F Comte

V Genon-Catalot

NONPARAMETRIC ADAPTIVE ESTIMATION FOR PURE JUMP L ÉVY PROCESSES

Keywords: Adaptive Estimation, Deconvolution, Lévy process, Nonparametric Projection Estimator

Nonparametric adaptive estimation for pure jump Lévy

processes.

Introduction

In recent years, the use of Lévy processes for modelling purposes has become very popular in many areas and especially in the field of finance (see e.g. [START_REF] Eberlein | Hyperbolic distributions in finance[END_REF], Barndorff-Nielsen and Shephard (2001), [START_REF] Cont | Financial modelling with jump processes[END_REF]; see also [START_REF] Bertoin | Lévy processes[END_REF] or [START_REF] Sato | Lévy processes and infinitely divisible distributions[END_REF] for a comprehensive study for these processes). The distribution of a Lévy process is usually specified by its characteristic triple (drift, Gaussian component and Lévy measure) rather than by the distribution of its independent increments. Indeed, the exact distribution of these increments is most often intractable or even has no closed form formula. For this reason, the standard parametric approach by likelihood methods is a difficult task and many authors have rather considered nonparametric methods. For Lévy processes, estimating the Lévy measure is of crucial importance since this measure specifies the jumps behavior. Nonparametric estimation of the Lévy measure has been the subject of several recent contributions. The statistical approaches depend on the way observations are performed. For instance, [START_REF] Basawa | Nonparametric estimation for nondecreasing Lévy processes[END_REF] consider non decreasing Lévy processes and observations of jumps with size larger than some positive ε, or discrete observations with fixed sampling interval. They build nonparametric estimators of a distribution function linked with the Lévy measure. More recently, Figueroa-López and Houdré (2006) consider a continuous-time observation of a general Lévy process and study penalized projection estimators of the Lévy density based on integrals of functions with respect to the random Poisson measure associated with the jumps of the process. However, their approach remains theoretical since these Poisson integrals are hardly accessible.

In this paper, we consider nonparametric estimation of the Lévy measure for real-valued Lévy processes of pure jump type, i.e. without drift and Gaussian component. We rely on the common assumption that the Lévy measure admits a density n(x) on R and assume that the process is discretely observed with fixed sampling interval ∆. Let (L t ) denote the underlying Lévy process and (Z ∆ k = L k∆ -L (k-1)∆ , k = 1, . . . , n) be the observed random variables which are independent and identically distributed. Under our assumption, the characteristic function of L ∆ = Z ∆ 1 is given by the following simple formula:

(1)

ψ ∆ (u) = E(exp iuZ ∆ 1 ) = exp (∆ R (e iux -1)n(x)dx)
where the unknown function is the Lévy density n(x). It is therefore natural to investigate the nonparametric estimation of n(x) using empirical estimators of the charasteristic functions and its derivatives and then recover the Lévy density by Fourier inversion. This approach is illustrated by [START_REF] Watteel | Nonparametric estimation of the canonical measure for infinitely divisible distributions[END_REF] and Neumann and Reiss (2000). However, these authors consider general Lévy processes, with drift and Gaussian component. Hence, at least two derivatives of the characteristic function are necessary to reach the Lévy density. Moreover, the way Fourier inversion is done in concrete is not detailed in these papers. In our case, under the assumption that R |x|n(x)dx < ∞, we get the simple relation:

(2) g * (u) = e iux g(x)dx = -i ψ ′ ∆ (u) ∆ψ ∆ (u) , with g(x) = xn(x). This equation indicates that we can estimate g * (u) by using empirical counterparts of ψ ∆ (u) and ψ ′ ∆ (u) only. Then, the problem of recovering an estimator of g looks like a classical deconvolution problem. We have at hand the methods used for estimating unknown densities of random variables observed with additive independent noise. This requires the additional assumption that g belongs to L 2 (R). However, the problem of deconvolution set by equation ( 2) is not standard and looks more like deconvolution in presence of unknown errors densities. This is due to the fact that both the numerator and the denominator are unknown and have to be estimated from the same data. This is why our estimator of ψ ∆ (u) is not a simple empirical counterpart. Instead, we use a truncated version analogous to the one used in [START_REF] Neumann | On the effect of estimating the error density in nonparametric deconvolution[END_REF] and Neumann and Reiss (2000).

Below, we show how to adapt the deconvolution method described in [START_REF] Comte | Penalized contrast estimator for adaptive density deconvolution[END_REF]. We consider an adequate sequence (S m , m = 1, . . . , m n ) of subspaces of L 2 (R) and build a collection of projection estimators (ĝ m ). Then using a penalization device, we select through a data-driven procedure the best estimator in the collection. We study the L 2risk of the resulting estimator under the asymptotic framework that n tends to infinity. Although the sampling interval ∆ is fixed, we keep it as much as possible in all formulae since the distributions of the observed random variables highly depend on ∆.

In Section 2, we give assumptions and some preliminary properties. Section 3 contains examples of models included in our framework. Section 4 describes the statistical strategy. We present the projection spaces and define the collection of estimators. Proposition 4.1 gives the upper bound for the risk of a projection estimator on a fixed projection space. This proposition guides the choice of the penalty function and allows to discuss the rates of convergence of the projection estimators. Afterwards, we introduce a theoretical penalty (depending on the unknown characteristic function ψ ∆ ) and study the risk bound of a false estimator (actually not an estimator) (Theorem 4.1). Then, we replace the theoretical penalty by an estimated counterpart and give the upper bound of the risk of the resulting penalized estimator (Theorem 4.2). Section 6 gives some conclusions and open problems. Proofs are gathered in Section 6. In the Appendix, a fondamental result used in our proofs is recalled.

Framework and assumptions.

Recall that we consider the discrete time observation with sample step ∆ of a Lévy process L t with Lévy density n and characteristic function given by [START_REF] Barndorff-Nielsen | Modelling by Lévy processes for financial econometrics[END_REF]. We assume that (L t ) is a pure jump process with finite variation on compacts. When the Lévy measure n(x)dx is concentrated on (0, +∞), then (L t ) has increasing paths and is called a subordinator. We focus on the estimation of the real valued function

(3) g(x) = xn(x),
and introduce the following assumptions on the function g:

(H1) R |x|n(x)dx < ∞. (H2(p)) For p integer, R |x| p-1 |g(x)|dx < ∞. (H3)
The function g belongs to L 2 (R). Note that (H1) is stronger than the usual assumption (|x| ∧ 1)n(x)dx < +∞, and is also a moment assumption for L t . Under the usual assumption, (H2(p)) for p ≥ 1 implies (H1) and (H2(k)) for k ≤ p.

Our estimation procedure is based on the random variables (4)

Z ∆ i = L i∆ -L (i-1
)∆ , i = 1, . . . , n, which are independent, identically distributed, with common characteristic function ψ ∆ (u).

The moments of Z ∆ 1 are linked with the function g. More precisely, we have:

Proposition 2.1. Let p ≥ 1 integer. Under (H2)(p), E|Z ∆ 1 | p < ∞. Moreover, setting, for k = 1, . . . p, M k = R x k-1 g(x)dx, we have E(Z ∆ 1 ) = ∆M 1 , E[(Z ∆ 1 ) 2 ] = ∆M 2 + ∆ 2 M 1
, and more generally, E[(Z ∆ 1 ) l ] = ∆ M l + o(∆) for all l = 1, . . . , p. Proof. By the assumption, the exponent of the exponential in (1) is p times differentiable and, by derivating ψ ∆ , we get the result. 2 Assumption (H1) yields the relation [START_REF] Barron | Risk bounds for model selection via penalization[END_REF], which is the basis of our estimation procedure. We need a precise control of ψ ∆ . For this, we introduce the assumption that, for m n an integer to be defined later, the following holds:

(H4) ∀x ∈ R, we have c ψ (1 + x 2 ) -∆β/2 ≤ |ψ ∆ (x)| ≤ C ψ (1 + x 2 ) -∆β/2
, for some given constants c ψ , C ψ and β ≥ 0. Note that an assumption of this type is also considered in [START_REF] Neumann | Nonparametric estimation for Lévy processes from low-frequency observations[END_REF].

For the adaptive version of our estimator, we need additional assumptions for g:

(H5) There exists some positive a such that |g * (x)| 2 (1 + x 2 ) a dx < +∞, and (H6)

x 2 g 2 (x)dx < +∞.

We must set independent assumptions for ψ ∆ and g, since there may be no relation at all between these two functions (see the examples). Note that, in Assumption (H5), which is a classical regularity assumption, the knowledge of a is not required.

P Z ∆ 1 (dz) = e -c∆ (δ 0 (dz) + n≥1 f * n (z) (c∆) n n! dz).
We have the following bound:

(7) 1 ≥ |ψ ∆ (u)| ≥ e -2c∆ .
On this example, it appears clearly that we can not link the regularity assumption on g and (H4) which holds with β = 0.

3.2. The Lévy gamma process. Let α > 0, β > 0. The Lévy gamma process (L t ) with parameters (β, α) is a subordinator such that, for all t > 0, L t has distribution Gamma with parameters (βt, α), i.e. has density:

(8) α βt Γ(βt) x βt-1 e -αx 1 x≥0 .
The characteristic function of Z ∆ 1 is equal to:

(9) ψ ∆ (u) = α α -iu β∆ .
The Lévy density is n(x) = βx -1 e -αx 1 I {x>0} so that g(x) = βe -αx 1 I {x>0} satisfies our assumptions. We have: [START_REF] Diggle | A Fourier approach to nonparametric deconvolution of a density estimate[END_REF] ψ n(x)dx = +∞ and g(x) = xn(x) belongs to L 2 (R)∩L 1 (R). The case δ = 0, which corresponds to the Lévy inverse Gaussian process does not fit in our framework. For 0 < δ < 1/2, we find

′ ∆ (u) ψ ∆ (u) = i∆ β α -iu , |ψ ∆ (u)| = α β∆ (α 2 + u 2 ) β∆/2
g * (x) = c Γ(δ + 1/2) (β -ix) δ+1/2 , and |ψ ∆ (x)| = exp -c ∆Γ(δ + 1/2) 1/2 -δ [(β 2 + x 2 ) -(δ-1/2)/2 -β -(δ-1/2) ] .
It is important to mention that ψ ∆ above does not satisfy assumption (H4) since

(11) |ψ ∆ (x)| ∼ x→+∞ K(β, δ) exp(-c∆ Γ(δ + 1/2) 1/2 -δ x -δ+1/2 )
where

K(β, δ) = exp c ∆Γ(δ+1/2) 1/2-δ β -(δ-1/2
) . Thus, it has an exponential rate of decrease.

3.4.

The bilateral Gamma process. This process has been recently introduced by Küchler and Tappe (2008). Consider X, Y two independent random variables, X with distribution Γ(β, α) and Y with distribution Γ(β ′ , α ′ ). Then, Z = X -Y has distribution bilateral gamma with parameters (β, α, β ′ , α ′ ), that we denote by Γ(β, α; β ′ , α ′ ). The characteristic function of Z is equal to:

(12) ψ(u) = α α -iu β α ′ α ′ + iu β ′ = exp ( R (e iux -1)n(x)dx), with n(x) = x -1 g(x), and, for x ∈ R, g(x) = βe -αx 1 (0,+∞)(x) -β ′ e -α ′ |x| 1 (-∞,0) (x).
The bilateral Gamma process (L t ) has characteristic function ψ t (u) = ψ(u) t . The method can be generalized and we may consider Lévy processes on R obtained by bilateralisation of two subordinators.

3.5. Subordinated Processes. Let (W t ) be a Brownian motion, and let (Z t ) be an increasing Lévy process (subordinator), independent of (W t ). Assume that the observed process is

L t = W Zt . We have ψ ∆ (u) = E(e iuL ∆ ) = E(e -u 2 2 Z ∆ ). As Z t is positive, we consider, for λ ≥ 0, ϑ ∆ (λ) = E(e -λZ ∆ ) = exp -∆ +∞ 0 (1 -e -λx )n Z (x)dx ,
where n Z denotes the Lévy density of (Z t ). Now let us assume that g Z (x) = xn Z (x) is integrable over (0, +∞). We have:

log(ϑ ∆ (λ)) = -∆ +∞ 0 1 -e -λx x xn Z (x)dx = -∆ +∞ 0 ( λ 0 e -sx ds)xn Z (x)dx = -∆ λ 0 +∞ 0 e -sx xn Z (x)dx ds.
Hence,

ψ ∆ (u) = exp -∆ u 2 /2 0 +∞ 0 e -sx g Z (x)dx ds .
Moreover, it is possible to relate the Lévy density n L of (L t ) with the Lévy density n Z of (Z t ) as follows. Consider f a non negative function on R, with f (0) = 0. Given the whole path (Z t ), the jumps δL s = W Zs -W Z s-are centered Gaussian with variance δZ s . Hence,

E( s≤t f (δL s )) = s≤t E( R f (u) exp (-u 2 /2δZ s ) du √ 2πδZ s ) = t R f (u)du +∞ 0 exp (-u 2 /2x) n Z (x)dx √ 2πx ) . This gives n L (u) = +∞ 0 exp (-u 2 /2x) n Z (x)dx √ 2πx
. By the same tools, we see that

E( s≤t |δL s |) = 2/πE( s≤t δZ s ) = t +∞ 0 √ xn Z (x)dx.
Therefore, if the above integral is finite, the process (L t ) has finite variation on compact sets and it holds that R |u|n L (u)du < ∞.

With (Z t ) a Lévy-Gamma process, g Z (x) = βe -αx 1 I x>0 . Then

+∞ 0

e -sx βe -αx dx = β/(α + s), and

ψ ∆ (u) = α α + u 2 2 ∆β .
This model is the Variance Gamma stochastic volatility model described by [START_REF] Madan | The Variance Gamma (V.G.) Model for Share Market Returns[END_REF]. As noted in [START_REF] Küchler | Bilateral Gamma distributions and processes in financial mathematics[END_REF], the Variance Gamma distributions are special cases of bilateral Gamma distributions. The condition

+∞ 0 √ xn Z (x)dx < ∞ holds.
We can compute, for instance using the norming constant for an inverse Gaussian density,

n L (u) = +∞ 0 exp (- 1 2 ( u 2 x + 2αx) βx -3/2 dx √ 2π = β(2α) 1/4 )|u| -1 exp (-(2α) 1/2 |u|) 4. Statistical strategy 4.1.
Notations. Subsequently we denote by u * the Fourier transform of the function u defined as u * (y) = e iyx u(x)dx, and by u , < u, v >, u * v the quantities

u 2 = |u(x)| 2 dx, < u, v >= u(x)v(x)dx with zz = |z| 2 and u ⋆ v(x) = u(y)v(x -y)dy.
Moreover, we recall that for any integrable and square-integrable functions u, u 1 , u 2 , (

) (u * ) * (x) = 2πu(-x) and u 1 , u 2 = (2π) -1 u * 1 , u * 2 . 4.2. 13
The projection spaces. As we use projection estimators, we describe now the projection spaces. Let us define

ϕ(x) = sin(πx) πx and ϕ m,j (x) = √ mϕ(mx -j),
where m is an integer, that can be taken equal to 2 ℓ . It is well known (see [START_REF] Meyer | Ondelettes et oprateurs[END_REF], p.22) that {ϕ m,j } j∈Z is an orthonormal basis of the space of square integrable functions having Fourier transforms with compact support included into [-πm, πm]. Indeed an elementary computation yields

(14) ϕ * m,j (x) = e ixj/m √ m 1 I [-πm,πm] (x).
We denote by S m such a space:

S m = Span{ϕ m,j , j ∈ Z} = {h ∈ L 2 (R), supp(h * ) ⊂ [-mπ, mπ]}.
We denote by (S m ) m∈Mn the collection of linear spaces, where

M n = {1, . . . , m n }
and m n ≤ n is the maximal admissible value of m, subject to constraints to be precised later.

In practice, we should consider the truncated spaces S

(n) m = Span{ϕ m,j , j ∈ Z, |j| ≤ K n }, where K n is an integer depending on n, and the associated estimators. Under assumption (H6), it is possible and does not change the main part of the study (see [START_REF] Comte | Penalized contrast estimator for adaptive density deconvolution[END_REF]). For the sake of simplicity, we consider here sums over Z.

4.3.

Estimation strategy. We want to estimate g such that (15)

g * (x) = -i ψ ′ ∆ (x) ∆ψ ∆ (x) = θ ∆ (x) ∆ψ ∆ (x) , with ψ ∆ (x) = E(e ixZ ∆ 1 ), θ ∆ (x) = -iψ ′ ∆ (x) = E(Z ∆ 1 e ixZ ∆ 1
). The orthogonal projection g m of g on S m is given by ( 16)

g m = j∈Z a m,j (g)ϕ m,j with a m,j (g) = R ϕ m,j (x)g(x)dx = ϕ m,j , g .
We have at hand the empirical versions of ψ ∆ and θ ∆ :

ψ∆ (x) = 1 n n k=1 e ixZ ∆ k , θ∆ (x) = 1 n n k=1 Z ∆ k e ixZ ∆ k .
Following [START_REF] Neumann | On the effect of estimating the error density in nonparametric deconvolution[END_REF] and [START_REF] Neumann | Nonparametric estimation for Lévy processes from low-frequency observations[END_REF], we truncate 1/ ψ∆ and set

(17) 1 ψ∆ (x) = 1 ψ∆ (x) 1 I | ψ∆ (x)|>κ ψ n -1/2 .
Now, for t belonging to a space S m of the collection (S m ) m∈Mn , let us define

(18) γ n (t) = 1 n n k=1 t 2 - 1 π∆ Z ∆ k e ixZ ∆ k t * (-x) ψ∆ (x) dx ,
Consider γ n (t) as an approximation of the theoretical contrast

γ th n (t) = 1 n n k=1 t 2 - 1 π∆ Z ∆ k e ixZ ∆ k t * (-x) ψ ∆ (x) dx ,
The following sequence of equalities, relying on [START_REF] Figueroa-López | Risk bounds for the nonparametric estimation of Lévy processes[END_REF], explains the choice of the contrast:

E( 1 2π∆ Z ∆ k e ixZ ∆ k t * (-x) ψ ∆ (x) dx) = 1 2π∆ θ ∆ (x) t * (-x) ψ ∆ (x) dx = 1 2π t * , g * = t, g .
Therefore, we find that E(γ th n (t)) = t 2 -2 g, t = t -g 2 -g 2 is minimal when t = g. Thus, we define the estimator belonging to S m by [START_REF] Neumann | On the effect of estimating the error density in nonparametric deconvolution[END_REF] ĝm = Argmin t∈Sm γ n (t)

This estimator can also be written

(20) ĝm = j∈Z âm,j ϕ m,j , with âm,j = 1 2πn∆ n k=1 Z ∆ k e ixZ ∆ k ϕ * m,j (-x) ψ∆ (x) dx, or âm,j = 1 2π∆ θ∆ (x) ϕ * m,j (-x) ψ∆ (x) dx.
E 1 ψ∆ (x) - 1 ψ ∆ (x) 2p ≤ C 1 |ψ ∆ (x)| 2p ∧ n -p |ψ ∆ (x)| 4p ,
where 1/ ψ∆ is defined by (17).

Neumann's result is for p = 1 but the extension to any p is straighforward. See also [START_REF] Neumann | Nonparametric estimation for Lévy processes from low-frequency observations[END_REF]. This lemma allows to prove the following risk bound.

21) E( g -ĝm 2 ) ≤ g -g m 2 + K E 1/2 [(Z ∆ 1 ) 4 ] πm -πm dx/|ψ ∆ (x)| 2 n∆ 2 .
where K is a constant.

It is worth stressing that (H4) is not required for the above result. Therefore, it holds even for exponential decay of ψ ∆ .

Proof of Proposition 4.1. First with Pythagoras Theorem, we have

(22) g -ĝm 2 = g -g m 2 + ĝm -g m 2 .
Let

a m,j (g) = 1 2π∆ θ ∆ (x) ϕ * m,j (-x) ψ ∆ (x) dx.
Then, using Parseval's formula and ( 14), we obtain

ĝm -g m 2 = j∈Z |â m,j -a m,j (g)| 2 = 1 2π∆ 2 πm -πm θ∆ (x) ψ∆ (x) - θ ∆ (x) ψ ∆ (x) 2 dx.
It follows that

E( ĝm -g m 2 ) ≤ c ∆ 2 πm -πm E θ∆ (x) 1 ψ∆ (x) - 1 ψ ∆ (x) 2 dx + πm -πm E| θ∆ (x) -θ ∆ (x)| 2 |ψ ∆ (x)| 2 dx ≤ c ∆ 2 πm -πm E(| θ∆ (x) -θ ∆ (x)| 2 1 ψ∆ (x) - 1 ψ ∆ (x) 2 dx (23) + πm -πm ∆ 2 |g * (x)ψ ∆ (x)| 2 E 1 ψ∆ (x) - 1 ψ ∆ (x) 2 + 1 n E[(Z ∆ 1 ) 2 ] |ψ ∆ (x)| 2 dx
The Schwarz Inequality yields

E | θ∆ (x) -θ ∆ (x)| 2 1 ψ∆ (x) - 1 ψ ∆ (x) 2 ≤ E 1/2 (| θ∆ (x)-θ ∆ (x)| 4 )E 1/2 1 ψ∆ (x) - 1 ψ ∆ (x) 4 .
Then, with the Rosenthal inequality E(| θ∆ (x) -θ ∆ (x)| 4 ) ≤ cE[(Z ∆ 1 ) 4 ]/n 2 and by using Lemma 4.1,

E 1 ψ∆ (x) - 1 ψ ∆ (x) 4 ≤ C |ψ ∆ (x)| 4 so that πm -πm E 1/2 (| θ∆ (x) -θ ∆ (x)| 4 )E 1/2 1 ψ∆ (x) - 1 ψ ∆ (x) 4 dx ≤ cE 1/2 [(Z ∆ 1 ) 4 ] n πm -πm dx |ψ ∆ (x)| 2 .
For the second term, we use Lemma 4.1, to get

E 1 ψ∆ (x) - 1 ψ ∆ (x) 2 ≤ Cn -1 |ψ ∆ (x)| 4 .
We obtain

(24) E( ĝm -g m 2 ) ≤ c n∆ 2 (E 1/2 [(Z ∆ 1 ) 4 ] + ∆ 2 g 2 1 + E[(Z ∆ 1 ) 2 ]) πm -πm dx |ψ ∆ (x)| 2 dx,
where g 1 = |g(x)|dx. Therefore, gathering [START_REF] Watteel | Nonparametric estimation of the canonical measure for infinitely divisible distributions[END_REF] and (24) implies the result. 2 Remark 4.1. In papers concerned with deconvolution in presence of unknown error densities, the error characteristic function is estimated using a preliminary and independent set of data. This solution is possible here: we may split the sample and use the first half to obtain a preliminary and independent estimator of ψ ∆ , and then estimate g from the second half. This would simplify the above proof, but not the study of the adaptive case. Then, the bias term satisfies

g -g m 2 = O(m -2a ).
Under (H4), the bound of the variance term satisfies

πm -πm dx/|ψ ∆ (x)| 2 n∆ = O m 2β∆+1 n∆ .
The optimal choice for m is O((n∆) 1/(2β∆+2a+1 ) and the resulting rate for the risk is (n∆) -2a/(2β∆+2a+1 ). It is worth noting that the sampling interval ∆ explicitely appears in the exponent of the rate. Therefore, for positive β, the rate is worse for large ∆ that for small ∆.

• Let us consider the example of the compound process. In this case β = 0, the upper bound of the mean integrated squared error is of order O((n∆) -2a/(2a+1) ), if g belongs to the Sobolev class S(a, L). Note that if g is analytic i.e. belongs to a class

A(γ, Q) = {f, (e γx + e -γx ) 2 |f * (x)| 2 dx ≤ Q},
then the risk is of order O(ln(n∆)/(n∆)) (choose m = O(ln(n∆))).

• For the Levy Gamma process, we have a more precise result since we have

|ψ ∆ (u)| = α β∆ (α 2 + u 2 ) β∆/2 , g * (x) = β α -ix . Therefore |x|≥πm |g * (x)| 2 dx = O(m -1 ) and [-πm,πm] dx/|ψ ∆ (x)| 2 = O(m 2β∆+1
). The resulting rate is of order (n∆) -1/(2β∆+2) for a choice of m of order O((n∆) 1/(2β∆+2) ).

• For the Bilateral Gamma process with (β, α) = (β ′ , α ′ ), we have

ψ ∆ (u) = α β∆ (α 2 + u 2 ) β∆ , g * (x) = β α 2 + x 2 . Therefore |x|≥πm |g * (x)| 2 dx = O(m -3 ) and [-πm,πm] dx/|ψ ∆ (x)| 2 = O(m 4β∆+1 ).
The resulting rate is of order (n∆) -3/(4β∆+4) for a choice of m of order O((n∆) 1/(4β∆+4) ).

These examples illustrate that the relevant choice of m depends on the unknown function, in particular on its smoothness. The model selection procedure proposes a data driven criterion to select m.

• Consider now the process described in Section 3.3. In that case, it follows from (11) that

[-πm,πm] dx/|ψ ∆ (x)| 2 = O(m δ+1/2 exp(κm 1/2-δ )) and |x|≥πm |g * (x)| 2 dx = O(m -2δ
). In this case, choosing κm 1/2-δ = ln(n∆)/2 gives the rate [ln(n∆)] -2δ which is thus very slow, but known to be optimal in the usual deconvolution setting (see Fan (1991)). This case is not considered in the following for the adaptative strategy since it does not satisfy (H4). 4.6. Study of the adaptive estimator. We have to select an adequate value of m. For this, we start by defining the term

(25) Φ ψ (m) = πm -πm dx |ψ ∆ (x)| 2 ,
and the following theoretical penalty

(26) pen(m) = κ(1 + E[(Z ∆ 1 ) 2 ]/∆) Φ ψ (m) n∆ .
We set m = arg min

m∈Mn {γ n (ĝ m ) + pen(m)} ,
and study first the "risk" of ĝ m. Moreover we need the following assumption on the collection of models M n = {1, . . . , m n }, m n ≤ n:

(H7) ∃ε, 0 < ε < 1, m 2β∆ n ≤ Cn 1-ε ,
where C is a fixed constant and β is defined by (H4). For instance, Assumption (H7) is fulfilled if:

(1) pen(m n ) ≤ C. In such a case, we have m n ≤ C(n∆) 1/(2β∆+1) .

(2) ∆ is small enough to ensure 2β∆ < 1. In such a case we can take M n = {1, . . . , n}.

Remark 4.2. Assumption (H7) raises a problem since it depends on the unknown β and concrete implementation requires the knowledge of m n . It is worth stressing that the analogous difficulty arises in deconvolution with unknown error density (see [START_REF] Comte | Deconvolution with estimated error[END_REF]). In the compound Poisson model, β = 0 and nothing is needed. Otherwise one should at least know if ψ ∆ is in a class of polynomial decay. The estimator ψ∆ may be used to that purpose and to provide an estimator of β (see e.g. [START_REF] Diggle | A Fourier approach to nonparametric deconvolution of a density estimate[END_REF]).

Let us define

θ (1) ∆ (x) = E(Z ∆ 1 1 I |Z ∆ 1 |≤kn √ ∆ e ixZ ∆ 1 ), θ (2)
∆ (x) = E(Z ∆ 1 1 I |Z ∆ 1 |>kn √ ∆ e ixZ ∆ 1 ) so that θ ∆ = θ (1) ∆ + θ (2)
∆ and analogously θ∆ = θ(1) ∆ + θ(2) ∆ . For any two functions t, s in S m , the contrast γ n satisfies:

γ n (t) -γ n (s) = t -g 2 -s -g 2 -2ν (1) n (t -s) -2ν (2) n (t -s) -2 4 i=1 R (i) n (t -s), (27)
with

ν (1) n (t) = 1 2π∆ t * (-x) θ(1) ∆ (x) -θ (1) ∆ (x) ψ ∆ (x) dx, ν (2) n (t) = 1 2π∆ t * (-x) θ ∆ (x) [ψ ∆ (x)] 2 (ψ ∆ (x) -ψ∆ (x))dx, R (1) n (t) = 1 2π∆ t * (-x)( θ∆ (x) -θ ∆ (x)) 1 ψ∆ (x) - 1 ψ ∆ (x) dx R (2) n (t) = 1 2π∆ t * (-x) θ ∆ (x) ψ ∆ (x) (ψ ∆ (x) -ψ∆ (x)) 1 ψ∆ (x) - 1 ψ ∆ (x) dx, R (3)
n (t) = 1 2π∆ t * (-x) θ(2) ∆ (x) -θ (2) ∆ (x) ψ ∆ (x) dx, R (4) n (t) = - 1 2π∆ t * (-x) θ ∆ (x) ψ ∆ (x) 1 I | ψ∆ (x)|≤κ ψ / √ n dx.
Using this decomposition and Talagrand's inequality, we can prove Theorem 4.1. Assume that assumptions (H1)-(H2)( 8)-(H3)-(H7) hold. Then

E( ĝ m -g 2 ) ≤ C inf m∈Mn g -g m 2 + pen(m) + K ln 2 (n) n∆ ,
where K is a constant.

Remark 4.3. Assumption (H6) is satisfied for the Levy-Gamma process. For the compound Poisson process, it is equivalent to x 4 f 2 (x)dx < +∞, where f denotes the density of Y i (see Section 3).

To get an estimator, we replace the theoretical penalty by:

pen(m) = κ ′ 1 + 1 n∆ 2 n i=1 (Z ∆ i ) 2 πm -πm dx/| ψ∆ (x)| 2 dx n .
In that case we can prove:

Theorem 4.2. Assume that assumptions (H1)-(H2)( 8)-(H3)-(H7) hold and let g = ĝ m be the estimator defined with m = arg min m∈Mn (γ n (ĝ m ) + pen(m)). Then

E( g -g 2 ) ≤ C inf m∈Mn g -g m 2 + pen(m) + K ′ ∆ ln 2 (n) n
where K ′ ∆ is a constant depending on ∆ (and on fixed quantities but not on n). Theorem 4.2 shows that the adaptive estimator automatically achieves the best rate that can be hoped. If g belongs to the Sobolev ball S(a, L), and under (H4), the rate is automatically of order O((n∆) -2a/(2β∆+2a+1) ). See Section 4.5.

Remark 4.4.

(1) It is possible to extend our study of the adaptive estimator to the case ψ ∆ having exponential decay. Note that the faster |ψ ∆ | decays, the more difficult it will be to estimate g.

(2) Few results on rates of convergence are available in the literature for this problem.

The results of [START_REF] Neumann | Nonparametric estimation for Lévy processes from low-frequency observations[END_REF] are difficult to compare with ours since the point of view is different.

ĝ m -g 2 ≤ g m -g 2 + 2ν (1) n (ĝ m -g m ) + 2ν (2) n (ĝ m -g m ) + 2 4 i=1 R (i) n (ĝ m -g m ) +pen(m) -pen( m).
Let us take expectations of both sides and bound each r.h.s. term.

|E(ν (1) n (ĝ m -g m ))| ≤ 1 16 E( g m -ĝ m 2 ) + 16E sup t∈Sm+S m, t =1 |ν (1) n (t)| 2 ≤ 1 8 E( g -ĝ m 2 ) + 1 8 g -g m 2 +16E sup t∈S m∨ m, t =1 |ν (1) n (t)| 2 -p 1 (m, m) + + 16E(p 1 (m, m)).
The same kind of bounds are obtained for ν

n and the residuals leading to

2 8 E( ĝ m -g 2 ) ≤ 14 8 g -g m 2 + 16 m ′ ∈Mn E sup t∈S m∨m ′ , t =1 |ν (1) n (t)| 2 -p 1 (m, m ′ ) + +16E sup t∈S m∨ m, t =1 |ν (2) n (t)| 2 -p 2 (m, m) + +16 2 i=1 E sup t∈S m∨ m, t =1 |R (i) n (t)| 2 -p 1 (m, m) +16 4 i=3 E sup t∈Sm n , t =1 |R (i) n (t)| 2 +pen(m) + E(48p 1 (m, m) + 16p 2 (m, m) -pen( m)). (28)
Next, definition of pen(.) comes from the following constraint:

(29) 48p 1 (m, m ′ ) + 16p 2 (m, m ′ ) ≤ pen(m ′ ) + pen(m).
This leads to

pen(m) + E(48p 1 (m, m) + 16p 2 (m, m) -pen( m)) ≤ 2pen(m).
First, we apply Talagrand's Inequality recalled in Lemma 6.1 to prove the following result:

Proposition 5.1. Under the assumptions of Theorem 4.1, define

p 1 (m, m ′ ) = (4E[(Z ∆ 1 ) 2 ] π(m∨m ′ ) -π(m∨m ′ ) |ψ ∆ (x)| -2 dx)/(πn∆ 2 ), then (30)
m ′ ∈Mn E sup t∈S m∨m ′ , t =1 |ν (1) n (t)| 2 -p 1 (m, m ′ ) + ≤ c n .
Next we prove:

Proposition 5.2. Under the assumptions of Theorem 4.1, define

p 2 (m, m ′ ) = 0 if -a + β∆ ≤ 0 and p 2 (m, m ′ ) = ( π(m∨m ′ ) -π(m∨m ′ ) |ψ ∆ (x)| -2 dx)/n otherwise. Then (31) E sup t∈S m∨ m , t =1 |ν (2) n (t)| 2 -p 2 (m, m) + ≤ c n .
For the residual terms, two type of results can be obtained.

Proposition 5.3. Under the assumptions of Theorem 4.1, for i = 1, 2,

E sup t∈S m∨ m, t =1 [R (i) n (t)] 2 -p 1 (m, m) ≤ C n∆ .
and Proposition 5.4. Under the assumptions of theorem 4.1, for i = 3, 4

E sup t∈Sm n , t =1 [R (i) n (t)] 2 ≤ c ln 2 (n) n∆ .
Then the choice pen(m) given by (26) gives, following (28) and (29),

1 4 E( ĝ m -g 2 ) ≤ 7 4 g -g m 2 + 2pen(m) + C ln 2 (n) n∆ ,
which is the result. 2

5.2. Proof of Proposition 5.1. Let ω t (z) = z1 I {|z|≤kn √ ∆} 2π∆ e izx t * (-x) ψ ∆ (x) dx
and notice that

ν (1) n (t) = 1 n n k=1 ω t (Z ∆ k ) -E(ω t (Z ∆ k )) .
To apply Lemma 6.1, we compute M 1 , H 1 and v 1 defined therein. First, we have

E sup t∈Sm, t =1 |ν (1) n (t)| 2 ≤ E   j∈Z |ν (1) n (ϕ m,j )| 2   = E   1 2π∆ 2 πm -πm θ(1) ∆ (x) -θ (1)
∆ (x) ψ ∆ (x) 2 dx   ≤ E[(Z ∆ 1 ) 2 ] 2πn∆ 2 Φ ψ (m),
where Φ ψ (m) is defined in (25). We can take, for m ⋆ = m ∨ m ′ ,

H 2 1 = E[(Z ∆ 1 ) 2 ] 2πn∆ 2 Φ ψ (m ⋆ ).
Then it is easy to see that if t = 1 and t ∈ S m ⋆ , then

|ω t (z)| ≤ k n 2π √ ∆ t * (-x) ψ ∆ (x) dx ≤ k n 2π √ ∆ Φ ψ (m ⋆ ) := M 1 .
Lastly, for t ∈ S m , t = 1, t = j∈Z t m,j ϕ m,j

Var(ω t (Z ∆ 1 )) ≤ 1 (2π) 2 ∆ 2 E e i(u-v)Z ∆ 1 (Z ∆ 1 1 I |Z ∆ 1 |≤kn √ ∆ ) 2 t * (-u)t * (v) ψ ∆ (u)ψ ∆ (-v) dudv = 1 (2π∆) 2 j,k t m,j t m,k E e i(u-v)Z ∆ 1 (Z ∆ 1 1 I |Z ∆ 1 |≤kn √ ∆ ) 2 ϕ * m,j (-u)ϕ * m,k (v) ψ ∆ (u)ψ ∆ (-v) dudv.
Denoting by

(32) h * ∆ (u) = E[e iuZ ∆ 1 (Z ∆ 1 1 I |Z ∆ 1 |≤kn √ ∆ ) 2 ]
, we obtain:

Var(ω t (Z ∆ 1 )) ≤ ≤ 1 (2π∆) 2   j,k h * ∆ (u -v) ϕ * m,j (-u)ϕ * m,k (v) ψ ∆ (u)ψ ∆ (-v) dudv 2   1/2 = 1 2π∆ 2 [-πm,πm] 2 h * ∆ (u -v) ψ ∆ (u)ψ ∆ (-v) 2 dudv 1/2
where the last equality follows from the Parseval equality. Next with the Schwarz inequality and the Fubini theorem, we obtain

Var(ω t (Z ∆ 1 )) ≤ 1 2π∆ 2 [-πm,πm] 2 |h * ∆ (u -v)| 2 |ψ ∆ (u)| 4 dudv 1/2 = 1 2π∆ 2 [-πm,πm] du ψ ∆ (u) 4 du |h * ∆ (z)| 2 dz 1/2 ≤ πm -πm dx/|ψ ∆ (x)| 4 2π∆ h * ∆ ∆ .
Now we use the following Lemma:

Lemma 5.1. Under the assumptions of Theorem 4.1,

h * ∆ /∆ ≤ 2 √ π x 2 g 2 (x)dx + E[(Z ∆ 1 ) 2 ] g 2 1/2 := ξ.
Thus, under (H5), ξ is finite. We set

v 1 = ξ πm ⋆ -πm ⋆ dx/|ψ ∆ (x)| 4 2π∆ .
Therefore, setting ǫ 2 = 1/2,

p 1 (m, m ′ ) = 4E[(Z ∆ 1 ) 2 /∆] Φ ψ (m ⋆ ) 2πn∆ (= 2(1 + 2ǫ 2 )H 2 1 ).
Using (H4) and the fact that E[(Z ∆ 1 ) 2 /∆] is bounded, we find

E sup t∈S m ⋆ , t =1 |ν (1) n (t)| 2 -p 1 (m, m ′ ) + ≤ C (m ⋆ ) 2β∆+1/2 n∆ e -K √ m ⋆ + k 2 n Φ ψ (m ⋆ ) n 2 ∆ e -K ′ √ n/kn .
Here K = K(c ψ , C ψ ). Moreover, we take (33)

k n = K ′ √ n/((2β∆ + 3) ln(n))
and we obtain

m ′ ∈Mn E sup t∈S m ⋆ , t =1 [ν (1) n (t)] 2 -p 1 (m, m ′ ) + ≤ K" n∆ . 5.3. Proof of Proposition 5.2. The study of ν (2)
n is slightly different.

E sup t∈Sm, t =1 |ν (2) n (t)| 2 ≤ 1 2πn∆ 2 πm -πm |θ ∆ (x)| 2 |ψ ∆ (x)| 4 dx = 1 2πn πm -πm |g * (x)| 2 |ψ ∆ (x)| 2 dx.
With assumptions (H4) and (H5), we can see that if -a + β∆ ≤ 0, then

πm -πm |g * (x)| 2 |ψ ∆ (x)| 2 dx ≤ πm -πm |g * (x)| 2 (1+x 2 ) a (1 + x 2 ) -a+β∆ c 2 ψ dx ≤ 1 c 2 ψ |g * (x)| 2 (1+x 2 ) a dx ≤ L c 2 ψ .
In that case, we simply take p 2 (m, m ′ ) = 0 and write

E sup t∈S m∨ m, t =1 [ν (2) n ] 2 (t) ≤ E sup t∈Sm n , t =1 [ν (2) n ] 2 (t) ≤ L nc 2 ψ .
Now we study the case -a + β∆ > 0 and find the constants

H = H 2 , v = v 2 , ǫ = ǫ 2 to apply Lemma 6.1. Consider ωt (z) = (1/2π∆) e izu t * (-u){θ ∆ (u)/[ψ ∆ (u)] 2 }du. As πm -πm |g * (x)| 2 |ψ ∆ (x)| 2 dx ≤ L c 2 ψ m -2a+2β∆ ,
we take

H 2 2 = L 2πc 2 ψ (m ⋆ ) -2a+2β∆
n .

Next, we have

M 2 = √ nH 2
and we use the rough bound v 2 = nH 2 2 . Moreover, we take ǫ 2 2 = (-2a+2β∆+2) ln(m ⋆ )/K 1 . There exists m 0 , such that for m ⋆ ≥ m 0 ,

2(1 + 2ǫ 2 2 )H 2 2 ≤ Φ ψ (m ⋆ )/n. We set p 2 (m, m ′ ) = Φ ψ (m ⋆ )/n. Introducing W n (m, m ′ ) = sup t∈S m∨m ′ , t =1 |ν (2) n | 2 (t) -p 2 (m, m ′ ) + ,
we find that

m ′ ∈Mn E(W n (m, m ′ )) = m ′ |m ⋆ ≤m 0 E(W n (m, m ′ )) + m ′ |m ⋆ >m 0 E(W n (m, m ′ )) ≤ m ′ |m ⋆ ≤m 0 [E( sup t∈S m ⋆ , t =1 |ν (2) n (t)| 2 -2(1 + 2ǫ 2 2 )H 2 2 ] + ) + m ′ |m ⋆ ≤m 0 |p 2 (m, m ′ ) -2(1 + 2ǫ 2 2 )H 2 2 | + m ′ |m ⋆ >m 0 E([ sup t∈S m ⋆ , t =1 |ν (2) n (t)| 2 -2(1 + 2ǫ 2 2 )H 2 2 ] + ).
Therefore

m ′ ∈Mn E(W n (m, m ′ )) ≤ 2 m ′ ∈Mn E([ sup t∈S m ⋆ , t =1 |ν (2) n (t)| 2 -2(1 + 2ǫ 2 2 )H 2 2 ] + ) + m ′ |m ⋆ ≤m 0 |p 2 (m, m ′ ) -2(1 + 2ǫ 2 2 )H 2 2 | ≤ 2 m ′ ∈Mn E([ sup t∈S m ⋆ , t =1 |ν (2) n (t)| 2 -2(1 + 2ǫ 2 2 )H 2 2 ] + ) + C(m 0 ) n .
Talagrand's Inequality again can be then applied and gives that

m ′ ∈Mn E([ sup t∈S m ⋆ , t =1 |ν (2) n (t)| 2 -2(1 + 2ǫ 2 2 )H 2 2 ] + ) ≤ C n .
The result for ν

n in this case follows then by saying as for ν

n that

E (W n (m, m)) ≤ m ′ ∈Mn E(W n (m, m ′ )). 5.4. Proof of Proposition 5.3. First define Ω(x) = Ω 1 (x) ∩ Ω 2 (x) with Ω 1 (x) = | θ∆ (x) -θ ∆ (x)| ≤ 8E 1/2 [(Z ∆ 1 ) 2 ](log 1/2 (n)n -1/2 , Ω 2 (x) = 1 ψ∆ (x) - 1 ψ ∆ (x) ≤ 1/(log 1/2 (n)n ω |ψ ∆ (x)| 2 ) .
Then split: R

n (t) = R (1,1) n (t) + R (1,2) n (t) where R (1,1) n (t) = 1 2π∆ t * (-x)( θ∆ -θ ∆ )(x) 1 ψ∆ (x) - 1 ψ ∆ (x) 1 I Ω(x) dx (1)
and R

(1,2) n (t) the integral on the complement of Ω(x).

E sup

t∈S m∨ m, t =1 |R (1) n (t)| 2 ≤ 2E sup t∈S m∨ m, t =1 |R (1,1) n (t)| 2 +2E sup t∈Sm n , t =1 |R (1,2) n (t)| 2 E sup t∈S m∨ m, t =1 |R (1,1) n (t)| 2 ≤ 1 2π∆ 2 E π(m∨ m) -π(m∨ m) | θ∆ (x) -θ ∆ (x)| 2 1 ψ∆ (x) - 1 ψ ∆ (x) 2 1 I Ω(x) dx ≤ 8(E[(Z ∆ 1 ) 2 ]/∆) 2πn∆ E π(m∨ m) -π(m∨ m) n -2ω dx |ψ ∆ (x)| 4 ≤ 4E[(Z ∆ 1 ) 2 ] π∆ E( Φ ψ (m ∨ m) n∆ ) ≤ E(p 1 (m, m)),
under the condition -2ω + (1 -ε) ≤ 0. Therefore we choose ω = (1 -ε)/2. Note that if β = 0 the decomposition is useless and the residual is straightforwardly negligible. On the other hand, Lemma (4.1) yields:

E 1/4 1 ψ∆ (x) - 1 ψ ∆ (x) 8 ≤ C ∆ n|ψ ∆ (x)| 4 .

Now, we find

E sup t∈Sm n , t =1 |R (1,2) n (t)| 2 ≤ 1 2π∆ 2 πmn -πmn P 1/2 (Ω(x) c )E 1/4 [( θ∆ (x) -θ ∆ (x)) 8 ]E 1/4 1 ψ∆ (x) - 1 ψ ∆ (x) 8 dx ≤ CE 1/4 [(Z ∆ 1 ) 8 ] 2πn 2 πmn -πmn P 1/2 (Ω(x) c ) |ψ ∆ (x)| 4 dx ≤ CE 1/4 [(Z ∆ 1 ) 8 ]n 2(1-ε)+1-b n 2 ≤ C ′ ∆ n if P(Ω(x) c ) ≤ n -2b and 2(1 -ε) -b ≤ 0.
We take b = 2(1 -ε). In fact,

P(Ω(x) c ) ≤ P(Ω 1 (x) c ) + P(Ω 2 (x) c ).
We use the Markov Inequality to bound P(Ω 2 (x) c ):

P(Ω 2 (x) c ) ≤ log p (n)n 2pω |ψ ∆ (x)| 4p E 1 ψ∆ (x) - 1 ψ ∆ (x) 2p ≤ log p (n)n 2pω-p .

The choice of p is thus constrained by 2pω

-p = -p(1 -2ω) < -4(1 -ε) that is p > 4(1 -ε)/ε, e.g. p = 5(1 -ε)/ε.
We use the decomposition of θ ∆ (x) = θ

∆ (x) + θ

∆ (x) with

k n √ ∆ = nE[(Z ∆ 1 ) 2 ] 8 log(n) .
We use the Bernstein Inequality to bound P(Ω 1 (x) c ). If X 1 , . . . , X n are i.i.d. variables with variance less than v 2 and such that |X i | ≤ c, then for S n = n i=1 X i , we have:

P(|S n -E(S n )| ≥ nǫ) ≤ 2 exp - nǫ 2 /2 v 2 + cǫ .
This yields

P(Ω 1 (x) c ) ≤ P | θ(1) ∆ (x) -θ (1) ∆ (x)| ≥ 4 E[(Z ∆ 1 ) 2 ] log(n)/n +P | θ(2) ∆ (x) -θ (2) ∆ (x)| ≥ 4 E[(Z ∆ 1 ) 2 ] log(n)/n ≤ n -16/3 + n 16E[(Z ∆ 1 ) 2 ] log(n) E(| θ(2) ∆ (x) -θ (2) ∆ (x)| 2 ) ≤ n -16/3 + E[(Z ∆ 1 ) 2 1 I |Z ∆ 1 |≥kn √ ∆ ] 16E[(Z ∆ 1 ) 2 ] log(n) ≤ n -16/3 + 8 4 E[(Z ∆ 1 ) 6 ] log 2 (n) 16E 3 [(Z ∆ 1 ) 2 ]n 2 ≤ n -16/3 + c n 2 ∆ 2 .
This gives the result of Proposition 5.3 for R

n . The study of R

n follows the same line and is omitted. E sup

t∈Sm n , t =1 |R (3) n (t)| 2 ≤ 1 4π 2 ∆ 2 E sup t∈Sm n , t =1 ( θ(2) ∆ (x) -θ (2)
∆ (x))

t * (-x) ψ ∆ (x) dx 2 ≤ 1 2π∆ 2 πmn -πmn E[| θ(2) ∆ (x) -θ (2) ∆ (x)| 2 ] dx |ψ ∆ (x)| 2 = 1 2π∆ 2 πmn -πmn Var(Z ∆ 1 1 I |Z ∆ 1 |≥kn √ ∆ ) n dx |ψ ∆ (x)| 2 ≤ E[(Z ∆ 1 ) 8 ]Φ ψ (m n ) 2πnk 6 n ∆ 4 ≤ KE[(Z ∆ 1 ) 8 ] ln 6 (n) n 2+ε ∆ 4 ,
using the choice of k n given by (33).

Next,

E sup t∈Sm n , t =1 |R (4) n (t)| 2 ≤ 1 2π∆ πmn -πmn |g * (x)| 2 P(| ψ∆ (x)| ≤ κ ψ / √ n)dx ≤ c n∆ . If |ψ ∆ (x)| ≥ 2κ ψ / √ n, then P(| ψ∆ (u)| ≤ κ ψ n -1/2 ) ≤ P(| ψ∆ (u) -ψ ∆ (u)| ≤ |ψ ∆ (u)| -κ ψ n -1/2 ) ≤ P(| ψ∆ (u) -ψ ∆ (u)| ≥ 1 2 |ψ ∆ (u)|) ≤ exp(-cn|ψ ∆ (u)| 2 )
for some c > 0, where the last inequality follows from Bernstein's Inequality. Now, it follows from (H4) that |ψ

∆ (u)| ≥ c ψ (1 + u 2 ) -∆β/2 . Therefore, for |u| ≤ πm n with m 2β∆ n ≤ Cn 1-ε by (H7), |ψ ∆ (u)| ≥ c ′ m -β∆ n ≥ 2κ ψ n -1/2 .
Moreover, with the previous remarks, exp (

-cn|ψ ∆ (u)| 2 ) ≤ exp(-cn ε ) and thus πmn -πmn |g * (x)| 2 P(| ψ∆ (x)| ≤ κ/ √ n)dx ≤ g * 2 exp(-cn ε ). Therefore E sup t∈Sm n , t =1 |R (4) n (t)| 2 ≤ c n∆ .
5.6. Proof of Lemma 5.1. Let us denote by P ∆ the distribution of Z ∆ 1 and define µ ∆ (dz) = ∆ -1 zP ∆ (dz). Let us set µ(dx) = g(x)dx. Equation [START_REF] Küchler | Bilateral Gamma distributions and processes in financial mathematics[END_REF] states that

µ * ∆ = µ * P * ∆ .
Hence, µ ∆ = µ ⋆ P ∆ . Therefore, µ ∆ has a density given by g(z -y)P ∆ (dy) = Eg(z -Z ∆ 1 ).

Moreover, we have, for any compactly supported function t:

1 ∆ E(Z ∆ 1 t(Z ∆ 1 )) = t(z)Eg(z -Z ∆ 1 )dz = E(t(x + Z ∆ 1 )g(x)dx.
Hence, we apply first Parseval formula:

h * ∆ 2 = |h * ∆ (x)| 2 dx = 2π h 2 ∆ (x)dx = 2π∆ z 2 1 I |z|≤kn √ ∆ E 2 (g(z -Z ∆ 1 ))dz ≤ 2π∆E z 2 1 I |z|≤kn √ ∆ g 2 (z -Z ∆ 1 )dz ≤ 2π∆E (x + Z ∆ 1 ) 2 g 2 (x)dx ≤ 4π∆E (x 2 + (Z ∆ 1 ) 2 )g 2 (z)dz ≤ 4π∆ x 2 g 2 (x) + E[(Z ∆ 1 ) 2 ] g 2 .
This ends the proof. 2 5.7. Proof of Theorem 4.2. Let us define the sets

Ω 1 = ∀m ∈ M n , πm -πm 1 ψ∆ (x) - 1 ψ ∆ (x) 2 dx ≤ k 1 πm -πm dx |ψ ∆ (x)| 2 and Ω 2 = 1 n n i=1 [Z ∆ i ] 2 E[(Z ∆ i ) 2 ] -1 ≤ k 2 .
Take 0 < k 1 < 1/2 and 0 < k 2 < 1. On Ω

[Z ∆ i ] 2 ≤ (1 + k 2 )E[(Z ∆ 1 ) 2 ] and E[(Z ∆ 1 ) 2 ] ≤ 1 1 -k 2 1 n n i=1 [Z ∆ i ] 2 .
Il follows that, on Ω 1 ∩ Ω 2 := Ω 1,2 , we can choose κ ′ large enough to ensure As m 2β∆+1 /(n∆) is bounded m 2pβ∆ n -p = O(n 2pβ∆/(2β∆+1)-p ) = O(n -p/(2β∆+1) ). Therefore, choosing p = 3/ε ensures that n 1-pε) = n -2 and P(Ω c 1 ) ≤ C/n 2 .

On the other hand, |ν (1) n (ϕ m,j ) + ν (2) n (ϕ m,j ) + )) ≤ c ′ n using the proof of Theorem 4.1 and the fact that pen(.) is less than O(n). The same line can be followed for the other terms. This result follows from the concentration inequality given in [START_REF] Klein | Concentration around the mean for maxima of empirical processes[END_REF] and arguments in Birgé and Massart (1998) (see the proof of their Corollary 2 page 354). It can be extended to the case where F is a unit ball of a linear space.

P[Ω c 2 ] ≤ 1 k p 2 E p [(Z ∆ 1 ) 2 ] E 1 n n i=1 [(Z ∆ i ) 2 -E[(Z ∆ 1 ) 2 ]]

4. 4 .

 4 Risk bound of the collection of estimators. First, we recall a key Lemma, borrowed from Neumann (1997) (see his Lemma 2.1): Lemma 4.1. It holds that, for any p ≥ 1,

Proposition 4 . 1 .

 41 Under Assumptions (H1)-(H2)(4)-(H3), then for all m:

(

4. 5 .

 5 Discussion about the rates. Let us study some examples and use (21) to get a relevant choice of m. We have g -g m 2 = |x|≥πm |g * (x)| 2 dx. Suppose that g belongs to the Sobolev class S(a, L) = {f, |f * (x)| 2 (x 2 + 1) a dx ≤ L}.

5 . Proofs 5 . 1 .

 551 Proof of Theorem 4.1. Writing that γ n (ĝ m) + pen( m) ≤ γ n (g m ) + pen(m) in view of (27) implies that

5. 5 .

 5 Proof of Proposition 5.4. First we study R (3) n .

48p 1 First, we prove that P(Ω c 1 , 2 )C

 112 (m, m) + 16p 2 (m, m) + pen(m) -pen( m) ≤ C(a, b)pen(m).This allows to extend the result of Theorem 4.1 as follows:∀m ∈ M n , E g -g 2 1 I Ω 1,2 ≤ C g -g m 2 + pen(m) + K ln 2 (n) n∆ .Next we need to prove that (34) E g -g 2 1 I Ω c ≤ c/n 2 by proving that P(Ω c 1 ) ≤ c/n 2 and P(Ω c 2 ) ≤ c/n.P((Ω 1 ) c ) ≤ p m p-1 n -p πm -πm dx/|ψ ∆ (x)| 4p (Φ ψ (m)) p ≤ m∈Mn C ′ p n -p m (p-1)-p(2β∆+1)+4pβ∆+1 = m∈Mn C ′ p m 2pβ∆ n -p≤ C"n 1-p+p(1-ε) ≤ C"n 1-pε .

p.

  Here the choice p = 4 gives P[Ω c 2 ] = O(1/n 2 ) with a simple variance inequality, provided that E[(Z ∆ 1 ) 8 ] < +∞. Next, we write thatg -g 2 = g -g m 2 + g m -ĝ m 2 ≤ g 2 + j∈Z |â m,j -a m,j (g)| 2and j∈Z |â m,j -a m,j (g)| 2 = j∈Z

6 . Appendix Lemma 6 . 1 . 1 + ǫ 2 - 1 ,

 661121 Let Y 1 , . . . , Y n be independent random variables, let ν n,Y (f ) = (1/n) n i=1 [f (Y i )-E(f (Y i ))] and let F be a countable class of uniformly bounded measurable functions. Then for ǫ 2 > 0E sup f ∈F |ν n,Y (f )| 2 -2(1 + 2ǫ 2 )H 2 + K 1 = 1/6, and sup f ∈F f ∞ ≤ M, E sup f ∈F |ν n,Y (f )| ≤ H, sup (Y k )) ≤ v.

  1 , we have, ∀m ∈ M n ,

	πm -πm and on Ω 2 , we find dx | ψ∆ (x)| 2 ≤ (2k 1 + 2)	πm -πm	dx |ψ ∆ (x)| 2 and	πm -πm	dx |ψ ∆ (x)| 2 ≤	2 1 -2k 1	πm -πm	dx | ψ∆ (x)| 2
	1	n
	n	i=1

  (ϕ m,j )| 2 + |ν (2) n (ϕ m,j )| 2 + It follows that, E( g 2 1 I Ω c 1,2 ) = g 2 P(Ω c 1,2) ≤ c/n, and for k = 3, 4,

							4
							R (k) n (ϕ m,j )| 2
							k=1
							4
					≤ C		k=1	|R (k) n (ϕ m,j )| 2 }
					= C	sup t∈S m , t =1	|ν (1) n (t)| 2 +	sup t∈S m , t =1	|ν (2) n (t)| 2
						4
					+	k=1	sup t∈S m , t =1	|R (k) n (t)| 2
		E	sup t∈S m , t =1	|R (k) n (t)| 2 1 I Ω c 1,2	≤ E	sup t∈Sm n , t =1	|R (k) n (t)| 2 ≤ C/n
	as it has been proved previously. Lastly,
	E	sup t∈S m , t =1	|ν (1) n (t)| 2 1 I Ω c 1,2	≤ E	sup t∈S m , t =1	|ν (1)
							1,2
							≤ c(	1 n∆	+ nP(Ω c 1,2

j∈Z {|ν

(1)

n n (t)| 2 -pen( m)

+ +E pen( m)1 I Ω c

Université Paris V, MAP5, UMR CNRS 8145. email: fabienne.comte@univ-paris5.fr and genon@mathinfo.univ-paris5.fr.


Dr Haruki Nagae

Jingzhao Xia

Dr Evgueni Kirillov

Dr Kosuke Higashida

Koya Shoji

Valentin Boiteau

Prof. Dr Wanbin Zhang
email: wanbin@sjtu.edu.cn

Jean-Francois Carpentier
email: jean-francois.carpentier@univ-rennes1.fr

Kazushi Mashima

Valentin Boi- Teau

Jean-François Carpentier

Mr Koya

Mr Valentin Boiteau

Asymmetric Allylic Alkylation of beta-Ketoesters via C-N Bond Cleavage of N-Allyl-N-methylaniline Derivatives Catalyzed by a Nickel-Diphosphine System

Keywords: Asymmetric Allylic Alkylation, Nickel Catalyst, β-Ketoesters, Allylic Amines, C-N Bond Cleavage

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

Asymmetric allylic alkylation (AAA), the asymmetric version of transition metal-assisted nucleophilic substitution of allylic compounds, namely the Tsuji-Trost reaction, 1,2 is one of the most efficient synthetic methods for producing optically active organic compounds. Allylic substrates bearing activated leaving groups such as carboxylates (acetates), 3-5 carbonates, 1f,6-9 phosphates, [START_REF] Han | Asymmetric Synthesis of Borylalkanes via Copper-Catalyzed Enantioselective Hydroallylation[END_REF] and halides, [START_REF] Schäfer | Chiral and achiral C-C bond formation reactions using allylic halides, previous reports are in references 1[END_REF] as well as allylic molecules having trichloroacetimidate, [START_REF] Shimizu | Chiral Brønsted Acid-Catalyzed Intramolecular SN2′ Reaction for Enantioselective Construction of A Quaternary Stereogenic Center[END_REF] trimethylsilyl, [START_REF] Kaib | Extremely Active Organocatalysts Enable a Highly Enantioselective Addition of Allyltrimethylsilane to Aldehydes[END_REF] carbamates and ureas, [START_REF] Abrams | Transition Metal Free Cycloamination of Prenyl Carbamates and Ureas Promoted by Aryldiazonium Salts[END_REF] exhibit high reactivity in AAA reactions. Nonetheless, allylic alcohols, 1g,1h,15-18 ethers, 1h,19 amines, 1h,15h,20-23 and C-H bond, [START_REF] Trost | A New Class of Non-C2-Symmetric Ligands for Oxidative and Redox-Neutral Palladium-Catalyzed Asymmetric Allylic Alkylations of 1,3-Diketones[END_REF] were recently applied directly as simple alternative allylic sources, although AAA reaction with such stable allylic sources has been developed almost exclusively by using easyto-handle precious metal-based catalysts, such as palladium, rhodium, and iridium complexes. In this context, some of us recently reported the AAA reaction of β-ketoesters with allylic alcohols to construct quaternary chiral centers mediated by non-precious nickel complex bearing chiral diphosphine ligand. 15b The substrate scope of the nickel-catalyzed AAA using allylic alcohols was quite narrow, however, and AAA was effective only when using 6-membered cyclic β-ketoesters: the use of 5-and 7-membered β-ketoesters afforded the corresponding products in quite low enantiomeric excess.

To further expand the scope of this nickel/chiral diphosphine system, we focused our attention on catalytic cleavage of the highly stable C-N bond of allylic amines. [START_REF] Zhao | C-N Bond Cleavage of Allylic Amines via Hydrogen Bond Activation with Alcohol Solvents in Pd-Catalyzed Allylic Alkylation of Carbonyl Compounds[END_REF][START_REF] Mukherjee | B. Chiral Counteranions in Asymmetric Transition-Metal Catalysis: Highly Enantioselective Pd/Brønsted Acid-Catalyzed Direct α-Allylation of Aldehydes[END_REF][START_REF] Murahashi | Palladium (0) Catalyzed 3-Aza-Cope Rearrangement of N-allylenamines[END_REF][START_REF] Trost | Dynamic Kinetic Asymmetric Allylic Amination and Acyl Migration of Vinyl Aziridines with Imido Carboxylates[END_REF] The leaving ability of the amine moiety of allylic amines is finely controllable by changing substituents on the nitrogen atom, and we previously reported the palladium-catalyzed allylic alkylation of enamines with allylic amines. [START_REF] Zhao | C-N Bond Cleavage of Allylic Amines via Hydrogen Bond Activation with Alcohol Solvents in Pd-Catalyzed Allylic Alkylation of Carbonyl Compounds[END_REF] Scheme 1 shows some landmark AAA reactions using allylic amines catalyzed by palladium complexes. In 2007, List et al. reported the first catalytic AAA reaction using allylic amine as an allyl source, and Pd(PPh 3 ) 4 associated with a chiral phosphoric acid, (R)-TRIP, as a catalyst (Scheme 1a). 21a One of us reported an AAA reaction using allylic amines and ketones catalyzed by [Pd(h 3 -C 3 H 5 )Cl] 2 with a chiral ferrocene-based phosphinooxazoline ligand, in which MeOH acted as the best solvent because the C-N bond of the allylic amines was activated by the hydrogen bond between the nitrogen atom of the allylic amine and the hydrogen atom of the methanol solvent (Scheme 1b). [START_REF] Zhao | C-N Bond Cleavage of Allylic Amines via Hydrogen Bond Activation with Alcohol Solvents in Pd-Catalyzed Allylic Alkylation of Carbonyl Compounds[END_REF] Tian et al. achieved an AAA reaction of malononitriles with allylic amines, involving a kinetic resolution catalyzed by [Pd(h 3 -C 3 H 5 )Cl] 2 / (S)-BINAP (Scheme 1c). 21b As shown in Scheme 1, AAA reactions with allylic amines were achieved by using precious palladium catalyst, and only one achiral allylation using allylic amine was accomplished by using nickel complex with achiral diphosphine ligand and metallic zinc as reductant. [START_REF] Sweeney | Catalytic C-C Bond Formation Using a Simple Nickel Precatalyst System: Base-and Activator-Free Direct C-Allylation by Alcohols and Amines[END_REF] Herein, we report a highly effective nickel-catalyzed AAA reaction using allylic amines with a substrate scope not limited to 6-membered β-ketoesters, but expanded to 5-and 7-membered benzo-fused β-ketoesters by further modification of the chiral diphosphine ligand (Scheme Tian, et al. in 2015. up to 99% ee with 36% yield

Ph

O R 1 + O R 1 R 2 Fe PPh 2 PPh 2 N O O N t Bu t Bu R 2 R 3 NH 2 R 1 CN CN + [Pd( h3-C 3 H 5)Cl]2 (1 mol%) (S)-BINAP (4 mol%) mesitylsulfonyl hydrazide (0.6 equiv) air, MS, dioxane, rt, 24 h R 2 R 3 CN CN R 1 c) S. K.

Results and Discussion

We began by searching for the best ligand among chiral diphosphines L1-8 (6 mol%) in combination with Ni(cod) 2 (5 mol%) as a catalyst precursor for the AAA of β-ketoester 1a with allylic amine 2a chosen as the model substrate; representative results obtained in diethyl ether solution (see Table S1 for solvent screening results) at 0 °C for 40 h are summarized in Table 1. The use of (S)-H 8 -BINAP (L1), which is the best ligand in AAA of allylic alcohols, 15b afforded product 3aa in 88% yield and 94% ee (entry 1). In the case of (S)-BINAP (L2), yield and enantioselectivity of 3aa were decreased to 74% and 91% ee, respectively (entry 2). Using (S)-Tol-BINAP (L3) and (S)-SEGPHOS (L4) yielded 3aa in quantitative yield with 92% ee and 94% ee, respectively (entries 3 and 4). On the other hand, the sterically hindered (S)-DTBM-SEGPHOS (L5) retarded the AAA reaction to give 3aa in only 12% yield with 14% ee (entry 5), while (S)-DIFLUORPHOS (L6) afforded 3aa in moderate yield with high enantioselectivity (entry 6). BIPHEP-type ligands, such as (S)-MeO-BIPHEP (L7) and (S)-Tol-MeO-BIPHEP (L8), were the most effective among the ligands investigated, giving 3aa in high yields (96% and 99%, respectively) and enantioselectivities (94% ee and 95% ee, respectively) (entries 7 and 8). We next examined the AAA reaction of various Nsubstituted allylic amine derivatives under typical conditions (5 mol% of Ni(cod) 2 , 6 mol% of (S)-Tol-MeO-BIPHEP (L8), diethyl ether, 0 °C, 40 h) to evaluate the leaving ability of the amine moiety; the results are summarized in Table 2. When N-cinnamyl-N-methylaniline (2a) was used, 3aa was obtained in 99% yield and with 95% ee (entry 1). On the other hand, N,N-diphenylamine 2b and N-phenylamine 2c derivatives resulted in lower yields of 3aa, 11% and 60%, respectively, yet still with high enantioselectivities (entries 2 and 3). The use of N-cinnamylamine (2d) afforded 3aa but the yield, 8%, and enantioselectivity, 61% ee, were lower than those obtained from 2a (entry 4 vs 1). Also, AAA reactions of N-cinnamyl-N,N-dialkylamines, such as N-cinnamyl-N,N-dimethylamine (2e), N-cinnamyl-N,N-diethylamine (2f), and N-cinnamylpyrrolidine (2g) derivatives, afforded 3aa in low to moderate yields with moderate enantioselectivities (entries 5-7). Accordingly, we selected N-phenyl-N-methyl derivative 2a as the best substituent pattern for the allylic amine. We assumed that allylic amines having an electron-donating substituent accelerated the protonation of allylic amine and subsequent formation of a cationic nickel complex having a lower energy barrier to cleave the C-N bond of allylic amine (vide infra). N-Cinnamyl-N,N-dialkylamine derivatives exhibit high basicity, but the leaving-group ability of N,N-dialkylamines is lower than that of N-alkylaniline derivatives; this is consistent with the lower yields observed from N,N-dialkylallylamines, such as 2d, 2e, and 2f, than from N-cinnamyl-N-methylaniline 2a. Hence, there seems to exist a balance between the basicity and the leaving-group ability of the amine moiety of allylic amine to achieve high catalytic activity.

O OMe O + Ph N R 1 R 2
Ni(cod)2 ( 5 Reaction conditions: 1a (0.50 mmol), 2 (0.60 mmol, 1.2 equiv), Ni(cod) 2 (5 mol%), L8 (6 mol%), and Et 2 O (1 mL). a Determined by 1 H NMR analysis using triphenylmethane as an internal standard. b Determined by HPLC analysis (see ESI).

With the optimized conditions in hands, we performed a large-scale reaction using 2.0 mmol of 1a and 2.4 mmol of 2a; this eventually resulted in the corresponding product 3aa in 99% yield and 95% ee (eq. 1), although a slightly longer reaction time, 50 h, was required to reach quantitative yield.

O OMe O + Ph N R 1 R 2 Ni(cod)2 (5 mol%) ( S )-Tol-MeO-BIPHEP (6 mol%) Et 2O, 0 °C, 50 h HNMeAr 1a (2.0 mmol) 2 (1.2 equiv) O OMe O 3aa (99% yield, 95% ee) Ph * (1)
Under the optimized conditions using N-cinnamyl-Nmethylaniline (2a), we evaluated the scope of nucleophilic substrates (Table 3). β-Ketoesters 1b and 1c, which have a fused anisyl ring, reacted smoothly to afford the corresponding products 3ba and 3ca in high yields and excellent enantioselectivities (94% ee and 96% ee), respectively (entries 1 and 2). The use of dimethyl-substituted β-ketoester 1d yielded the corresponding product 3da in good yield with 93% ee (entry 3). On the other hand, the AAA reaction of β-ketoester 1e having an ethyl ester was slow and required a longer reaction time, 80 h, to give 3ea in 99% yield with 95% ee; this suggests that the steric hindrance at the ester group directly affected the reactivity, but not the enantioselectivity (compare Table 2, entry 1 with Table 3, entry 4). β-Ketoester 1f without an aromatic ring yielded product 3fa in moderate yield with high enantiomeric excess (96% ee) (entry 5). In contrast to the high enantioselectivities observed for β-ketoesters with a 6membered ring, contrivances by tuning the ligand and substrate substituents were required for 5-membered ring substrates. In fact, with the (S)-Tol-MeO-BIPHEP (L8)-based catalyst, β-ketoester 1g gave 3ga in 99% yield, but with quite low enantioselectivity (21% ee, entry 6). After reexamining the ligand system for the AAA reaction of β-ketoester having a 5-membered ring (Table S3), we selected (S)-H 8 -BINAP (L1) and obtained slightly better enantioselectivities (34% ee at 0 °C for 40 h) (entry 7). More importantly, when using ligand L1, we found that a bulky ester moiety improved both reactivity and enantioselectivity: hence, the reaction of βketoester 1h bearing the sterically hindered COOCEt 3 ester group operated at temperatures as low as 0 °C and -20 °C for 60 h and gave the corresponding AAA product 3ha in 99% yield with 82% ee and 88% ee, respectively (entries 8 and 9); operating at -30 °C further increased the enantioselectivity to 92% ee, but the yield of the product dropped to 59% (entry 10). The use of methyl and fluoro-substituted substrates 1i and 1j at -20 °C for 60 h afforded the corresponding products 3ia and 3ja in quantitative yields with 82% ee and 90% ee, respectively (entries 11 and 12). Dimethoxy-substituted βketoester 1k gave 3ka in 74% yield and 79% ee (entry 13). A broader scope was demonstrated by using the 7-membered ring β-ketoester 1l using ligand L8 at 0 °C, which resulted in 95% yield of 3la with 77% ee (entry 14). [START_REF]For results using other nucleophiles[END_REF] We also investigated other nucleophiles such as β-diketone 1m, β-ketoamide 1n, and β-diketonitrile 1o. The use of 1m gave the corresponding product 3ma in 86% yield with low enantioselectivity (26% ee), possibly due to the difference of coordination ability between the -CO 2 Me and -C(O)Me moieties (entry 15). When 1n was used as nucleophile, no reaction was observed, possibly because of the sterically hindered piperidine moiety and the higher pKa value of β-ketoamide than that of β-ketoester and β-diketone (entry 16). [START_REF] Bordwell | Carbon Acids. 8. The Trimethylammonio Group as a Model for Assessing the Polar Effects of Electron-Withdrawing Groups[END_REF] Interestingly, βdiketonitrile 1o reacted smoothly and afforded the corresponding product 3oa in 94% yield with very high enantioselectivity (entry 17). It is unclear why the small nitrile group improved the selectivity but we assume this may also be related to the different coordination ability of the -CN moiety.

Next, we examined AAA of substituted allylic amines with an N-methyl aniline moiety, and the results are shown in Table 4. The steric hindrance on the aryl ring of cinnamyl moiety affected the reactivity: p-and m-tolyl substituted allylic amines 2h and 2i afforded the corresponding products 3ah and 3ai in excellent yield with high enantioselectivity, respectively (entries 1 and 2), but the more sterically hindered o-tolyl substituted allylic amine 2j gave 3aj in slightly lower yield and enantiomeric excess (77% and 90% ee, entry 3). More critical steric effects were observed when 2k, which has a methyl group on 2-position of the cinnamyl moiety, was used as allyl source, as no reaction was observed (entry 4). Thus, the steric hindrance around the C=C bond is quite essential. This is consistent with the X-ray single crystal diffraction analysis of 4b which evidenced coordination of the C=C bond to the nickel center, most likely as the first step of the AAA reaction (vide infra). p-Methoxy, and p-fluoro substituted allylic amines 2l and 2m afforded the corresponding products 3al and 3am quantitatively with high enantioselectivity after 40 h, respectively (entries 5-6). Allylic amine 2n, having a strong electron-withdrawing p-CF 3 group, exhibited low reactivity to give 3an in 51% yield, but the high enantioselectivity was maintained (entry 7). By prolonging the reaction time to 120 h, the corresponding product 3an was obtained in 87% yield with almost the same enantioselectivity (entry 8). N-Allyl-Nmethylaniline (2o) and N-(2-buten-1-yl)aniline (2p) worked well and delivered products 3ao and 3ap in 99% and 95% yield after 40 h, respectively. Allylic amines 2o and 2p have less bulky substituents (H and Me, respectively) than Ph group, which decreased the enantioselectivity to 74% and 81% ee, respectively (entries 9 and 10). Ni(cod)2 ( 5 To investigate the reaction mechanism, we prepared olefinic complexes 4a and 4b that, respectively, correspond to and mimic putative intermediates (Scheme 2). Complex 4a was prepared by treating (L8)Ni(cod) (L8 = (S)-Tol-MeO-BIPHEP) with allylic amine 2a. The 31 P{ 1 H} NMR spectrum of 4a in C 6 D 6 displayed two equal-intensity doublets centered at δ 29.41 and 24.20 ppm with J P-P = 47.4 Hz due to two nonequivalent phosphorus atoms of the chiral diphosphine ligand bound to Ni(0) center. In the 1 H NMR spectrum of 4a, the allylic methylene hydrogens were observed at δ 3.5-3.3 ppm as a multiplet signal, and two olefinic hydrogens were observed at δ 4.6-4.4 ppm and 4.2-4.1 ppm as multiplet signals.

In addition, one olefinic carbon of 4a was observed at δ 68.6 ppm as a doublet of doublets (J C-P = 16.7, 2.0 Hz), and the other olefinic carbon of 4a was observed at δ 56.9-56.5 ppm along with an olefinic carbon signal of COD of (L8)Ni(cod). Because all attempts to crystallize complex 4a failed to date, we isolated the related complex 4b in 23% yield as orange crystals from the reaction of Ni(cod) 2 with 1,1'bis(diphenylphosphino)ferrocene (DPPF) in the presence of an excess amount of 2a. Its structure was characterized by NMR spectroscopy and X-ray single-crystal analysis. Figure 1 shows the molecular structure of 4b. The two phosphorus atoms of DPPF are bonded to the Ni(0) center with a P1-Ni-P2 bite angle of 106.84(3)°, which is almost the same as that (107.13(9)°) of (DPPF)Ni 0 (H 2 C=CHCH 2 OH). 15b The C41-C42 bond length of 1.417(4) Å is consistent with other C=C Although the steric characteristics of the isolated olefin-DPPF complex 4b differ from those of complex 4a, [START_REF] Dierkes | The bite angle makes the difference: a practical ligand parameter for diphosphine ligands[END_REF][START_REF]complexes 4a and 4b were calculated as 51.1% Vbur[END_REF][31] which is the best catalyst identified but not isolable, we conducted control experiments with 4b. Complex 4b and the in situ mixture of Ni(cod) 2 and DPPF featured almost the same catalytic performance for the allylic alkylation reaction of 1a with 2a to afford 3aa (Scheme 3). In addition, we observed no reaction of (DPPF)Ni(cod) with β-ketoester 1a (eq. 2) and no oxidative addition of the C-N bond of the coordinated allylic amine over complex 4b in Et 2 O at room temperature in the absence of substrate (eq. 3). These observations indicated that the coordination of 2a to '(DPPF)Ni' to form 4b proceeded smoothly as the first step of the reaction, but was not the ratedetermining step. The small rate dependency, less than 1, on [Ni] suggests that free COD coordinates to nickel to form an inert (L8)Ni(cod) species and prevents generation of the active complex 4a, (L8)Ni(2a). 33 In fact, we observed that the addition of 20 mol% of COD to the reaction mixture remarkably decreased the AAA reaction rate (Figure S4). Based on the plot of ln(k obs /T) versus 1/T (Figure 2), the activation parameters were determined: DH ‡ =12.8±2.5 kcal mol -1 , DS ‡ = -142±36 J K -1 mol -1 , and DG ‡ (298 K) = 22.8±5.1 kcal mol -1 . The large negative DS ‡ value suggests a highly ordered transition state, and the low activation Gibbs free energy, 22.8±5.1 kcal mol -1 , was consistent with the result of the DFT calculation, DG ‡ (298 K) = 26.4 kcal mol -1 (vide infra) although we employed a different diphosphine ligand, DPPF, instead of (S)-Tol-MeO-Biphep (L8) for DFT calculation. DFT calculations performed on the model Ni-DPPF system allowed us to rationalize a mechanism for AAA reactions catalyzed by the isolated olefin-Ni(0) complex 4b (Scheme 4), and key structures for some transition state are collected in Figure 3. After the first ligand exchange between one COD and DPPF to give (DPPF)Ni(cod) (Scheme 2), reaction of (DPPF)Ni(cod) with allylic amine 2a reversibly forms 4b which is stabilized by 6.6 kcal mol -1 . Protonation of the amine moiety of 4b by β-ketoester 1f proceeds endergonically by 22.9 kcal mol -1 to generate intermediate A and enolate 1f -. Electron-withdrawing groups on the nitrogen atom of allylic amine increase the energy gap between 4b with 1f and A with 1f -, which decreases the rate of the protonation step (4b→A).

Cleavage of the C-N bond of A affords cationic nickel π-allyl complex B via transition state TS1cat outer along with release of N-methylaniline. The energy barrier 4b→TS1cat outer is 26.4 kcal mol -1 , a much lower value than that of the estimated energy barrier of the alternative pathway (A'→TS1'→B', 33.6 kcal mol -1 ) where a neutral nickel allyl complex B' (i.e., without preliminary protonation) is generated via oxidative addition of the C-N bond. It is a reliable pathway by which the cationic complex B reacts with enolate 1f -to give an enolate adduct B-1f and neutral nickel alkyl species D. Complex B reacts with an outer-sphere enolate 1f -to give an olefin complex C including a quaternary carbon at the α-position of the carbonyl moiety via stable transition state TS2 outer . The activation barrier of the outer-sphere pathway (24.1 kcal mol -1 ), involving the B ↔ D pre-equilibrium step through a low-lying TS B-D , is slightly lower than C-N bond cleavage step. We excluded another possible pathway in which complex D reacts with inner-sphere enolate to give complex C via TS2 inner due to its much higher energy barrier (34.9 kcal mol -1 ) than TS2 outer . [START_REF]pathway via neutral nickel complexes, TS1neut outer , TS1neut inner , and neutral nickel complex TS1cat inner . All reaction pathways calculated are shown in Figure S6 in the supporting information[END_REF] Finally, an exchange reaction of complex C by allylic amine 2a affords the corresponding intermediate 4b.

On the basis of kinetic data showing first-order rate dependence on [2a], the C-C bond formation step (D→TS2 outer →C) is suggested to be the rate-determining step; of note, the corresponding calculated activation barrier is slightly lower than that of 4b→TS1cat outer →B (24.1 vs. 26.4 kcal mol -1 , respectively) but this slight difference (2.3 kcal•mol -1 ) falls within the error of the DFT method. Based on these control experiments and DFT calculations, we propose a catalytic cycle of the AAA reaction mediated by Ni(0) species, as shown in Scheme 5. In the nickel-catalyzed AAA reaction, the first step is coordination of allylic amine 2 onto the Ni(0)-diphosphine complex to form olefin complex 4. The amine moiety of complex 4 is then protonated by βketoester 1 to give cationic olefin complex A and enolate 1 -. Cleavage of the C-N bond of A affords B via transition state TS1, as the rate-determining step. Then, an outer-sphere enolate anions 1 -reacts with B to form olefin complex C with a quaternary carbon at the α-position of the carbonyl moiety. Lastly, an olefin exchange reaction between C and 2 proceeds to give the corresponding product 3 and regenerates catalytic intermediate 4.

It is noteworthy to compare the current nickel catalyst system with the palladium catalyst system derived from [Pd(h 3 -C 3 H 5 )Cl] 2 and a chiral ferrocene-based phosphinooxazoline ligand. [START_REF] Zhao | C-N Bond Cleavage of Allylic Amines via Hydrogen Bond Activation with Alcohol Solvents in Pd-Catalyzed Allylic Alkylation of Carbonyl Compounds[END_REF] In the latter system, activation of the allylic amine is proposed to proceed by protonation from methanol used as a solvent; yet, the use of methanol as solvent in our nickel system decreased the conversion of 1a and 2a to afford only 8% yield of 3aa (ie, 92% of 1a and 2a left) with 30% ee (Table S1).

Conclusion

We developed an efficient version of the AAA reaction with a catalyst system made of cheap and readily available Ni(0) and a chiral diphosphine ligand using allylic amines as the allyl source. Some stoichiometric reactions as well as DFT calculations revealed the reaction mechanism, in which deprotonation of β-ketoesters by the N-methylaniline moiety of allylic amines is the key step to smoothly cleave the C-N The C-N bond cleavage of this nickel-catalyzed AAA reaction is completely different from that of palladiumcatalyzed systems. Although our Ni(0) system is sensitive to the choice of chiral diphosphine ligands, the demonstrated scope of effective AAA reactions is broad. Related AAA reactions using other allyl sources and substrates such as enamines are on-going projects in our laboratory Experimental Section General: All reactions and manipulations involving air-and moisture-sensitive organometallic compounds were operated using the standard Schlenk or dry box techniques under argon atmosphere. MeOH was dried and deoxygenated by distillation over sodium benzophenone ketyl under argon. Benzonitrile was distilled from the calcium hydride. Super dehydrated DMF was purchased from Wako Pure Chemical Industries, Ltd. and used as received. Alternatively, Et 2 O, acetonitrile, THF, toluene and CH 2 Cl 2 were dried and deoxygenated by using Grubbs column (Glass Counter Solvent Dispending System, Nikko Hansen & Co, Ltd.). [START_REF] Pangborn | Safe and Convenient Procedure for Solvent Purification[END_REF] Other chemicals were purchased and used without further purification. 1 H NMR (400 MHz), [START_REF] Kaib | Extremely Active Organocatalysts Enable a Highly Enantioselective Addition of Allyltrimethylsilane to Aldehydes[END_REF] C{ 1 H} NMR (100 MHz) and [START_REF] Huo | Hydrogen-Bond-Activated Palladium-Catalyzed Allylic Alkylation via Allylic Alkyl Ethers: Challenging Leaving Groups[END_REF] F{ 1 H} NMR (376 MHz) spectra were measured on Bruker Avance III-400 spectrometers. All 1 H NMR chemical shifts were reported in ppm (δ) relative to tetramethylsilane at δ 0.00 ppm or referenced to the chemical shifts of residual solvent resonances (THF-d 8 was used as internal standard, δ 3.57 ppm). All 13 C{ 1 H} NMR chemical shifts were reported in ppm (δ) relative to carbon resonances of CDCl 3 at δ 77.16 ppm, or αcarbon of THF-d 8 at δ 67.21 ppm. All 19 F{ 1 H} NMR chemical shifts were reported in ppm (δ) relative to carbon resonances in α,α,α-trifluorotoluene at δ -63.90 ppm. HPLC spectra were recorded on a JASCO UV-2075. Optical rotation values were recorded on an Anton Paar MCP100 polarimeter at 589 nm (sodium lamp) and are given in 10 -1 deg cm 2 g -1 . Mass spectra were obtained on JEOL JMS-700. All melting points were recorded on BUCHI Melting Point M-565. All Infrared spectroscopy were recorded on Jasco FT/IR-4200. Flash column chromatography was performed using silica gel 60 (0.040-0.0663 nm, 230-400 mesh ASTM). X-ray crystal data were collected with a Rigaku RAXIS-RAPID Imaging Plate diffractometer. The elemental analyses were recorded by using Perkin Elmer 2400 at the Faculty of Engineering Science, Osaka University.

General procedure for Ni-catalyzed AAA reaction of βketoesters with allylamines: Ni(cod) 2 (0.025 mmol, 5.0 mol%), Diphosphine ligand (0.030 mmol, 6.0 mol%), and βketoester (0.50 mmol, in the case of solid compounds) were added to a 20 mL Schlenk flask in an argon filled glove box. Then, dry Et 2 O (1.0 mL) and β-ketoester (0.50 mmol, in the case of liquid compounds) were added to the reaction mixture under argon. After 10 min, the allylic amine (0.60 mmol, 1.2 equiv) was added to the reaction mixture under argon atmosphere at -30-0 °C. The reaction mixture was stirred at 0 °C. After consumption of β-ketoester, as monitored by TLC, the reaction mixture was quenched with ethyl acetate (40 mL) followed by washing with 6N hydrochloric acid (20 mL × 2) and saturated aqueous solution of NaHCO 3 (30 mL). The organic layer was dried over Na 2 SO 4 . The filtrate was concentrated in vacuo, and the residue was purified by flash column chromatography on silica gel to afford the corresponding AAA product.

Computational studies General Considerations:

The calculations were performed using the Gaussian 09 [START_REF] Frisch | Revision D.01[END_REF] program employing B3PW91 38 functional, and using a standard double-ξ polarized basis set, namely the LANL2DZ set, augmented with a single polarization f function on iron and nickel (2.462 and 3.130, respectively) and a single polarization d function on phosphorous (0.364). The solvent effects, in our case for Et 2 O, were taken into account during all the calculations by means of the SMD model. [START_REF] Marenich | Universal Solvation Model Based on Solute Electron Density and on a Continuum Model of the Solvent Defined by the Bulk Dielectric Constant and Atomic Surface Tensions[END_REF] All stationary points were fully characterized via analytical frequency calculations as either true minima (all positive eigenvalues) or transition states (one imaginary eigenvalue). The IRC procedure was used to confirm the nature of each transition state connecting two minima. [START_REF] Gonzales | An improved algorithm for reaction path following[END_REF] Zero-point vibrational energy corrections (ZPVE) were estimated by a frequency calculation at the same level of theory, to be considered for the calculation of the total energy values.

ASSOCIATED CONTENT

Supporting Information

Details of experimental procedures, optimization of reaction conditions, kinetic studies, NMR spectra, HPLC chart for all new compounds, and Cartesian coordinates of DFT-computed structures (PDF) were included in a Supporting Information. X-ray crystallographic data for complex 4b (CIF).

  .List, et al. in 2007. b) W.Zhang, et al. in 2011.

  1 (0.50 mmol), 2a (0.60 mmol, 1.2 equiv), Ni(cod) 2 (5 mol%), chiral diphosphine ligand (6 mol%), and Et 2 O (1 mL). a Isolated yield. b Determined by HPLC analysis (see ESI). c Determined by 1 H NMR analysis using triphenylmethane as an internal standard. d 0.5 mL of toluene and 0.5 mL of Et 2 O were used as solvents due to the low solubility.

  1a (0.50 mmol), 2 (0.60 mmol, 1.2 equiv), Ni(cod) 2 (5 mol%), L8 (6 mol%), and Et 2 O (1 mL). a Isolated yield. b Determined by HPLC analysis (see ESI).

Figure 1 .

 1 Scheme 2. Synthesis of Olefinic Complexes 4a and 4b

Scheme 3 .

 3 Scheme 3. Allylic Alkylation Reaction Catalyzed by The Isolated Complex 4b and an in situ Mixture of Ni(cod) 2 with DPPF

Figure 2 .

 2 Figure 2. Eyring Plot for The AAA Reaction of 1a and 2a Performed in Et 2 O Using An in situ Generated Catalyst System From Ni(cod) 2 and (S)-Tol-MeO-Biphep (L8)

  35

Scheme 4 .Figure 3 .

 43 Scheme 4. DFT-Computed Reaction Pathways of Nickel-Catalyzed AAA Reaction of β-Ketoesters with Allylic Amines

  Scheme 5. Proposed Reaction Mechanism for The Nickel-Diphosphine-Catalyzed AAA Reaction Using Allylic Amine and β-Ketoester

Table 1 . Screening of Chiral Diphosphine Ligands

 1

	O	O						Ni(cod)2 (5 mol%)	O	O
		OMe	+	Ph	N Me	Ph	Et 2O, 0 °C, 40 h chiral ligand (6 mol%)	*	OMe
								HNMePh	Ph
	1a				2a				3aa
		entry		chiral ligand				yield [%]	a	ee [%]	b
		1		L1: (S)-H8 -BINAP			88	94
		2		L2: (S)-BINAP				74	91
		3		L3: (S)-Tol-BINAP			>99	92
		4		L4: (S)-SEGPHOS			98	94
		5		L5: (S)-DTBM-SEGPHOS		12	14
		6		L6: (S)-DIFLUORPHOS		52	92
		7		L7: (S)-MeO-BIPHEP		96	94
		8		L8: (S)-Tol-MeO-BIPHEP		99	95
							R	O
		PPh 2			PAr 2	R	O	PAr 2	MeO	PAr 2
		PPh 2			PAr 2	R	O	PAr 2	MeO	PAr 2
							R	O
		L1		L2: Ar = Ph		L4: R = H, Ar = Ph	L7: Ar = Ph
				L3: Ar = 4-MeC 6 H 4	L5: R = H, Ar = 3,5-t Bu 2 -4-OMeC 6 H 2	L8: Ar = 4-MeC 6 H 4
							L6: R = F, Ar = Ph
	Reaction conditions: 1a (0.50 mmol), 2a (0.60 mmol, 1.2
	equiv), Ni(cod) 2 (5 mol%), chiral diphosphine ligand (6 mol%), and Et 2 O (1 mL). a Determined by 1 H NMR analysis using triphenylmethane as an internal standard. b Determined
	by HPLC analysis (see ESI).

Table 2 . Screening of Substituents on Nitrogen Atom of Allylamines 2

 2

Table 3 .

 3 β-Ketoesters Substrate Scope

					Ni(cod)2 (5 mol%)
			Me		chiral ligand (6 mol%)
	+	Ph	N	Ph	Et 2O, temp., time
					HNMePh

Table 4 . Allylic Amines Substrate Scope

 4

	O	O				O
					Me
		OMe	+	Ph	N	R 5
						R 5
	1a				2	3

)-C(sp 3 ) Bond

ACKNOWLEDGMENT

This work was supported by JSPS KAKENHI Grant Numbers JP15H05808 and JP15K21707 in Precisely Designed Catalysts with Customized Scaffolding (No. 2702) and National Natural Science Foundation of China (No. 21620102003). V. B., an internship student from INP-ENSIACET, thanks the INP-ENSIACET for financial support. We appreciate to Prof. Dr. Sensuke Ogoshi for his fruitful discussion. E.K. is grateful to ENSCR for computational facilities.