ] -Σ ii Σ i ′ i ′ , B 1b (ij, i ′ j ′ ) = ω E[|ξ j ξ j ′ | 2 ] -Σ jj Σ j ′ j ′ , B 2a (ij, i ′ j ′ ) = (θ -ω)|Σ ii ′ | 2 , B 2b (ij, i ′ j ′ ) = (θ -ω)|Σ jj ′ | 2 , B 3a (ij, i ′ j ′ ) = (τ -ω)| E[ξ i ξ i ′ ]| 2 , B 3b (ij, i ′ j ′ ) = (τ -ω)| E[ξ j ξ j ′ ]| 2 .
Here, the constant τ equals

τ = lim n 1 n tr(I + A n )(I + A n ) T = 1 + 2ym 1 (λ) + m 4 (λ) . ( 3

.20)

The limiting covariance matrix Γ has a complicated expression. However, the variance of a diagonal element R ii has a much simpler expression if moreover,

E[ξ 2 (i)] = 0 for all 1 ≤ i ≤ M , var(R ii ) = [θ + β ′ i ω]Σ 2 ii , with β ′ i = E[ξ(i) 4 ] Σ 2 ii -2. (3.21) In particular, if ξ(i) is Gaussian, β ′ i = 0.
3.3. CLT for extreme eigenvalues. We are in order to introduce the main result of the paper. Let the spectral decomposition of Σ,

Σ = U    α 1 I n 1 • • • 0 0 . . . 0 • • • 0 α K I n K    U * , (3.22)
where U is an unitary matrix. Following Section 2, for each spiked eigenvalue

α k / ∈ [1 - √ y, 1 + √ y],
let {λ n,j , j ∈ J k } be the n k packed eigenvalues of the sample covariance matrix which all tend almost surely to λ k = φ(α k ). Let R(λ k ) be the Gaussian matrix limit of the sequence of matrices of random forms [R n (λ k )] n given in Proposition 3.1 (real variables case) and Proposition 3.2 (complex variables case), respectively. Let

R(λ k ) = U * R(λ k )U . (3.23) Theorem 3.1 For each spike eigenvalue α k / ∈ [1 - √ y, 1 + √ y], the n k -dimensional real vector √ n{λ n,j -λ k , j ∈ J k } ,
converges weakly to the distribution of the n k eigenvalues of the Gaussian random matrix

1 1 + ym 3 (λ k )α k R kk (λ k ).
where

R kk (λ k ) is the k-th diagonal bloc of R(λ k ) corresponding to the indexes {u, v ∈ J k }.
One strking fact from this theorem is that the limiting distribution of such n k packed sample extreme eigenvalues are generally non Gaussian and asymptotically dependent. Indeed, the limiting distribution of a single sample extreme eigenvalue λ n,j is Gaussian if and only if the corresponding population spike eigenvalue is simple.

Examples and numerical results

. This section is devoted to describe in more details the content of Theorem 3.1 with several meaningful examples together with extended numerical computations.

4.1.

A special Gaussian case from Paul [START_REF] Paul | Asymptotics of the leading sample eigenvalues for a spiked covariance model[END_REF]. We consider a particular situation examined in Paul [START_REF] Paul | Asymptotics of the leading sample eigenvalues for a spiked covariance model[END_REF]. Assume that the variables are real Gaussian, Σ diagonal whose eigenvalues are all simple. In other words, K = M and n k = 1 for all 1 ≤ k ≤ M . Hence, U = I M . Following Theorem 3.1, for any

λ k = φ(α k ), √ n(λ n,k -λ k )
converges weakly to the Gaussian variable (1+

ym 3 (λ k )α k ) -1 R(λ k ) kk .
This variable is zero-mean. For the computation of its variance, we remark that by Eq.(3.16

) var R(λ k ) kk = 2θα 2 k ,
where

θ = 1 + 2ym 1 (λ k ) + ym 2 (λ k ) = (α k -1 + y) 2 (α k -1) 2 -y .
Taking into account (6.6), we get finally, for 1

≤ k ≤ M √ n(λ n,k -λ k ) D =⇒ N (0, σ 2 k ) , σ 2 k = 2α 2 k [(α k -1) 2 -y] (α k -1) 2 .
This coincides with Theorem 3 of [START_REF] Paul | Asymptotics of the leading sample eigenvalues for a spiked covariance model[END_REF].

4.2. More general Gaussian variables case. In this example, we assume that all variables are real Gaussian, and the coordinates of ξ are independent. As in [START_REF] Baik | Eigenvalues of large sample covariance matrices of spiked population models[END_REF], we fix y = 0.5. The critical interval is then

[1 - √ y, 1 + √ y] = [0.293, 1.707] and the limiting support [a y , b y ] = [0.086, 2.914]. Consider K = 4 spike eigenvalues (α 1 , α 2 , α 3 , α 4 ) = (4, 3, 0.2, 0.1) with respective multiplicity (n 1 , n 2 , n 3 , n 4 ) = (1, 2, 2, 1). Let λ n,1 ≥ λ n,2 ≥ λ n,3 and λ n,4 ≥ λ n,5 ≥ λ n,6
be respectively, the three largest and the three smallest eigenvalues of the sample covariance matrix. Let, as in Section 4.1,

σ 2 α k = 2α 2 k [(α k -1) 2 -y] (α k -1) 2 . (4.1)
We have (σ 2 α k , k = 1, . . . , 4) = (30.222, 15.75, 0.0175, 0.00765). Following Theorem 3.1, taking into account Section 4.1 and Proposition 3.1, we have

• For j = 1 and 6, δ n,j = √ n[λ n,j -φ(α k )] D =⇒ N (0, σ 2 α k ). (4.2)
Here, for j = 1, k = 1 , φ(α 1 ) = 4.667 and σ 2 α 1 = 30.222 ; and for j = 6, k = 4 , φ(α 4 ) = 0.044 and σ 2 α 4 = 0.00765. • For j = (2, 3) or j = (4, 5), the two-dimensional vector δ n,j = √ n[λ n,j -φ(α k )] converges weakly to the distribution of (ordered) eigenvalues of the random matrix

G = σ α k W 11 W 12 W 12 W 22 .
Here, because the initial variables (ξ(i))'s are Gaussian, by Eqs. (3.16)-(3.17), we have var(W 11 ) = var(W 22 ) = 1, var(W 12 ) = 1 2 , so that (W ij ) is a real Gaussian Wigner matrix. Again, the variance parameter σ 2 α k is defined as previously but with k = 2 for j = (2, 3) and k = 3 for j = (4, 5), respectively. Since the joint distribution of eigenvalues of a Gaussian Wigner matrix is known [see 11], we get the following (unordered) density for the limiting distribution of δ n,j :

g(δ, γ) = 1 4σ 3 α k √ π |δ -γ| exp - 1 2σ 2 α k (δ 2 + γ 2 ) . (4.3)
Experiments are conducted to compare numerically the empirical distribution of the δ n,j to their limiting value. To this end, we fix p = 500 and n = 1000. We repeat 1000 independent simulations to get 1000 replications of the six random variates {δ n,j , j = 1, . . . , 6}. Based on these replications, we compute • a kernel density estimate for two univariate variables δ n,1 and δ n,6 , denoted by f n,1 f n,6 respectively.

• a kernel density estimate for two bivariate variables (δ n,2 , δ n,3 ) and (δ n,4 , δ n,5 ), denoted by f n,23 f n,45 respectively.

The kernel density estimates are computed using the R software implementing an automatic bandwidth selection method from [START_REF] Sheather | A reliable data-based bandwidth selection method for kernel density estimation[END_REF].

Figure 1 compare the two univariate density estimates f n,1 and f n,6 to their Gaussian limits (4.2). As we can see, the simulations confirm well the found formula. ) sample eigenvalue from 1000 independent replications, compared to their Gaussian limits (dashed lines). Gaussian entries with p = 500 and n = 1000.

To compare the bivaiate density estimates f n,23 and f n,45 to their limiting densities given in (4.3), we choose to display their contour lines. This is done in Figure 2 for f n,23 and Figure 3 for f n,45 . Again we see that the theoretical result is well confirmed.

4.3.

A binary variables case. As in the previous example, we fix y = 0.5 and adopt the same spike eigenvalues (α 1 , α 2 , α 3 , α 4 ) = (4, 3, 0.2, 0.1) with multiplicities (n 1 , n 2 , n 3 , n 4 ) = (1, 2, 2, 1). Let the σ 2 α k 's be as defined in (4.1). Again we assume that all the coordinates are independent but this time we consider binary entries. To cope with the eigenvalues, we set

ξ(i) = √ α k ε i , η(j) = ε ′ j
where (ε i ) and (ε ′ j ) are two independent sequences of i.i.d. binary variables taking values {+1,-1} with equiprobability. We remark that Eε i = 0, Eε 2 i = 1 and

β i = E[ξ 4 (i)]/[ Eξ 2 (i)] 2 -3 = -2.
This last value denotes a departure from the Gaussian case. As in the previous example, we examine the limiting distributions of the three largest and the three smallest eigenvalues {λ n,j , j = 1, . . . , 6} of the sample covariance matrix. Following Theorem 3.1, we have

• For j = 1 and 6, δ n,j = √ n[λ n,j -φ(α k )] D =⇒ N (0, s 2 α k ), s 2 α k = σ 2 α k y (α k -1) 2 .
Compared to the previous Gaussian case, as the factor y/(α k -1) 2 < 1, the limiting Gaussian distributions of the largest and the smallest eigenvalue are less dispersed.

• For j = (2, 3) or j = (4, 5), the two-dimensional vector δ n,j = √ n[λ n,j -φ(α k )] converges weakly to the distribution of (ordered) eigenvalues of the random matrix

G = σ α k W 11 W 12 W 12 W 22 . (4.4)
Here, because the initial variables (ξ(i))'s are binary, hence β i = -2 (which is zero for Gaussian variables), by Eqs. (3.16)-(3.17), we have var(W 12 ) = 1 2 but var(W 11 ) = var(W 22 ) = y/(α k -1) 2 . Therefore, the matrix W = (W ij ) is no more a real Gaussian Wigner matrix. Again, the variance parameter σ 2 α k is defined as previously but with k = 2 for j = (2, 3) and k = 3 for j = (4, 5), respectively. Unfortunately and unlike the previous Gaussian case, the joint distribution of eigenvalues of W is unknown analytically. We then compute empirically by simulation this joint density using 10000 independent replications. Again, as y/(α k -1) 2 < 1, these limiting distributions are less dispersed than previously.

The kernel density estimates f n,1 , f n,6 , f n,23 and f n,45 are computed as in the previous case using p = 500, n = 1000 and 1000 independent replications.

Figure 4 compare the two univariate density estimates f n,1 and f n,6 to their Gaussian limits. Again, we see that simulations confirm well the found formula. However, we remark a bigger difference than in the Gaussian case.

The bivaiate density estimates f n,23 and f n,45 are then compared to their limiting densities in Figures 5 and6, respectively. Again we see that the theoretical result is well confirmed. We remark that the shape of these bivariate limiting distributions are rather different from the previous Gaussian case. We remind the reader that the limiting bivariate densities are obtained by simulations of 10000 independent G matrices given in (4.4). 5. Some extensions. It is possible to extend the spiked population model introduced in Section 2 to a much greater generality. Let us consider a population p × p covariance matrix

V = cov(x) = Σ 0 0 T p ,
where Σ is as previously while T p is now an arbitrary Hermitian matrix. As M will be fixed and p → ∞, the limit F of the E.S.D of the sample covariance matrix depends on the sequence of (T p ) only. With some ambiguity, we again call the eigenvalues α k 's of Σ spike eigenvalues in the sense that they do not contribute to this limit.

In the following, we assume for simplicity that Σ as well as T p are diagonal, and when p → ∞, the empirical distribution of the eigenvalues of T p converges weakly to a probability measure H(dt) on the real line. Therefore, the limit F of the E.S.D. is characterized by an explicit formula for its Stieltjies transform, see Bai and Silverstein [START_REF] Bai | No eigenvalues outside the support of the limiting spectral distribution of large dimensional sample covariance matrices[END_REF].

The previous model of Section 2 corresponds to the situation where H(dt) is the Dirac measure at the point 1. A more involved example which will be analyzed later by numerical computations is the following. The core spectrum of V is made with two eigenvalues ω 1 > ω 2 > 0, nearly p/2 times for each, and V has a fixed number M of spiked eigenvalues distinct from the ω j 's. In this case, the limiting distribution

H is 1 2 (δ {ω 1 } (dt) + δ {ω 2 } (dt)
), a mixture of two Dirac masses. The sample eigenvalues {λ n,j } are defined as previously. Assume that a spiked eigenvalue α k is "sufficiently separated" from the core spectrum of V , so that for some function ψ to be determined, there is a point ψ(α k ) outside the support of F to which converge almost surely n k packed sample eigenvalues {λ n,j , j ∈ J k }. In such case, the analysis we have proposed is also valid yielding a CLT analogous to Theorem 3.1: the n k -dimensional real vector √ n{λ n,j -ψ(α k ), j ∈ J k } , converges weakly to the distribution of the n k eigenvalues of some Gaussian random matrix. In particular, if n k = 1, this limiting distribution is Gaussian. We do not intent to provide here all details in this extended situation. However, let us indicate how we can determine the almost sure limit ψ(α k ) of the packed eigenvalues. From the almost sure convergence and since ψ(α k ) is outside the support of F , with probability tending to one, λ n,j solve the determinant equation (3.8).

With A n = X * 2 (λI -X 2 X * 2 ) -1 X 2 , we have K n (λ) = S 11 + X 1 A n X * 1 = 1 n ξ 1:n (I + A n )ξ * 1:n ,
which tends almost surely to [1 + ym 1 (λ)] Σ. Therefore, any limit λ of a λ n,j fulfills the relation λ -

[1 + ym 1 (λ)] α = 0 , (5.1)
for some eigenvalue α of Σ.

Let m(λ) be the Stieltjies transform of the limiting distribution F and m(λ) the one of yF (dt) + (1 -y)δ {0} (dt). Clearly, λm(λ) = -1 + y + yλm(λ). Moreover, it is known that, see e.g. [START_REF] Bai | No eigenvalues outside the support of the limiting spectral distribution of large dimensional sample covariance matrices[END_REF],

λ = - 1 m(λ) + y t 1 + tm(λ) H(dt). (5.2)
As m 1 (λ) = -1 -λm(λ) by definition, Equation (5.1) reads as

λ = [1 -y -yλm(λ)] α = -λm(λ)α.
It follows then 1 + αm(λ) = 0 (generally, λ = 0). Combining with (5.2), we get finally

λ = ψ(α) = α 1 + y t α -t H(dt) . (5.3)
In particular, for the original spiked population model with H(dt) = δ {1} (dt), we recover the relation given in (2.4). We conclude the section by giving some numerical results of the above mentioned example of an extended spiked population model. Then, we consider (ω 1 , ω 2 ) = (1, 10), (α 1 , α 2 , α 3 ) = (5, 4, 3) with respective multiplicity (1, 2, 1), and the limit ratio y = 0.2. Note that these spiked eigenvalues are now between the dominating eigenvalues (1 and 10). On the other hand, the support of the limiting distribution of the E.S.D. can be determined following the method given in [START_REF] Bai | No eigenvalues outside the support of the limiting spectral distribution of large dimensional sample covariance matrices[END_REF], and we get two disjoint intervals: supp F = [0.395, 1.579] ∪ [4.784, 17.441].

For simulation, we use p = 500, n = 2500 and the eigenvalues of the population covariance matrix V are 1 (248), 3 (1), 4 (2), 5 (1) and 10 (248). We simulate 500 independent replications of the sample covariance matrix with Gaussian variables. A example of these 500 replication is displayed in Figure 7.

For each replication, the four eigenvalues at the middle (of indexes 249, 250, 251, 252) are extracted. Let us denote these 4 eigenvalues by λ n,1 , λ n,2 , λ n,3 , λ n,4 . By (5.3), we know that the almost sure limits of these sample eigenvalues are respectively

ψ(α k ) = α k 1 + 1 10(α k -1) + 1 α k -10
= (4.125, 3.467, 2.721).

The next figure 8 displays the empirical densities of

δ n,j = √ n(λ n,j -ψ(α k )), 1 ≤ j ≤ 4,
from the 500 independent replications. The graphs of δ n,1 and δ n,4 confirm a limiting zero-mean Gaussian distribution corresponding to single spike eigenvalues 5 and 3. In contrary, the limiting distributions of δ n,2 and δ n,3 , related to the double spike eigenvalue 4, are not zero-mean Gaussian. We note that δ n,2 and -δ n,3 have approximately the same distribution. Indeed, their joint distribution converges to that of the eigenvalues of 2 × 2 Gaussian Wigner matrix. 6. Proofs of Propositions 3.1, 3.2 and Theorem 3.1. Before giving the proofs, some preliminary results and useful lemmas are introduced. Note that these proofs are based on a CLT for random sequilinear forms which is itself introduced and proved in Section 7.

m 2 (λ) = x 2 (λ -x) 2 F y (dx) , (6.2)
m 3 (λ) = x (λ -x) 2 F y (dx) . (6.3)
It is easily seen that

λ λ -x F y (dx) = 1 + m 1 (λ) , λ 2 (λ -x) 2 = 1 + 2m 1 (λ) + m 2 (λ) . If a real constant α / ∈ [1 - √ y, 1 + √ y], then φ(α) / ∈ [a y , b y ] and we have m 1 • φ(α) = 1 α -1 , (6.4)
m 2 • φ(α) = (α -1) + y(α + 1) (α -1)[(α -1) 2 -y] , (6.5)
m 3 • φ(α) = 1 (α -1) 2 -y . (6.6)
Let us mention that all these formula can be obtained by derivation of the Stieltjes transform of the Marčenko-Pastur law F y (dx)

m(z) = 1 x -z F y (dx) = 1 2yz {1 -y -z + (y + 1 -z) 2 -4y}, z / ∈ [a y , b y ].
Here, √ u denotes the square root with positive imaginary part for u ∈ C.

The following lemma gives the law of large numbers for some useful statistics related to the random matrix A n introduced in Eq. (3.9). Lemma 6.1 We have

1 n trA n P -→ ym 1 (λ) , (6.7) 1 n trA n A * n P -→ ym 2 (λ) , (6.8) 1 n n i=1 a 2 ii P -→ y[1 + m 1 (λ)] λ -y[1 + m 1 (λ)] 2 .
(6.9)

Proof. Let β n,j , j = 1, . . . , p be the eigenvalues of S 22 = X 2 X * 2 . The first equality is easy. For the second one, We have

1 n trA n A * n = 1 n tr(λI -X 2 X * 2 ) -1 X 2 X * 2 (λI -X 2 X * 2 ) -1 X 2 X * 2 = p n p j=1 β 2 n,j (λ -β n,j ) 2 P -→ y x 2 (λ -x) 2 F y (dx) .
For (6.9), let e i ∈ C n be the column vector whose i-th element is 1 and others are 0 and X 2i denote the matrix obtained from X 2 by deleting the i-th column of X 2 . We have

X 2 = X 2i + 1 n η i η * i . Therefore,
a ii = e * i X * 2 (λI-X 2 X * 2 ) -1 X 2 e i = 1 n η * i (λI-X 2 X * 2 ) -1 η i = - 1 n η * i (X 2i X * 2i -λI) -1 η i 1 + 1 n η * i (X 2i X * 2i -λI) -1 η i .
Using Lemma 2.7 of Bai and Silverstein [START_REF] Bai | No eigenvalues outside the support of the limiting spectral distribution of large dimensional sample covariance matrices[END_REF],

E| 1 n η * i (X 2i X * 2i -λI) -1 η i - 1 n tr(X 2i X * 2i -λI) -1 | 2 ≤ K n 2 E|η(1)| 4 Etr(X 2i X * 2i -λI) -2
which gives that

a ii P -→ - y 1 x-λ F y (dx) 1 + y 1 x-λ F y (dx) = y[1 + m 1 (λ)] λ -y[1 + m 1 (λ)] . (6.10)
Further, it is easy to verify that lim

n→∞ E tr(X * 2 (λI -X 2 X * 2 ) -1 X 2 ) 4 n < ∞
which implies, together with inequality 3.3.41 of [START_REF] Horn | Matrix Analysis[END_REF] that

sup n Ea 4 11 = sup n 1 n n i=1 Ea 4 ii ≤ sup n E tr(X * 2 (λI -X 2 X * 2 ) -1 X 2 ) 4 n < ∞.
Therefore, the family of the random variables {a 2 11 } indexed by n is uniformly integrable. Combining with (6.10), we get

E 1 n n i=1 a 2 ii -( y[1 + m 1 (λ)] λ -y[1 + m 1 (λ)] ) 2 ≤ E|a 2 11 -( y[1 + m 1 (λ)] λ -y[1 + m 1 (λ)] ) 2 | → 0.
Thus (6.9) follows.

6.2. Proof of Proposition 3.1. We apply Theorem 7.1 by considering K = 1 2 M (M + 1) bilinear forms

u(i)(I + A n )u(j) T , 1 ≤ i ≤ j ≤ M with u(i) = (ξ 1 (i), . . . , ξ n (i)).
More precisely, with ℓ = (i, j), we are substituting u(i) T for X(ℓ), and u(j) T for Y (ℓ), respectively. Consequently, x ℓ1 = ξ 1 (i) and y ℓ1 = ξ 1 (j) for the application of Theorem 7.1. We have, by Lemma 6.1,

θ = τ = lim n 1 n tr(I + A n ) 2 = 1 + 2ym 1 (λ) + ym 2 (λ) , ω = lim n 1 n n i=1 [(I + A n ) ii ] 2 = 1 + 2ym 1 (λ) + y[1 + m 1 (λ)] λ -y[1 + m 1 (λ)] 2 .
Following Theorem 7.1, R n converges weakly to a symmetric random matrix with zero-mean Gaussian variables R = (R ij ) with the following covariance function, assuming 1

≤ i ≤ j ≤ M , cov(R ij , R i ′ j ′ ) = ω E ξ(i)ξ(j)ξ(i ′ )ξ(j ′ ) -Σ ij Σ i ′ j ′ +(θ -ω) E[ξ(i)ξ(j ′ )] E[ξ(i ′ )ξ(j)] +(θ -ω) E[ξ(i)ξ(i ′ )] E[ξ(j)ξ(j ′ )] . (6.11) 6.3.
Proof of Proposition 3.2. The aim is to apply Theorem 7.

3 to K = 1 2 M (M + 1) sesquilinear forms u(i)(I + A n )u(j) * , 1 ≤ i ≤ j ≤ M with u(i) = (ξ 1 (i), . . . , ξ n (i)).
More precisely, with ℓ = (i, j), we are substituting u(i) * for X(ℓ), and u(j) * for Y (ℓ), respectively. Consequently, x ℓ1 = ξ 1 (i) and y ℓ1 = ξ 1 (j) for the application of Theorem 7.3. Again by Lemma 6.1,

θ = lim n 1 n tr(I + A n ) 2 = 1 + 2ym 1 (λ) + ym 2 (λ) , ω = lim n 1 n n i=1 [(I + A n ) ii ] 2 = 1 + 2ym 1 (λ) + y[1 + m 1 (λ)] λ -y[1 + m 1 (λ)] 2 .
Here we need an additional condition which is specific to the complex case. Assume Therefore,

τ = lim n 1 n tr(I + A n )(I + A n ) T = 1 + 2ym 1 (λ) + m 4 (λ) .
Consequently by Theorem 7.3, R n converges weakly to a zero-mean Hermitian random matrix R = (R ij ). Moreover, the joint distribution of the real and imaginary parts of the upper-triangular bloc {R ij , 1 ≤ i ≤ j ≤ M } is a 2K-dimensional Gaussian vector with covariance matrix

Γ = Γ 11 Γ 12 Γ 21 Γ 22 (6.12)
where

Γ 11 = 1 4 3 j=1 {2ℜ(B j ) + B ja + B jb } , Γ 22 = 1 4 3 j=1 {-2ℜ(B j ) + B ja + B jb } , Γ 12 = 1 2 3 j=1 ℑ(B j ) ,
and for 1

≤ i ≤ j ≤ M and 1 ≤ i ′ ≤ j ′ ≤ M , B 1 (ij, i ′ j ′ ) = ω E[ξ i ξ j ξ i ′ ξ j ′ ] -Σ ij Σ i ′ j ′ , B (ij, i ′ j ′ ) = (θ -ω)Σ ij ′ Σ i ′ j , B 3 (ij, i ′ j ′ ) = (τ -ω) E[ξ i ξ i ′ ] E[ξ j ξ j ′ ] , B 1a (ij, i ′ j ′ ) = ω E[|ξ i ξ i ′ | 2 ] -Σ ii Σ i ′ i ′ , B 1b (ij, i ′ j ′ ) = ω E[|ξ j ξ j ′ | 2 ] -Σ jj Σ j ′ j ′ , B 2a (ij, i ′ j ′ ) = (θ -ω)|Σ ii ′ | 2 , B 2b (ij, i ′ j ′ ) = (θ -ω)|Σ jj ′ | 2 , B 3a (ij, i ′ j ′ ) = (τ -ω)| E[ξ i ξ i ′ ]| 2 , B 3b (ij, i ′ j ′ ) = (τ -ω)| E[ξ j ξ j ′ ]| 2 . 6.4. Proof of Theorem 3.1. Let α k / ∈ [1 - √ y, 1 +
√ y] be fixed. Following Section 3.1, we can assume that the n k packed sample eigenvalues {λ n,j , j ∈ J k } are solutions of the equation |λ -K n (λ)| = 0. As λ n,j → λ k almost surely, we define

δ n,j = √ n(λ n,j -λ k ) .
We have

λ nj I -K n (λ n,j ) = λ k I + 1 √ n δ n,j I -K n (λ k ) -[K n (λ n,j ) -K n (λ k )]. Furthermore, using A -1 -B -1 = A -1 (B -A)B -1 , we have K n (λ n,j ) -K n (λ k ) = 1 n ξ 1:n X * 2 [λ k + 1 √ n δ n,j ]I -S 22 -1 -(λ k I -S 22 ) -1 X 2 ξ * 1:n = - 1 √ n δ n,j 1 n ξ 1:n X * 2 [λ k + 1 √ n δ n,j ]I -S 22 -1 (λ k I -S 22 ) -1 X 2 ξ * 1:n = - 1 √ n δ n,j [ym 3 (λ k )Σ + o P (1)] .
Combining these estimations and (3.10)-(3.11) , we have

λ nj I -K n (λ n,j ) = λ k I -[1 + ym 1 (λ k )]Σ - 1 √ n R n (λ k ) + 1 √ n δ n,j [I + ym 3 (λ k )Σ] + o P ( 1 √ n ) . (6.13)
By Section 3.1, R n (λ k ) converges in distribution to a M ×M random matrix R(λ k ) with Gaussian entries with a fully identified covariance matrix. We now follow a method devised in [START_REF] Bai | A note on limiting distribution of the eigenvalues of a class of random matrice[END_REF] and [START_REF] Bai | Advances in Spectrum Analysis and Array Processing, chapter Estimation of directions of arrival of signals: Asymptotic results[END_REF] for limiting distributions of eigenvalues or eigenvectors from random matrices. First, we use Skorokhod strong representation so that on an appropriate probability space, the convergence R n (λ k ) → R(λ k ) as well as (6.13) take place almost surely. Multiplying both sides of (6.13) by U from the left and by U * from the right yields

U [λ nj I -K n (λ n,j )]U * =     . . . 0 0 0 (λ k -[1 + ym 1 (λ k )]α u )I nu 0 0 0 . . .     - 1 √ n U R n (λ k )U * + 1 √ n     . . . 0 0 0 δ n,j (1 + ym 3 (λ k )α u )I nu 0 0 0 . . .     + o( 1 √ n ) .
First, in the right side of the equation and using a bloc decomposition induced by (3.22), we see that all the non diagonal blocs tend to zero. Next, for a diagonal bloc with index u = k, by definition λ k -[1 + ym 1 (λ k )]α u = 0, and this is the limit of that diagonal bloc since the contributions from the remaining three terms tend to zero. As

λ k -[1 + ym 1 (λ k )]α k = 0 by definition, the k-th diagonal bloc reduces to - 1 √ n [U R n (λ k )U * ] kk + 1 √ n δ n,j (1 + ym 3 (λ k )α k )I n k + o( 1 √ n ).
For n sufficiently large, its determinant must equals to zero,

- 1 √ n [U R n (λ k )U * ] kk + 1 √ n δ n,j (1 + ym 3 (λ k )α k )I n k + o( 1 √ n ) = 0 , or equivalently, |-[U R n (λ k )U * ] kk + δ n,j (1 + ym 3 (λ k )α k )I n k + o(1)| = 0.
Therefore, δ n,j tends to a solution of

|-[U R n (λ k )U * ] kk + λ(1 + ym 3 (λ k )α k )I n k | = 0 ,
that is, an eigenvalue of the matrix (1+ym 3 (λ k )α k ) -1 R kk (λ k ). Finally, as the index j is arbitrary, all the J k random variables √ n{λ n,j -λ k , j ≤ J k } converge almost surely to the set of eigenvalues of the above matrix. Of cause, this convergence also holds in distribution on the new probability space, hence on the original one.

7.

A CLT for random sesquilinear forms. The aim of this section is to establish a CLT for random sesquilinear forms as one of the central tools used in the paper. These results are independent from the previous sections and should have their own interest.

Consider a sequence {(x i , y i ) i∈N } of i.i.d. complex-valued, zero-mean random vectors belonging to C K × C K with finite moment of fourth-order. We write

x i = (x ℓi ) =    x 1i . . . x Ki    , X(ℓ) = (x ℓ1 , . . . , x ℓn ) T , 1 ≤ ℓ ≤ K , (7.1)
with a similar definition for the vectors

{Y (ℓ)} 1≤ℓ≤K . Set ρ(ℓ) = E[x ℓ1 y ℓ1 ]. Theorem 7.1 Let {A n = [a ij (n)]
} n be a sequence of n × n Hermitian matrices and the vectors {X(ℓ), Y (ℓ)} 1≤ℓ≤K are as defined in (7.1). Assume that the following limit exist

ω = lim n→∞ 1 n n u=1 a 2 uu (n) , θ = lim n→∞ 1 n tr A 2 n = lim n→∞ 1 n n u,v=1 |a uv (n)| 2 , τ = lim n→∞ 1 n tr A n A T n = lim n→∞ 1 n n u,v=1 a 2 uv (n) .
Then, the M -dimensional complex-valued random vectors

Z n = (Z n,ℓ ), Z n,ℓ = 1 √ n [X(ℓ) * A n Y (ℓ) -ρ(ℓ) tr A n ] , 1 ≤ ℓ ≤ K , (7.2)
converges weakly to a zero-mean complex-valued vector W whose real and imaginary parts are Gaussian. Moreover, the Laplace transform of W is given by

Ee c T W = exp[ 1 2 c T Bc], c ∈ C K , (7.3)
For the general complex case, we need a special device. Write Z n = U n + iV n . Following Theorem 7.1, (U n , V n ) converges weakly to a 2K-dimensional Gaussian vector with some covariance matrix Γ. The aim is to identify Γ. We have

E exp[t T U n + s T V n ] → exp 1 2 (t T , s T )Γ t s , t ∈ R K , s ∈ R K . (7.7)
On the other hand, from U n = 1 2 (Z n + Zn ) and V n = 1 2i (Z n -Zn ), we have a second expression

E exp[t T U n + s T V n ] = E exp ( t 2 + s 2i ) T Z n + ( t 2 - s 2i ) T Zn .
Interestingly enough, the last transform can be found by application of Theorem 7.1 to the random sesquilinear forms Proof. For the vector of sesquilinear formes Z n in (7.8), one can check that the limiting matrix B in Theorem 7.1 is to be replaced by

Z n = Z n Z n . ( 7
B =      3 j=1 B j 3 j=1 B ja 3 j=1 B jb 3 j=1 B j .     
Then following this theorem, for c = (

t T 2 + s T 2i , t T 2 -s T 2i ) T , E exp[ c T Z n ] → exp 1 2 ( t T 2 + s T 2i , t T 2 - s T 2i ) B t 2 + s 2i t 2 -s 2i .
By identifying this formula to Eq.(7.7), we get the required form of Γ. First, by a classical procedure of truncation and renormalization (see Section 7.2 for details), we can, without loss of generality, assume that there is a sequence

ε n ↓ 0 such that 1 ≤ i ≤ n, x i ∨ y i ≤ ε n n 1 4 . (7.12)
We will use the method of moments. Define, while dropping the index n in the coefficients of A n ,

ξ n = 1 √ n K ℓ=1 c ℓ [X(ℓ) * A n Y (ℓ) -ρ(ℓ) tr A n ] = 1 √ n e a e ψ e ,
where e = (u, v) ∈ {1, . . . , n} 2 and

ψ e =      K ℓ=1 c ℓ [x u y u -ρ(ℓ)], e = (u, u), K ℓ=1 c ℓ x u y v , e = (u, v), u = v.
Let k ≥ 1 a given integer. We have

n k 2 ξ k n = e 1 ,...,e k a e 1 • • • a e k ψ e 1 • • • ψ e k .
To each term in the sum we associate a directed graph G by drawing an arrow u → v for each factor e j = (u, v). The set of vertexes is then a subset of {1, . . . , n}. Therefore, to a loop u → u corresponds the product a uu ψ u→u (ℓ) =

a uu K ℓ=1 c ℓ [x u (ℓ)y u (ℓ) -ρ(ℓ)] and to an edge u → v with u = v corresponds the product a uv ψ u→v = a uv K ℓ=1 c ℓ x u (ℓ)y v (ℓ). In other words, n k 2 ξ k n = G a G ψ G , a G = e∈G a e , ψ G = e∈G ψ e .
We now consider the collection of connected sub-graphs of G. These connected sub-graphs can be classified into two types.

• Type-I sub-graphs. We call C a Type-I connected sub-graph of G if C contains loops only. In particular C has a unique vertex. The set of all the m 1 Type-I connected sub-graphs is denoted by F 1 , and the degrees of their vertexes by µ 1 , • • • , µ m 1 , respectively.

If µ j = 2 for some vertex j in a sub-graph C, then Ea G ψ G = 0 because of independence. Therefore we need only consider those graphs G whose m 1 Type-I sub-graphs have all its vertexes with degrees µ j ≥ 4. The contributions from all these sub-graphs to the moment part ψ G are then bounded by

E C∈F 1 ψ C ≤ K(ε n n 1 4 ) m 1 P i=1 (µ i -4) . (7.13)
• Type-II sub-graphs. A connected sub-graph containing at least one arrow u → v with u = v is called a Type-II sub-graph. The set of all these m 2 components is denoted by F 2 . For each C s ∈ F 2 , let u s be the number of its vertexes whose degrees are denoted by γ js , j = 1, • • • , u s . As in Type-I, we can also omit the case where γ js = 1 for some vertex j. Contributions from all the m 2 Type-II components to ψ G are then bounded by

E Cs∈F 2 ψ Cs ≤ K(ε n n 1
s 1 ,••• ,sm 1 C∈F 1 |a C | = s 1 ,••• ,sm 1 m 1 i=1 |a s i s i | µ i 2 ≤ m 1 i=1 n k=1 |a kk | µ i 2 ≤ Kn m 1 . ( 7
→ v with u = v. Note that v 1 |a v 1 v 2 | 2 ≤ ρ(A n ) 2 .
(7.17)

This, via induction, implies that we have for the tree part

v 1 ,••• ,vu s us-1 j=1 |a e j | 2 ≤ ρ(A n ) 2us-2 n,
and for the remaining sub-graph

v 1 ,••• ,vu s ts j=us |a e j | 2 ≤ ρ(A n ) 2ts-2us+2 n us-1 .
In the second inequality above, we use the fact that t s > u s as all degrees of vertex of Type-II are no less than 2. It follows that

v 1 ,••• ,vu s ts j=1 |a e j | ≤   v 1 ,••• ,vu s us-1 j=1 |a e j | 2 v 1 ,••• ,vu s ts j=us |a e j | 2   1 2 ≤ ρ(A n ) ts n us 2 , (7.18)
which gives, together with (7.16), that The last inequality results from the fact that by construction, the exponent of ε n is nonnegative. Consequently, the contributions from such graphs can be neglected.

G |a G | = G C∈F 1 |a C | Cs∈F 2 |a Cs | ≤ Kn u 1 2 +•••+ um 2 2 +m 1 . ( 7
n -k 2 E G a G ψ G ≤ n -k 2 G |a G || Eψ G | ≤ Kn -k 2 (ε n n 1 4 ) 2k-4m 1 -2 m 2 P s=1 us G |a G | ≤ Kε 2k-4m 1 -2

b).

If not all vertexes of connected sub-graph have a loop, then To see it, as an example, we consider the following

| v 1 ,v 2 ,v 3 a v 1 v 2 a v 1 v 3 a v 2 v 2 a v 3 v 3 | = | v 2 ,v 3 b v 2 v 3 a v 2 v 2 a v 3 v 3 | ≤ ( v 2 ,v 3 |b v 2 v 3 | 2 ) 1 2 ( v 3 a 2 v 3 v 3 ) 1 2 ( v 2 a 2 v 2 v 2 ) 1 2 = O(n 3 2 ), where b v 2 v 3 = v 1 a v 1 v 2 a v 1 v 3 = v 1 a v 2 v 1 a v 1 v 3 = (A n A n ) v 2 v 3 .
For general case, we can verify the order by induction. Using (7.13), (7.21) and (7.25), similar to (7.20), we get u s , we know that the graphs leading to a non negligible term are as follows: then degrees of vertexes of all its Type-I components must be four; its Type-II components all fall into Case 1 such that all its vertexes are of degree two. More precisely, we know that only the following isomorphic classes give a dominating term:

n -k 2 E G a G ψ G ≤ Kn -m 2 2 ε 2k-4m 1 -2
• k 1 double loops u → u with terms a 2 uu K ℓ=1 c ℓ (x ℓu y ℓu -ρ(ℓ)) In addition, the degrees of vertexes satisfy

4(k 1 + k 2 + k 3 ) = 2k
which implies that k must be even. Therefore, let k = 2p be an even integer. We notice that here, the relations on the edges, namely 2(k 1 + k 2 + k 3 ) = k, hold automatically. Thus, we can claim that

Eξ 2p n = 1 n p k 1 +k 2 +k 3 =k (2p)! k 1 !k 2 !k 3 ! × C 1 × C 2 × C 3 + o(1) , (7.28)
where 7.2. Truncation. The truncation and renormalization under the forth-moment condition is by now standard, see e.g. [START_REF] Bai | Methodologies in spectral analysis of large dimensional random matrices. a review[END_REF]. For our purpose and for ease of presentation, we give full details in the case of K = 1. The general case goes through in a same manner.

C 1 = k 1 j=1 E   a 2 u j u j

Figure 1 .

 1 Figure 1. Empirical density estimates (in solid lines) from the largest (top: b fn,1 ) and the smallest (bottom: b fn,6) sample eigenvalue from 1000 independent replications, compared to their Gaussian limits (dashed lines). Gaussian entries with p = 500 and n = 1000.

Figure 2 .Figure 3 .

 23 Figure 2. Limiting bivariate distribution from the second and the third sample eigenvalues. Top: contour lines of the empirical kernel density estimates b fn,23 from 1000 independent replications with p = 500, n = 1000 and Gaussian entries. Bottom: Contour lines of their limiting distribution given by the eigenvalues of a 2 × 2 Gaussian Wigner matrix.

Figure 4 .Figure 5 .Figure 6 .

 456 Figure 4. Empirical density estimates (in solid lines) from the largest (top: b fn,1 ) and the smallest (bottom: b fn,6) sample eigenvalue from 1000 independent replications, compared to their Gaussian limits (dashed lines). Binary entries with p = 500 and n = 1000.

Figure 7 .Figure 8 .

 78 Figure 7. An example of p = 500 sample eigenvalues (top) and a zoomed view on [0,5] (bottom). The limiting distribution of the E.S.D has support [0.395, 1.579] ∪ [4.784, 17.441]. The four eigenvalues {λn,j, 1 ≤ λ ≤ 4} in the middle, related to spiked eigenvalues are marked with a blue point. Gaussian entries with n = 2500.

6. 1 .

 1 Preliminary results and useful lemmas. For λ / ∈ [a y , b y ], we definem 1 (λ) = x λ -x F y (dx),(6.1)

  j ) + B ja + B jb } , (B j ) + B ja + B jb } ,

7. 1 .

 1 Proof of Theorem 7.1. It is sufficient to establish the CLT for the sequence of linear combinations of random Hermitian forms K ℓ=1 c ℓ X(ℓ) * A n Y (ℓ), where the coefficients (c ℓ ) ∈ C K are arbitrary. Notice that E[X(ℓ) * A n Y (ℓ)] = ρ(ℓ) tr(A n ) where ρ(ℓ) = E[x ℓ1 y ℓ1 ].

  [START_REF] Sheather | A reliable data-based bandwidth selection method for kernel density estimation[END_REF]) and (7.14) by noticing the relation i µ i + j,s γ js = 2k, the overall contribution from random variables has a bound | Eψ G | ≤ K(ε n n of the weight part a G will be established. Since n j=1 |a jj | w = O(n) holds for any positive integer w, thus

v 1 ,

 1 ••• ,vu s ts j=1 |a e j | ≤ ρ(A n ) ts n

2 ;• k 2 2 K ℓ=1 x ℓu y ℓv K ℓ=1 x ℓv y ℓu ; • k 3 ℓ=1 c ℓ x ℓu y ℓv 2 .

 22232 simple cycles u → v, v → u with terms |a uv | double arrows u → v, u → v with terms a 2 uv K

c 2 = k 2 j=1E |a u j v j | 2 K 2 u 2 ,c ℓ x ℓ1 y ℓ2 2 . 1 1 α k 2 2 α k 3 3 a 2 u j 1 u j 1 |a u j 2 v j 2 | 2 a 2 u j 3 v j 3 +

 222222121223 ℓ (x ℓu j y ℓu j -ρ(ℓ))2   , {u j } ⊂ {1, 2, . . . , n} C ℓ=1 c ℓ (x ℓu j y ℓv j ) K ℓ=1 c ℓ (x ℓv j y ℓu j ) , {u j , v j } ⊂ {1, 2, . . . j v j K ℓ=1 c ℓ x ℓu j y ℓv j 2   , {u j v j } ⊂ {1, 2, . . . , n}.Letα 1 = E K ℓ=1 c ℓ (x ℓ1 y ℓ1 -ρ(ℓ)) 20) or (7.27) again, along with inclusion-exclusion principle, (7.28) turns intoEξ 2p n = 1 n p (2p -1)!! × k 1 +k 2 +k 3 =k (p)! k 1 !k 2 !k 3 ! (k 1 ,k 2 ,k 3 (j 1 ,j 2 ,j 3 )=(1,1,1) α k

  .16) For a given Type-II component C s with t s edges, e 1 , • • • , e ts and u s vertexes, v 1 , • • • , v us , we extract a spanning tree from C s and assume its edges are e 1 , • • • , e us-1 , without loss of generality. However, we need to distinguish two situations for the remaining sub-graph after extraction of the spanning tree. Let ρ(A n ) be the spectral norm of the matrix A n .Case 1. The remaining sub-graph has at least one edge u

  Case 2. The remaining sub-graph does not contain any edge u → v with u = v, i.e., all remaining edges are loops. Since the degree of each vertex of Type-II component is no less than two, there must exist at least two vertexes whose degrees are more than two. Thus (7.14) turns into But, at this point, there must exist a vertex such that its degree exceeds three and so (7.21) correspondingly changes into

														m 2
														P	us
											n			s=1	.	(7.20)
						E	Cs∈F 2	ψ Cs ≤ K(ε n n	1 4 ) s=1 m 2 P	us P j=1	(γ js -2)-2m 2	.	(7.21)
	We now need to consider two possibilities.
	a). If all vertexes of connected sub-graph have a loop, then similar to (7.18), we
	have
	v 1 ,••• ,vu s	ts j=1	|a e j | ≤	  v 1 ,••• ,vu s	us-1 j=1	|a e j | 2	v 1 ,••• ,vu s	ts j=us	|a e j | 2	 	1 2	≤ ρ(A n ) ts n	us+1 2 .
	and then (7.19) becomes
											u 1 +1 2 +•••+	um 2 +1 2	+m 1 .	(7.22)
						E	C∈F 2	ψ C ≤ K(ε n n	1 4 ) s=1 m 2 P	us P j=1 (γ js -2)-4m 2	.	(7.23)
	By (7.22) and (7.23), similar to (7.20), we get
														m 2
	n -k 2	E	G	a G ψ G ≤ Kn -m 2 2 ε 2k-4m 1 -2 n	P s=1	us-4m 2	≤ Kn -m 2 2 .	(7.24)

G |a G | ≤ Kn

  So the contribution from this kind of graphs can also be neglected. Here we remind the reader that (7.20) is obtained by assuming all m 2 Type-II component belonging to case 1 and that (7.24) or (7.26) holds if all m 2 Type-II component belong to case 2. If some Type-II components of the graph G belong to case 1 and the others pertain to case 2, by the similar argument to the above, one can show that Therefore, if some item involves the connected sub-graph of case 2, the contribution from this item can then be omitted.In summary, in conjunction with (7.20) and the meanings of 2k, 4m 1 , 2

		n	m 2 P s=1	us-2m 2	≤ Kn -m 2 2 .	(7.26)
	n -k 2	E
					m 2
					s=1

G a G ψ G = o(1).

(7.27)

Research was (partially) completed while J.-F. Yao was visiting the Institute for † The research of this author was supported by CNSF grant 10571020 and NUS grant R-155-000-061-112 AMS 2000 subject classifications: Primary 62H25, 62E20; secondary 60F05,

where the matrix B = B 1 + B 2 + B 3 with

The proof of the theorem is postponed to the end of the section. First, we describe some specific applications of the theorem with own interest. Note that by definition, the three matrices B j 's are symmetrical (complex-valued).

Consider first the real variables case with i.i.d random vectors {(x i , y i ) i∈N } from R K × R K , and a sequence of symmetric matrices {A n = [a ij (n)]} n . We are then considering K random bilinear forms and consequently, θ = τ . The matrix B given above is then exactly the limiting covariance matrix of the Gaussian vector W .

Corollary 7.1 Under the same conditions as in Theorem 7.1 but with real random vectors {(x i , y i ) i∈N } and symmetric matrices {A n } n , the sequence of vectors (Z n ) n converges weakly to a zero-mean K-dimensional Gaussian vector with covariance matrix B.

An interesting application to the case (x i ) = (y i ) gives the following CLT for random quadratic forms in a straightforward way.

Let the vectors {X(ℓ)} 1≤ℓ≤K are as defined in (7.1). Assume the following limit exist

Then, the M -dimensional random vectors

converges weakly to a zero-mean Gaussian vector with covariance matrix

where

If all the diagonal elements of the matrices (A n ) are null, then ω = 0. The limiting covariance matrix D takes a much simpler form:

For ease of the presentation, we need to define more limiting quantities. For

. We introduce the following matrices

(7.9)

Here, the indexes are 1 ≤ ℓ, ℓ ′ ≤ K. By definition, all these matrices are real and symmetrical. Let us also define the 2K × 2K matrices

Theorem 7.3 Consider the M -dimensional complex-valued random vectors Z n = (Z n,ℓ ) defined in Theorem 7.1. Under the the same conditions as in that theorem, the real and the imaginary parts (U n , V n ) of Z n converges weakly to a 2K-dimensional Gaussian vector with covariance matrix

We aim to the replacement of the entries of x and y with truncated, centralized, normalized variables. Let

and then, because of the arbitrariness of ε, there exists a positive sequence ε n such that n P(|x 4 ) → 0, and ε n → 0.

It follows that

), for all n > n h with

Now, for each n, let γ n be the larger of ρ n and ε n . However, in the following we still use the notation ε n instead of γ n . By Markov inequality and Burkholder inequality

where we have used the inequality

which, together with (7.33), gives that

. Moreover, it is not difficult to see that we can substitute ρ ′ = cov(x 1 / E|x 1 | 2 , y 1 ) for ρ without alternating the weak limit.

The truncation, centralization and normalization of y can be completed with similar argument as above. In the sequel, for simplicity, we shall suppress all superscripts on the variables and suppose that |x i | ≤ ε n n


E Kirillov
email: evgueni.kirillov@univ-rennes1.fr

J.-F Carpentier

E Bunel

Carboxylic Acid Derivatives via Catalytic Carboxylation of Unsaturated Hydrocarbons: Whether the Nature of Reductant May Determine the Mechanism of CO 2 Incorporation?

Carboxylic acid derivatives via catalytic carboxylation of

unsaturated hydrocarbons: whether the nature of a reductant may determine the mechanism of CO2 incorporation?

Introduction

In a global context aiming at a more sustainable development, catalytic activation of CO 2 mediated by transition metal complexes with complete integration of the whole CO 2 molecules into a Organometallics, Materials and Catalysis Dept., Institut des Sciences Chimiques de Rennes, UMR 6226 CNRS -Université de Rennes 1, F-35042 Rennes Cedex, France Corresponding author: evgueni.kirillov@univ-rennes1.fr b Chemical Sciences and Engineering Division, Argonne National Laboratory, Argonne, IL 60349-4837, United States 2 the final products has become a highly topical research area. Many international leading research groups have been working on such catalytic CO 2 conversion into valuable chemicals, or on general methods for fixation of CO 2 . Besides two very recent major contributions, [START_REF]Carbon Dioxide as Chemical Feedstock[END_REF] many reviews have been released on topics relevant to catalytic CO 2 transformation and its utilization as C1 building block. 2 Reactions of particular interest include catalytic reduction (hydrogenation, hydrosilylation) of CO 2 , 3 synthesis of carboxylic acids, 4 of acrylic acid derivatives from olefins/alkynes, 5 and of polycarbonates from epoxides; 6 also, much work has been devoted to computational aspects of some catalytic reactions implying CO 2 , 7 the stoichiometric chemistry of CO 2, 8 and of other processes. 9

Among the above processes, formation of C-C bonds by carboxylation of readily available unsaturated hydrocarbons (alkenes, dienes, styrenics, alkynes) offers a direct access to valuable saturated and unsaturated carboxylic acids and their derivatives. 10 Modern industrial processes for production of carboxylic acids do not involve reactions with CO 2 . Only the Monsanto/Cativa processes implicate indirectly CO 2 as a component for the water-gas shift reaction. Otherwise, carboxylic acids and their derivatives are traditionally synthesized by multi-step oxidation protocols starting from petrochemicals (propylene, butadiene, toluene, xylenes, naphthalene) or by carbonylation of alkenes with CO/H 2 O-alcohols (e.g. Koch acids, Reppe chemistry, hydroesterifications (Lucite process)). 11 The direct catalytic carboxylation of unsaturated hydrocarbons bestows, in principle, obvious benefits in terms of energy and raw materials with respect to regular multistep oxidation protocols. This is fundamentally different from other activation processes implying severe reduction of CO 2 (sometimes complete when hydrocarbonstype products are targeted), which are highly energy-intensive (reduction of C=O bonds) and inherently poorly atom-efficient.

This Perspective article discusses a few existing strategies for the direct coupling of unsaturated hydrocarbons (alkenes, alkynes) with CO 2 towards carboxylic acids.; a special attention is drawn to potential catalytic processes enabling the atom-economic reductive carboxylation of alkenes (Scheme 1). The article is focused on unraveling the essential problems of catalytic coupling of CO 2 with unsaturated hydrocarbons and it analyzes the available information about the nature of key intermediates and side-products of the operative (actual and putative) catalytic mechanisms.

Scheme 1. Catalytic reductive carboxylation of unsaturated hydrocarbons into carboxylic acids and esters.

State-of-the-Art and Main Features of Carboxylation of Unsaturated Hydrocarbons

Some Thermodynamic Considerations

The high thermodynamic stability of CO 2 makes unfavorable its conversion to other molecules; 7a,12 therefore, the corresponding processes require significant energy input in a form of heat, 13 or, more common, as highly energetic co-reagents. For instance, some non-extensive thermochemical calculations on relevant reactions involving CO 2 (Table 1) demonstrated that formation of the corresponding acids from CO 2 and alkane/alkene is generally an endothermic process (reactions (2) and (3), respectively). Yet, although a significant thermodynamic constraint is also characteristic for the reduction of CO 2 with H 2 into formic acid (reaction ( 1)), the related reaction conducted in the presence of ethylene, to deliver eventually propionic acid, is calculated to be quite thermodynamically favorable (reaction ( 7)). Because a saturated acid forms in the latter coupling reaction, the corresponding energy is virtually driven by the exergonic ethylene hydrogenation reaction (4). Similarly, highly negative enthalpies and Gibbs energies are estimated 4 for the reactions implying strong reductants such as hydrosilanes or hydroboranes (reactions (8)( 11)), which provide the corresponding carboxylic acids esters. The above thermochemical estimations unequivocally illustrate the feasibility of alkene carboxylation reactions towards the corresponding carboxylic acids or some of their derivatives, provided they are carried out in the presence of an appropriate reducing agent (H 2 or hydrosilane/hydroborane, respectively). Thus, a quest for elaboration of new systems competent for catalyzing these processes has been ongoing.

Direct carboxylation of unsaturated hydrocarbons (alkenes, alkynes, dienes) to carboxylic acids. Role of the (reverse) water-gas shift reaction ((r)WGSR).

Straightforward production of carboxylic acids from parent alkenes can be realized via two principal catalytic ways -hydrocarboxylation and hydroxycarbonylation -, both of them using end-products of the water-gas shift reaction (WGSR), 14 namely, CO/H 2 O and CO 2 /H 2 combinations, respectively.

It has been previously disclosed that carboxylic acids (as mixtures of linear and branched products) can be obtained via Pd-catalyzed hydroxycarboxylation of -olefins with CO/H 2 O. 15 The so-called hydride mechanism, elucidated by Van Leeuwen and coworkers (Scheme 2), 16 presumed that the Pd-acyl species, formed via insertion of CO into the Pd-alkyl intermediate, reacts with H 2 O eliminating the corresponding carboxylic acid and, thus, regenerating the catalyst. In spite of the obvious thermodynamic preference of the WGSR (H = 9.8 kcal•mol 1 ), 14 this step (even if operational) seemed not to play any crucial role in the process. 16,[START_REF] Zudin | Pd-catalyzed hydrocarbonylation of ethylene to diethyl ketone, WGSR serves as a source of H 2 , while the concomitantly produced CO 2 remains intact[END_REF] CO/H 2 O (only formation of the linear product is illustrated). 16 5

On the other hand, there are only a few reports disclosing the straightforward synthesis of carboxylic acids from CO 2 . In seminal contributions, it has been demonstrated that the "direct" carboxylation of ethylene with CO 2 , mediated by homogeneous catalytic systems ((Ph 3 P) 2 PdCl 2 /HCl or (Ph 3 P) 3 RhCl/HBr), required very harsh conditions (120180 °C, 150700 atm) and resulted in the formation of propionic acid in 38% yield, with ethanol and ethyl propionate as the major byproducts. 18 Very recently, another catalytic protocol for hydrocarboxylation of both linear and cyclic alkenes has been disclosed by Leitner et al;19 intermediates, as well as that the principal hydroxycarbonylation cycle parallels the one of the mechanism of hydroxycarboxylation using CO/H 2 O. 16 The overall reaction occurs in acetic acid under, however, still drastic conditions (60 bar CO 2 , 10 bar H 2 , 180 °C).

Scheme 3. Proposed mechanism for the Rh-catalyzed hydroxycarbonylation-"hydrocarboxylation" of olefins. 19 Interestingly, under virtually similar conditions (CO 2 /H 2 , 3060 bar, 130140 °C, NMP as solvent and Li salts as additive) upon using H 4 Ru(CO) 12 20 and Ru 3 (CO) 12 /phosphine 21 as catalysts, domino hydroformylation/reduction reactions of alkenes have been observed, which resulted in isolation of the corresponding alcohols (mixtures of linear and branched products). The possible intermediacy of the carboxylic acid formation step for these processes has not been established.

Yet, electrochemical coupling of CO 2 with -olefins (propylene and 1-butene) mediated by Ni(II) complexes has been reported to afford mixtures of two major products, namely, a substituted succinic acid and the corresponding unsaturated monocarboxylic acid (Scheme 4). 22 Under the same conditions, butadiene has been transformed into a mixture of mono-and dicarboxylic acids:

predominantly, 3-pentenoic, 2-ethenyl-succinic and cis-/trans-3-hexene-1,6-dioic acids. The mechanism of formation of these products remains unclear. By a similar electroreductive coupling of both terminal and internal alkynes and CO 2 , catalyzed by Ni complexes, unsaturated carboxylic

acids have been prepared. 23 R R = CH 3 , CH 2 CH 3 e -, CO 2 [Ni 2+ ] O O O O R O O + X = H,CH 3 [Ni] = NH N NH Ni 2+ 2(BF 4 ) - X Scheme 4.
Ni-catalyzed electrocarboxylation of propylene and 1-butene. 22 Benzonitrile-promoted electrocarboxylation of styrene into 3-phenylpropionic acid has been also documented to operate presumably via a radical mechanism. 24

Acrylates from alkenes and CO 2 : from stoichiometric studies towards catalysis

The catalytic coupling of alkenes with CO 2 to produce acrylic acids [START_REF]Acrylic acid is produced by a multi-step oxidation of propylene. The market of acrylic acid derivatives is around 2.5 MT/year with annual expansion rate of 4%[END_REF] has long been of interest from both fundamental and industrial points of view. In pioneering studies aimed at identifying potential catalytic pathways for the straightforward production of acrylic acid from ethylene and CO 2 , several Ni and Pd complexes have been recognized as effective precatalysts.

The key step of such catalytic processes is presumably the oxidative coupling reaction of the alkene with CO 2 onto low valent metal centers in the catalytically active species, which affords metallacyclic product 1-M (Scheme 5). [START_REF] Papai | [END_REF][START_REF] Plessow | A revisited mechanism has been described recently[END_REF] Such intermediacy has been experimentally established for Ni(0), 28 Fe(0) 29 and Pd(0) 30 (with dienes) precursors in the pioneering studies of Hoberg and coworkers and others. 31,[START_REF]A thermally stable metallalactone complex of Ti(IV) has been reported to be the product of the reaction of the parent Ti(II) precursor Cp * 2 Ti( 2 C 2 H 4 ) and CO 2 : S. A. Cohen[END_REF][START_REF] Alt | Zr(II)-metallocenes have been used for the synthesis of the corresponding Zr(IV) zirconalactone complexes: (a) H. G[END_REF]

{L} n M R 2 {L} n M O O R 2 R 1 R 2 R 1 R 1 1-M CO 2 {L} n M O O R 2 R 1 H {L} n M R 2 R 1 O HO O HO R 2 R 1
-H elimination reductive elimination oxidative addition / C-C coupling Scheme 5. Hypothetic mechanism for the synthesis of acrylic acids from alkenes and CO 2 via formation of metallalactone 1-M (M = Ni, Fe). 28,29 While it had been proposed that the metallalactone derivative 1-M can follow -H and reductive elimination reactions giving rise to the corresponding acrylic acid (Scheme 4), it has eventually appeared that the metallacycles are essentially robust; in fact, cleavage of the MO and MC bonds does not occur without decomposition of these complexes. Only in one case, transformation of several nickelalactone complexes into hydrido-acrylate and then into acrylate derivatives, however without further reductive elimination step, induced by addition of dppm ligand, has been documented. 34 It has been experimentally demonstrated that some additional components can provoke release of the carboxylic fragment (Scheme 6). For instance, acidic hydrolysis yielded the corresponding carboxylic acid (Scheme 6, eq (1)), while treatment of nickelalactone 1-Ni with CH 3 OH under acidic conditions provided the corresponding ester (eq (2)). 28b,d,35 In particular, stoichiometric reactions of 1-Ni with ethylene, styrenes and other functionalized alkenes (eq (3)) followed by hydrolysis resulted in isolation of mainly linear carboxylic acids. 36 In this case, it was proposed that the insertion of the substrate C=C double bond into the NiC bond of 1-Ni resulted in the ring expansion of the five-membered metallacycle, the derived thereof seven-membered intermediate being more prone for β-H elimination reaction. The reductive elimination reaction towards the corresponding carboxylic acid and regeneratation of the {L} n Ni(0) catalytic species appeared, however, to be unfavorable for the above systems, which hampered catalytic turnover; nonetheless, this step was found operational for the Pd-and Ni-catalyzed catalytic coupling of CO 2 with dienes (functionalization/telomerization) 37 or with diynes. 38

{L} 2 Ni O O H + R 2 R 1 CH 3 OH/ HCl CH 3 I (1)
(2) 40 and others 41 on the reactions between unsubstituted nickelalactones 1-Ni bearing different ligands L with an excess of CH 3 I as the metallalactone cleaving agent (Scheme 6, eq (4)). However, the low yields (256% vs Ni) of methyl acrylate product over 348 h and significant concomitant decomposition of the Ni species make these reactions catalytically impractical so far. Additional insights in the nature of possible intermediates in this process and hypothetic decomposition pathways have been recently gained from combined computational 42 and experimental studies. 43 For example, for the reactions involving CH 3 I as co-reagent, the corresponding activation barrier G  298 for the β-H elimination of methyl acrylate was computed in the reasonable range 1525 kcal•mol 1 (depending on the theoretical method and model). On the other hand, a high endothermicity (G 298 = +525 kcal•mol 1 ) was computed for the HI reductive elimination step (which can in principle be surmounted by an introduction of a base), required to regenerate a {L} n Ni(0) catalyst; this set up another obstacle for catalytic turnover.

R 1 = R 2 = H {L} n Ni I H -HI {L} n Ni 0 (3) 1-Ni O HO R 2 R 1 O MeO R 2 R 1 O MeO -H elimination catalysis {L} n Ni I OMe O - R 1 = R 2 = H R 3 1) 2) H + O HO R 3 O HO R 3 or R 3 = H, Ph R 3 = CO 2 Me,
By analogy, addition of stoichiometric amounts of different reagents (B(C 6 F 5 ) 3 , 44 Na + [B(m-(CF 3 ) 2 Ph) 4 ]  , 45 X 2 (X = Cl, Br, I), 46 Ph 2 Zn 47 ) to 1-Ni complexes has been shown to eventually promote cleavage of the nickelalactone cycle and elimination of the acrylate fragment.

Yet, the carboxylation reactions of Mo(0)-and W(0)-ethylene complexes, affording the corresponding acrylates 2 and 3, originally reported by Carmona and coworkers, constitute another model for coupling of CO 2 with olefins (Scheme 7). 48 As proposed from DFT calculations, the formation of 2-W and 3-W is intermediated by formation of a metallacyclic species analogous to 1-M. 49 Thus, the corresponding barrier for the β-H elimination reaction yielding 2-W and 3-W was estimated to be significantly lower (computed E  298 = ca. 10 kcal•mol 1 ) than those typically found in Ni-based systems.

Scheme 7. Reaction between Mo(0)-and W(0)-ethylene complexes and CO 2 . 48 Of particular interest is the reactivity of hydride complex (Triphos)MoH 4 (PPh 3 ) (Triphos = Ar 2 PCH 2 CH 2 ) 2 PPh; Ar = Ph) disclosed by Bernskoetter and coworkers (Scheme 8). 50 Several products, formate, acrylate and propionate complexes can be derived from the corresponding precursor, the selectivity of the reaction depending on the ratio of the C 2 H 4 /CO 2 feed. For a completely different system based on a cationic Pd-methoxycarbonyl complex (Scheme 9), it has been established that methyl acrylate is readily released upon exposure to an ethylene atmosphere under mild conditions (room temperature; 1 bar pressure). 53 Nature of the solvent influenced the yield of the reaction; significantly better yields than those having been obtained in CH 2 Cl 2 /MeCN mixture or in neat CH 2 Cl 2 were achieved in DMF. According to the DFT model, the essential role of DMF is the stabilization of intermediates and transition states via formation of relatively strong metal-solvent bonds that grants more favorable pathway in terms of free energy. The rate-determining step for this process was suggested to be the CC bond coupling reaction, that is, the insertion of ethylene into the PdC bond of the methoxycarbonyl complex Under somewhat more drastic conditions (30 bar ethylene, 30 bar CO 2 ) complex

[{dppe} 2 Pd(COOMe)] + [SO 3 CF 3 ]  , along with stoichiometric production of methyl acrylate, has been found to catalyze the formation of ethyl acrylate. 53 A putative mechanism involving the intermediacy of Pd-hydrido species was proposed (Scheme 10). However, the results of both stoichiometric and theoretical studies have ruled out the proposed mechanism. 53 No alternative mechanism for this catalytic reaction matching the experimental observations has been suggested so far.

Scheme 10. Disclaimed putative hydride mechanism for Pd-catalyzed synthesis of ethyl acrylate from ethylene and CO 2 . 53

Catalytic coupling of CO 2 with alkenes/alkynes/dienes in the presence of organic/organometallic substrates as reductants

The lack of catalytic activity systematically observed in the above studies about alkene carboxylation into acrylic acid derivatives, and also fundamental thermodynamic calculations, 39 suggest unfavorable overall thermodynamics of this type of reaction. That is, the thermodynamic driving force of the coupling reaction of alkene with CO 2 is directly related to the formation of a species with a strong MO bond (a metallalactone or a carboxylate), but release of a "free" carboxylic acid and regeneration of a reduced metal species, which is mandatory to enable the catalytic turnover, are disfavored.

A workaround has been identified, which consists in implementing in the reaction (super)stoichiometric amounts of "energetic" reactants (organometallics or related organoelementals) that compensate the initially unfavorable thermodynamics of the reaction by the more favored formation of the corresponding metal/organoelemental carboxylate derivatives, and simultaneously regenerate the low valent metal active species.

A breakthrough along these lines has been achieved with alkoxy/phenoxy, amido and borate salts of Na, which enabled the synthesis of sodium acrylates with a Ni-based catalytic system (Scheme 11); 54 so far, the maximal TON reached ca. 100 (10 bar CO 2 , 5 bar ethylene, THF, 100 °C; 300 equiv o-F-C 6 H 4 -ONa vs. Ni).

As it has been corroborated by theoretical investigations, Erreur ! Signet non défini.a, 55 the key factor in this system is the presence of a base with high proton affinity (typically, RO  ), which modifies a transition state leading to the β-H elimination reaction in the nickelalactone and reduces the barrier of this rate determining step (the corresponding calculated barrier G  298 drops down to 1216 kcal•mol 1 ). Also, as long as the corresponding carboxylic acid is produced in a form of a sodium salt, this allows maintaining favorable thermodynamics of the global process (computed G 298 = ca. 15 kcal•mol 1 ). The first catalytic reductive hydrocarboxylation of styrenes under mild conditions (1 bar CO 2 ; room temperature; THF), using an excess of Et 2 Zn as nucleophilic interceptor, 56 has been successfully achieved by Rovis and coworkers. 57 The NiR 2 /additive/Et 2 Zn (R = 1,5-COD, acac; additive = Cs 2 CO 3 , DBU, pyridine) catalytic system has been surmised to operate via a completely different mechanism, which excludes formation of nickelalactone intermediates and involves distinct hydrometallation and carboxylation steps leading regioselectively to saturated -substituted carboxylic acid derivatives (Scheme 12). Several limitations have been identified for the Ni-based system: (1) it is poorly active in carboxylation of both linear olefins, cyclic dienes and -substituted styrenes;

{L} n Ni R 2 {L} n Ni O O R 2 R 1 R 2 R 1 R 1 1-Ni CO 2 {L} n Ni R 2 R 1 O O O O R 2 R 1 -H elimination reductive elimination oxidative addition / C-C coupling M + X - {L} n Ni O O R 2 R 1 M X -
(2) the activity of the catalytic system strongly depends on the nature of the additive; 3) also, the nature of the reducing agent plays one of the key roles, as the system is not efficient if FeCl 2 /bis(iminopyridine)-catalyzed reductive hydrocarboxylation of styrenes has been also reported to proceed under mild conditions (1 bar CO 2 ; room temperature; THF) using excess EtMgBr as reductant. 59 The proposed mechanism (Scheme 13) involves an organomagnesiation step, followed by carboxylation of the resulting secondary Grignard reagent to yield mostly the corresponding -substituted carboxylic acid derivatives. This Fe-based system is only active with There are also a few reports on reductive carboxylation of functionalized alkenes: terminal alkene hydroboration followed by Cu-catalyzed carboxylation in the presence of KOtBu, 60 and reductive carboxylation of allyl esters and halides on Ni catalysts carried out using metallic Zn and Mn powders as stoichiometric reducing agents. 61 Dienes and alkynes are substantially more reactive substrates that can be converted to the corresponding unsaturated carboxylic acids derivatives. Thus, the ring-closing carboxylation of bis (1,3-dienes) 62 and hydrocarboxylation of 1,3-dienes 63 and allenes, 64 catalyzed by Ni and Pd complexes have been reported to proceed in the presence of superstoichiometric amounts of organometallic reagents (R 2 Zn (R = Et, Ph) and Et 3 Al, respectively). Similarly, both terminal and internal alkynes and some functional derivatives have been used in Ni-and Cu-catalyzed reactions with CO 2 in the presence of different reagents (R 2 Zn and RZnX (R = Me, Et, Bu, Bn, Ph; X = Cl, Br, I), M 2 CO 3 (M = K, Cs)), 65,66 and in related bora- 67 and silacarboxylation 68 and Ag(I) complexes 69 and heterogeneous Ag-doped MOFs 70 has been also developed.

The coupling of styrene derivatives with CO 2 in MeOH into the corresponding methyl esters catalyzed by Ni hydrides has been described by Garcia and coworkers. 71 The proposed hydroesterification mechanism (Scheme 14) parallels that of hydroxycarbonylation (Scheme 2); 16 it also incorporates a step of formation of an acyl-metal intermediate from the corresponding Ni-alkyl species and CO generated from CO 2 . 72 The CO 2 origin of the carbonyl group in the final product has been corroborated by 13 C labelling experiments. A similar mechanistic pathway has been proposed by Beller and coworkers for the Rucatalyzed hydroesterification of many industrially important olefins; 73 the reduction of CO 2 to CO in MeOH (solvent) was also suggested.

There are only a few examples describing the use of hydrosilanes as reductants in carboxylation of unsaturated substrates (typically, alkynes). 74 For example, the hydrocarboxylation of disubstituted alkynes catalyzed by (NHC)CuF (NHC = N-heterocyclic carbene) has been reported to proceed under mild conditions (1 atm CO 2 , 65100 °C), giving the corresponding silyl esters of ,-unsaturated carboxylic acids (Scheme 15). 75 Stoichiometric studies revealed that the rate-determining step of the process is the insertion of CO 2 into the CuC bond of the alkenyl intermediate. Remarkably, the reaction of the Cu(I)-carboxylate intermediate with hydrosilane occurred readily, already at room temperature; different hydrosilanes have been tested as reducing agents, exhibiting the following order of reactivity: (EtO) 3 SiH  PMHS (= polymethylhydrosiloxane) > (iPrO) 3 SiH > Ph 2 SiH 2 .

{L} n Cu H {L} n Cu R CO 2 R {L} n Cu-O 2 C R R R 3 Si-O 2 C R R R R R 3 Si

Perspectives 20

Depending on the nature of the catalytic system used for the carboxylation of alkenes, three distinct types of mechanisms can be operational: (1) a rWGSR-mediated hydroxycarbonylation that affords straightforwardly saturated carboxylic acids; (2) the formation of acrylates via oxidative addition of CO 2 and alkene onto a low-valent metal center, followed by and reductive eliminations steps; (3) the production of metallic salts/organoelemental derivatives of the corresponding saturated acids via an hydrocarboxylation mediated by metal-hydrido or -alkyl species.

For the first process, the key and indispensable step is the rWGSR utilizing CO 2 /H 2 mixtures as CO surrogates. Since dihydrogen is widely preferred as reductant in organic synthesis and catalysis [START_REF]Organic Chemistry Portal[END_REF] as an inexpensive, clean and sustainable reagent, 77 it obviously grants atom-economy for the given reaction. The use of dihydrogen as reducing agent in this process admittedly limits its applicability to homogeneous catalytic systems that are capable of maintaining efficiently the rWGSR step in the putative catalytic cycle (Scheme 3). While many heterogeneous, both singleand multicomponent systems are known to catalyze this reversible process, 78 only for a few homogeneous systems (typically based on Rh and Ru) the rWGSR has been proposed to operate as a part of global carbonylation processes utilizing CO 2 /H 2 mixtures as CO surrogates. 2a,j,19,20,21 In this regard, the identification of other homogeneous catalytic systems, based preferably on more abundant, less expensive metals (Fe, Co, Ni etc) competent for catalyzing the rWGSR under mild conditions, would also be of high industrial relevance. Unfortunately, to date, the complete deficiency of experimental (kinetic) and computational data for this process does not allow a rational optimization/development; identification of the optimal conditions remains still serendipitous.

For the second and third processes, efficient high-energy reductants are mandatory in order to compensate the unfavorable thermodynamics of the carboxylation reactions: (1) for cleavage of the metallacycle in the divalent Ni (II) /Pd (II) intermediates formed in the oxidative coupling step of unsaturated hydrocarbon with CO 2 ; and/or (2) for regeneration of the low valent transition metal species from the corresponding carboxylate intermediates; and (3) for conversion via transmetallation of metal-carboxylate intermediates to the catalytically active metal-alkyl orhydrido species. These reactions thus result in release of the carboxylate products as metallic (Na + , Zn 2+ or Mg 2+ , respectively) salts, which have to be treated acidically in order to recover the corresponding pure carboxylic acids. Therefore, this way of achieving the carboxylation process is plagued by the consumption of a stoichiometric amount of an expensive, difficult-to-handle and waste-generating organometallic reagent.

On the other hand, use of dihydrogen as reductant is hardly compatible with both mechanisms (Schemes 1113). Indeed, the pivotal step of the reductive carboxylation process, namely, the reduction of metal-carboxylate intermediates with dihydrogen to give the corresponding carboxylic acid and a metal-hydrido species, is thermodynamically difficult (BDE 298 (HH bond) = 104 kcal•mol 1 , BDE 298 (SiH bond) = 80 kcal•mol 1 ). 79 A modified scenario can be envisaged, by implementing reduction with a molecular hydrosilane or the more economically-viable polymethylhydrosiloxane (PMHS); this would provide direct access to valuable carboxylic acid silyl esters. Readily available, easy-to-handle and relatively inexpensive (although admittedly incompatible with the production of commodities) hydrosilanes are widely used as reducing organic agents for many functional groups. 80 Given the high propensity of hydrosilanes to induce formation of MH bonds in metal complexes via the simultaneous, thermodynamically favored formation of SiO bonds (BDE 298 (SiO bond) = 190 kcal•mol 1 ) 79 and their highly tunable reactivity by the substituents at the silicon atom, these are eminently competitive reducing agents to organometallic substrates; they are also a frequent alternative to dihydrogen in many metal-catalyzed processes. PMHS, a low molecular weight (M n = 1,7003,200 g•mol 1 ) polymeric byproduct of the silicone industry, can also be used as a very cheap and environmentally friendly reducing agent. PMHS is more air-and moisture-stable than other hydrosilanes and can be stored for long periods of time without loss of activity. The mechanism of action and reactivity pattern of PMHS are the same as those featured by usual hydrosilanes; application of PMHS for reduction of C=O groups in aldehydes and ketones has been reported. 81 The envisioned mechanism for reductive carboxylation of unsaturated hydrocarbons with hydrosilanes (Scheme 16, Cycle A) is composed of a series of elementary chemical reactions which, individually, are all well-documented: (i) alkene insertion into the M-H bond of active metal-hydrido species 1 to afford metal-alkyl 2; [START_REF]This reaction constitutes the principal step of the mechanism of polymerization of alkenes using Ziegler-Natta, metallocene and post-metallocene catalysts[END_REF] (ii) insertion of CO 2 into the M-alkyl bond of 2 yielding metal-carboxylate 3; 8 (iii) a transelementation reaction of 3 with a hydrosilane delivering the corresponding alkyl silyl ester 4. [START_REF]To our knowledge, this reaction has been intentionally substantiated only in a few cases: (a) hydrosilane-mediated reduction of Ir-carboxylate complexes into the corresponding hydrides[END_REF] A competing mechanism may be envisioned (Scheme 16, Cycle B), which involves rapid insertion of CO 2 into the M-H bond of 1 to give a metal-formate intermediate of the type 5, which upon reaction with the hydrosilane would release the corresponding silyl formate 6 as a by-product besides the desired 4. The latter process, that is hydrosilylation of CO 2 , is well documented in the literature and it is known to be catalyzed by complexes and salts of non-transition elements. In both cases (Scheme 16, Cycles A and B), hydrolysis of silyl esters 4 and 6 under mild conditions in a distinct, final workup step shall yield the corresponding alkyl-carboxylic and formic acids, respectively, and nontoxic (poly)siloxane byproducts (use of PMHS). However, silyl esters primarily resulting from the above global process are also much valuable precursors and synthetic intermediates. For instance, they are employed as intermediates and cross-linking reagents in the synthesis of poly(silyl esters), some biodegradable polymers widely used for medical applications (drug delivery systems, surgical devices, recyclable materials). 84 Synthesis of silyl esters generally relies on condensation of carboxylic acids with chlorosilanes with concomitant elimination of HCl. 85 An alternative synthetic protocol exists, which is based on dehydrogenative coupling of carboxylic acids with hydrosilanes catalyzed by transition metal complexes (Ru-Pd, Pt, Cu). 84b,c

Conclusions

In this Perspective, we have highlighted advances and recent breakthroughs in the "direct" catalytic synthesis of carboxylic acids and their derivatives starting from unsaturated hydrocarbons (alkenes, alkynes, dienes) and CO 2 . All of the existing systems are plagued either by a poor catalytic productivity and/or the use of expensive and waste-generating stoichiometric reductants.

Efforts must be paid to optimize the cost, safety and global sustainability of such processes. These objectives could be achieved by two interlinked approaches: (1) implementation of catalytic systems that are competent for carboxylation of unsaturated hydrocarbons (alkenes) using H 2 as reducing agent under mild conditions (temperature, pressure); and/or (2) implementation of relatively inexpensive and efficient reducing agents in the catalytic systems (e.g., molecular hydrosilanes or the polymeric analogue, polymethylhydrosiloxane (PMHS)). Understanding of the nature and role of key intermediates and mechanisms of CO 2 activation from experimental approaches and multi-scale theoretical simulations is the key for elaborating processes that enable efficient transformation of CO 2 into value-added chemicals. This is also of high interest for future renewable energy technologies and addressing the global issue of energy transition.

  Scheme 2. Proposed mechanism for the Pd catalyzed hydroxycarbonylation of olefins with

(

  Scheme 9. Computed mechanism for methyl acrylate formation from the reaction of complex [{dppe} 2 Pd(COOMe)] + [SO 3 CF 3 ]  with ethylene in DMF.53

  Scheme 11. Proposed mechanism for the Ni-catalyzed formation of acrylic acid derivatives from alkenes and CO 2 in the presence of inorganic or organometallic salts M + X  (and an additional base). Erreur ! Signet non défini.

  reactions using B 2 (pin) 2 /LiOtBu and Et 3 SiB(pin)/NaOtBu co-reagents. The synthesis of propiolic acids by carboxylation of terminal alkynes in the presence of bases M 2 CO 3 (M = K, Cs) catalyzed by Cu(I)

Scheme 14 .

 14 Scheme 14. Proposed mechanism for the Ni-catalyzed hydroesterification of styrenes.71

  Scheme 15. Proposed mechanism for the (NHC)CuF/R 3 SiH-catalyzed hydrocarboxylation of alkynes.75

  3

Scheme 16 .

 16 Scheme 16. Putative catalytic cycle (A) for the formation of silyl esters by reductive carboxylation of alkenes, and CO 2 hydrosilylation (B) as anticipated side reaction.

Table 1 .

 1 Calculated Enthalpies and Free Energies (in kcal•mol 1 ) for some basic reactions implying CO 2 and ethylene. a

	Reaction	H	G
	1. CO 2 + H 2  HCOOH	+7.4 (+5.1) b	+12.5 (+12.0) b
	2. CO 2 + CH 3 CH 3  CH 3 CH 2 COOH	+9.7	+16.9
	3. CO 2 + CH 2 =CH 2  CH 2 =CHCOOH	+5.8 (+5.7) b	+20.4 (+18.0) b
	4. CH 2 =CH 2 + H 2  CH 3 CH 3 5. CH 2 =CH 2 + Et 3 SiH  Et 4 Si 6. CH 2 =CH 2 + HBpin  EtBpin 7. CO 2 + CH 2 =CH 2 + H 2  CH 3 CH 2 COOH 8. CO 2 + Et 3 SiH  HCOOSiEt 3 9. CO 2 + HBpin  HCOOBpin 10. CO 2 + CH 2 =CH 2 + Et 3 SiH  CH 3 CH 2 COOSiEt 3 11. CO 2 + CH 2 =CH 2 + HBpin  CH 3 CH 2 COOBpin	36.3 32.6 35.6 26.7 16.0 14.8 49.9 48.3	23.4 15.3 19.3 7.5 11.7 5.0 23.4 21.7

a M06/6-31

(d,p)

in gas phase. b taken from ref. 12b (G3B3 in gas phase).

  Ni (L = DBU), which would have resulted in formation of a metal-hydrido-acrylate intermediate (Scheme 5), is not only kinetically problematic (computed G  298 = ca. 40 kcal•mol 1 ) but also thermodynamically unfavorable (computed G 298 = ca. +10 kcal•mol 1 ). Moreover, the subsequent reductive elimination step involving coupling of the metal-hydride and acrylate groups was also found to feature a high activation barrier, which, depending on the mechanism of elimination, can vary between 20 and 40 kcal•mol 1 . It has been predicted for the given system that preliminary cleavage of the NiO bond and decoordination of the carboxylate group can release the ring strain and stabilize the transition state leading to β-H elimination. This estimate has been further corroborated experimentally by Rieger and coworkers

CN, CH=CH 2 , =C=CH 2 (4) Scheme 6. Stoichiometric reactions of nickelalactones 1-Ni. 10 As demonstrated by DFT studies, 39 the direct β-H elimination reaction from metallalactone 1-

  Reactivity of (Triphos)MoH 4 (PPh 3 ) with ethylene and CO 2 .

	Mo H H P P Ph Ph Ph H H	PPh 3 P Ph Ph	C 2 H 4 (4 atm) -C 2 H 6 -H 2	P P Ph Ph Ph	Mo	P Ph Ph	CO 2 -2 C 2 H 4	P P Ph Ph Ph	P Ph O C O Ph Mo
	C 2 H 4 :CO 2 (1:1 mixture)			-PPh 3						PPh 3
		H
	P Ph Ph P Ph O	H Mo O	PPh 3 P Ph Ph		P P Ph Ph Ph O	Mo H O	PPh 3 P Ph Ph	H 2	P P Ph Ph Ph O	Mo H O	PPh 3 P Ph Ph
	Scheme 8.

50

Selective formation of similar acrylate products in reactions of zerovalent Mo-Triphos complexes (Triphos)Mo(C 2 H 4 )(N 2 ) 2 has been found to take place with different rates depending on the nature of aryl substituents on the ligand (the following rates order has been established: Ph > m-

Me 2 -Ph  p-CF 3 -Ph > p-Me-Ph  p-F-Ph > p-MeO-Ph > m-tBu 2 -Ph)

.

51

Yet, the above acyclic metal-hydrido-carboxylate products of Mo and W did not reveal any propensity to reductively eliminate the corresponding carboxylic acids. Alike 1-Ni, the monomeric tungsten acrylate ((MeO) 3 P) 5 Mo(H)(OC(O)CH=CH 2 ) upon treatment with CH 3 I slowly released methyl acrylate again with low yield.

52

  Et 2 Zn is replaced with iPrOH, Ph 3 SiH or H 2 , while Me 2 Zn and Ph 2 Zn afforded different products Proposed mechanism for the Ni(COD) 2 /Cs 2 CO 3 /Et 2 Zn-catalyzed hydrocarboxylation of styrenes.57

(no alkylated product and benzoic acid, respectively). Very recently, the mechanism has been computationally revisited, asserting transient Ni-hydrides as catalytically active species, whereas the other possible pathway involving formation of nickelalactone has been found to be kinetically disfavored.

58

Scheme 12.

  ethyl Grignard co-reagents; many other traditional hydride sources (NaBH 4 , LiAlH 4 , NaHBEt 3 , nBuLi, Et 2 Zn, Et 3 Al, iBu 2 AlH) were found completely inefficient.

			{L} n Fe
		Ar	Ar	H
	{L} n Fe Et		{L} n Fe H
	EtMgBr		Hydrometallation	Ar
				Ar
	Ar		{L} n Fe	H
	XMg	H		Ar
	CO 2 Carboxylation	XMgO 2 C
	Scheme 13. Proposed mechanism for FeCl 2 /bis(iminopyridine)/EtMgBr-catalyzed carboxylation of
	styrenes. 59