M Élina Bec

Claire Lacour

ADAPTIVE POINTWISE ESTIMATION FOR PURE JUMP L ÉVY PROCESSES

Keywords: Adaptive Estimation, High frequency, Pure jump Lévy process, Nonparametric Kernel Estimator

HAL is

Introduction

Consider (L t , t ≥ 0) a real-valued Lévy process with characteristic function given by:

(1) ψ t (u) = E(exp iuL t ) = exp (t R (e iux -1)N (x)dx).

We assume that the Lévy measure admits a density N and that the function g(x) = xN (x) is integrable. Under these assumptions, (L t , t ≥ 0) is a pure jump Lévy process without drift and with finite variation on compact sets. Moreover E(|L t |) < ∞ (see [START_REF] Bertoin | Lévy processes[END_REF]). Suppose that we have discrete observations (L k∆ , k = 1, ..., n) with sampling interval ∆. Our aim in this paper is the nonparametric adaptive kernel estimation of the function g(x) = xN (x) based on these observations under the asymptotic framework n tends to ∞. This subject has been recently investigated by several authors. [START_REF] Figueroa-López | Risk bounds for the non-parametric estimation of Lévy processes[END_REF] use a penalized projection method to estimate the Lévy density on a compact set separated from 0. Other authors develop an estimation procedure based on empirical estimations of the characteristic function ψ ∆ (u) of the increments (Z ∆ k = L k∆ -L (k-1)∆ , k = 1, . . . , n) and its derivatives followed by a Fourier inversion to recover the Lévy density. For low frequency data (∆ is fixed), we can quote [START_REF] Watteel | Nonparametric estimation of the canonical measure for infinitely divisible distributions[END_REF], or [START_REF] Jongbloed | Parametric estimation for subordinators and induced OU processes[END_REF] for a parametric study. Still in the low frequency framework, [START_REF] Neumann | Nonparametric estimation for Lévy processes from low-frequency observations[END_REF] estimate ν(x) = x 2 N (x) in the more general case with drift and volatility, and [START_REF] Comte | Nonparametric adaptive estimation for pure jump Lévy processes[END_REF] use model selection to build an adaptive estimator. An adaptive method to estimate linear functionals is also given in [START_REF] Kappus | Nonparametric adaptive estimation of linear functionals for low frequency observed Lévy processes[END_REF]. [START_REF] Belomestny | Statistical inference for time-changed Lévy processes via composite characteristic function estimation[END_REF] addresses the issue of inference for timechanged Lévy processes with results in term of uniform and pointwise distance. One can also cite [START_REF] Gugushvili | Nonparametric inference for discretely sampled Lévy processes[END_REF] or [START_REF] Nickl | A Donsker theorem for Lévy measures[END_REF] for recent works at fixed ∆.

In the high frequency context, which is our concern in this paper, the problem is simpler since, for any fixed u, ψ ∆ (u) → 1 when ∆ → 0. This implies that ψ ∆ (u) need not to be estimated and can simply be replaced by 1 in the estimation procedures. This is what is done in [START_REF] Comte | Nonparametric estimation for pure jump Lévy processes based on high frequency data[END_REF]. These authors start from the equality:

(2)

E Z ∆ k e iuZ ∆ k = -iψ ∆ (u) = ∆ψ ∆ (u)g * (u),
obtained by differentiating (1). Here g * (u) = e iux g(x)dx is the Fourier transform of g, well defined since we assume g integrable. Then, as ψ ∆ (u) 1, equation ( 2) writes

E Z ∆ k e iuZ ∆ k
∆g * (u). This gives an estimator of g * (u) as follows:

1 n∆ n k=1 Z ∆ k e iuZ ∆ k .
Now, to recover g, the authors apply Fourier inversion with cutoff parameter m. Here, we rather introduce a kernel to make inversion possible:

1 n∆ n k=1 Z ∆ k K * (uh)e iuZ ∆ k
which is in fact the Fourier transform of 1/(nh∆) n k=1 Z ∆ k K((x -Z ∆ k )/h). At the end, in the high frequency context, a direct method without Fourier inversion can be applied. Indeed, a consequence of (2) is that the empirical measure:

μn (dz) = 1 n∆ n k=1 Z ∆ k δ Z ∆ k (dz)
weakly converges to g(z)dz (note that the idea of exploiting this weak convergence is already present in [START_REF] Figueroa-López | Nonparametric estimation of time-changed lévy models under high-frequency data[END_REF]). This suggests to consider kernel estimators of g of the form

(3) ĝh (x) = K h μn (x) = 1 n∆ n k=1 Z ∆ k K h (x -Z ∆ k )
where K h (x) = (1/h)K(x/h) and K is a kernel such that K = 1. Below, we study the quadratic pointwise risk of the estimators ĝh (x) and evaluate the rate of convergence of this risk as n tends to infinity, ∆ = ∆(n) tends to 0 and h = h(n) tends to 0. This is done under Hölder regularity assumptions for the function g. Note that a pointwise study involving a kernel estimator can be found in van [START_REF] Van Es | A kernel type nonparametric density estimator for decompounding[END_REF] for more specific compound Poisson processes, but the estimator is different from ours, as well as the observation scheme. In Figueroa-López (2011) a pointwise central limit theorem is given for the estimation of the Lévy density, as well as confidence intervals. Still in the high frequency context, but for integrated distance, we can cite [START_REF] Ueltzhöfer | An oracle inequality for penalised projection estimation of lévy densities from high-frequency observations[END_REF], and [START_REF] Duval | Adaptive wavelet estimation of a compound Poisson process[END_REF] for the estimation of a compound Poisson process with low conditions on ∆. [START_REF] Bücher | Nonparametric inference on Lévy measures and copulas[END_REF] deal with the multivariate case.

In this paper, we study local adaptive bandwidth selection (which the previous authors do not consider). For a given non-zero real x 0 , we select a bandwidth ĥ(x 0 ) such that the resulting adaptive estimator ĝĥ (x 0 ) (x 0 ) automatically reaches the optimal rate of convergence corresponding to the unknown regularity of the function g. The method of bandwidth selection follows the scheme developped by [START_REF] Goldenshluger | Bandwidth selection in kernel density estimation: oracle inequalities and adaptive minimax optimality[END_REF] for density estimation. The advantage of our kernel method is that it allows us to estimate the Lévy density at a fixed point, with a local adaptive choice. This method is easy to implement, and we show its good numerical performance on different examples. Moreover our contribution includes an alternative proof for a lower bound result (see Figueroa-López (2009a)) which proves the optimality of the rate for this pointwise estimation. We also study the framework of irregular sampling.

In Section 2, we give notations and assumptions. In Section 3, we study the pointwise mean square error (MSE) of ĝh (x 0 ) given in (3) for g belonging to a Hölder class of regularity β and we present the bandwidth selection method together with both lower and upper risk bound for our adaptive estimator. The rate of convergence of the risk is (log(n∆)/n∆) β/(2β+1) which is expected in adaptive pointwise context. Examples and simulations in our framework are discussed in Section 4. The case of irregular sampling is addressed in Section 5 and proofs are gathered in Section 6.

Notations and assumptions

We present the assumptions on the kernel K and on the function g required to study the estimator given by (3). First, we set some notations. For any functions u, v, we denote by u * the Fourier transform of u, u * (y) = e iyx u(x)dx and by u 2 , < u, v >, u v the quantities

u 2 = |u(x)| 2 dx 1/2 , < u, v >= u(x)v(x)dx with zz = |z| 2 and u v(x) = u(y)v(x -y)dy.
We shall also use

u 1 = |u(x)|dx and u ∞ = sup x∈R |u(x)|.
For a positive real β, β denotes the largest integer strictly smaller than β. Let us also define the following functional space: Definition 2.1. (Hölder class) Let β > 0, L > 0 and let l = β . The Hölder class H(β, L) on R is the set of all functions f : R -→ R such that derivative f (l) exists and verifies:

|f (l) (x) -f (l) (y)| ≤ L|x -y| β-l , ∀x, y ∈ R.
We can now define the assumptions concerning the target function g, defined by g(x) = xN (x), where N is the Lévy density. G1: g ∈ L 2 G2: g * is differentiable almost everywhere and its derivative belongs to L 1 G3(p): For p integer, |x| p-1 |g(x)|dx < ∞ G4(β): g ∈ H(β, L) G5: g exists and is uniformly bounded

The first assumption is natural to use Fourier analysis, as well as G3(1). Assumption G3(p) ensures that E|Z ∆ 1 | p < ∞. G4 is a classical regularity assumption in nonparametric estimation; it allows to quantify the bias (see [START_REF] Tsybakov | Introduction to nonparametric estimation[END_REF]). Note that G5 and G3(2) imply G1. Moreover G5 implies that g ∈ H(1, L ) so we can assume β ≥ 1. Now let us describe which kind of kernel we choose for our estimator. For m ≥ 1 an integer, we say that K : R → R is a kernel of order m if functions u → u j K(u), j = 0, 1, ..., m are integrable and satisfy

K(u)du = 1, u j K(u)du = 0, j ∈ {1, ..., m}.
Let us define the following conditions

K1: K belongs to L 1 ∩ L 2 ∩ L ∞ and K * ∈ L 1 K2(β):
The kernel K is of order β and |x| β |K(x)|dx < +∞ These assumptions are standard when working on problems of estimation by kernel methods. Note that there is a way to build a kernel of order l. Indeed, let u be a bounded integrable function such that u ∈ L 2 , u * ∈ L 1 and u(y)dy = 1, and set for any given integer l,

K(t) = l k=1 l k (-1) k+1 1 k u t k . (4)
The kernel K defined by ( 4) is a kernel of order l which also satisfies K1 (see [START_REF] Kerkyacharian | Nonlinear estimation in anisotropic multi-index denoising[END_REF] and [START_REF] Goldenshluger | Bandwidth selection in kernel density estimation: oracle inequalities and adaptive minimax optimality[END_REF]). As usual, we define K h by

∀x ∈ R K h (x) = 1 h K x h .
In all the following we fix x 0 ∈ R, x 0 = 0.

Risk bound

3.1. Risk bound for a fixed bandwidth. In this subsection, the bandwidth h is fixed, thus we omit the subscript h for the sake of simplicity: we denote ĝ = ĝh , defined in (3). The usual bias variance decomposition of the Mean Squared Error yields:

M SE(x 0 , h) := E[( g(x 0 ) -g(x 0 )) 2 ] = E[( g(x 0 ) -E[ g(x 0 )]) 2 ] + (E[ g(x 0 )] -g(x 0 )) 2 .
But the bias needs further decomposition:

b(x 0 ) 2 := (E[ g(x 0 )] -g(x 0 )) 2 ≤ 2b 1 (x 0 ) 2 + 2b 2 (x 0 ) 2 with the usual bias, b 1 (x 0 ) = K h g(x 0 ) -g(x 0 ),
and the bias resulting from the approximation of ψ ∆ (u) by 1,

b 2 (x 0 ) = E[ g(x 0 )] -K h g(x 0 ).
We can provide the following bias bound:

Lemma 3.1. Under G3(1), G4(β), G5 and if the kernel K satisfies K1 and K2(β)

|b(x 0 )| 2 ≤ c 1 h 2β + c 1 ∆ 2 with c 1 = 2 L/ β ! |K(v)||v| β dv 2 and c 1 = 2(2 g ∞ g 1 K 1 ) 2 .
Moreover, the variance is controlled as follows:

Lemma 3.2. Under G1, G2, G3(2) and if the kernel satisfies K1, we have

Var[ g(x 0 )] ≤ c 2 1 nh∆ + c 2 1 nh with c 2 = (g * ) 1 K 2 2 /(2π) and c 2 = K 2 2 g 2 2 .
Lemmas 3.1 and 3.2 lead us to the following risk bound: Proposition 3.1. Under G2, G3(2), G4(β), G5 and if K satifies K1 and K2(β), we have

M SE(x 0 , h) ≤ c 1 h 2β + c 2 1 nh∆ + c 2 1 nh + c 1 ∆ 2 . ( 5
)
Recall that ∆ = ∆(n) is such that lim n→+∞ ∆ = 0, thus 1/nh is negligible compared to 1/nh∆. For the two first terms the optimal choice of h is h opt ∝ ((n∆)

-1 2β+1
) and the associated rate has classical order O (n∆)

-2β 2β+1 . Next, a sufficient condition for ∆ 2 ≤ (n∆) -2β 2β+1 for all β is (C*) ∆ = O(n -1/3 ).
Proposition 3.2. Under the assumptions of Proposition 3.1 and under condition (C*), the choice h opt ∝ ((n∆)

-1 2β+1
) minimizes the risk bound ( 5) and gives M SE(x 0 , h opt ) = O((n∆)

-2β 2β+1 ). As a consequence E[( g(x 0 )/x 0 -N (x 0 )) 2 ] = O((n∆) -2β 2β+1 ).
We can link this result to the one of Figueroa-López (2011) who proves that his projection estimator N is such that ( N (x 0 ) -N (x 0 ))(n∆) α tends to a normal distribution for any 0 < α < β/(2β + 1). Note that when condition (C*) is not satisfied, the rate of convergence is spoiled. For example, if ∆ ∼ n -γ with 0 < γ < 1, then the rate of convergence becomes (n∆)

-2 min β 2β+1 , γ 1-γ .
The rate obtained in Proposition 3.2 turns out to be the optimal minimax rate of convergence over the class H(β, L). This result is proved in Figueroa-López (2009a) in the more general case of estimators based on the whole path of the process up to time n∆. In our case of discrete sampling, another proof is given in Section 6.3, where we prove the following result: Theorem 3.1. Assume ∆ = O(1) and ∆ -1 = O(n). Let x 0 = 0. There exists C > 0 such that for any estimator ĝn (x 0 ) based on observations Z ∆ 1 , . . . , Z ∆ n , and for n large enough,

sup g∈H(β,L) E g (ĝ n (x 0 ) -g(x 0 )) 2 ≥ C(n∆) -2β 2β+1 .
Obviously, the result is also true replacing g by the Lévy density N .

3.2. Bandwidth selection. As β is unknown, we need a data-driven selection of the bandwidth. We follow ideas given in [START_REF] Goldenshluger | Bandwidth selection in kernel density estimation: oracle inequalities and adaptive minimax optimality[END_REF] for density estimation. We introduce a set of bandwidth of the form H = { j M , 1 ≤ j ≤ M } with M an integer to be specified later. Actually it is sufficient to control h∈H h -w for some w so that more general set of bandwiths are possible. We set:

V (h) = C 0 log(n∆) nh∆ (6)
with C 0 to be specified later. Note that V (h) has the same order as the variance multiplied by log(n∆). We also define ĝh,h (x 0 ) = K h ĝh (x 0 ) = K h ĝh (x 0 ). This auxiliary estimator can also be written

ĝh,h (x 0 ) = 1 n∆ n k=1 Z ∆ k K h K h (x 0 -Z ∆ k ).
Lastly we set, as an estimator of the bias,

A(h, x 0 ) = sup h ∈H |ĝ h,h (x 0 ) -ĝh (x 0 )| 2 -V (h ) + .
Heuristically, this term has the same order as sup h ∈H E(ĝ h,h (x 0 ) -ĝh (x 0 )) 2 because the distance to the expectation is canceled by V (h ). And, if h tends to 0,

E(ĝ h,h (x 0 )) - E(ĝ h (x 0 )) tends to E(ĝ h (x 0 )) -g(x 0
). The precise link with the bias is detailed in the proofs. Then, the adaptive bandwidth h is chosen as follows: ĥ = ĥ(x 0 ) ∈ arg min

h∈H {A(h, x 0 ) + V (h)}.
This can be seen as a bias-variance trade-off since V (h) is close to the variance. Before to study the performance of our final estimator ĝĥ (x 0 ), let us clarify the observation context. We still work in the high frequency framework, and we have seen that we need conditon (C*). Thus, the assumption on the observation step is the following S: ∆ → 0 and n∆ → ∞. Moreover ∆ ≤ 1 and ∆ = O(n -1/3 ) We can now state the following oracle inequality.

Theorem 3.2. We use a kernel satisfying K1 and a set of bandwidth H = { j M , 1 ≤ j ≤ M } with M = O((n∆) 1/3 ). Assume that g satisfies G1, G2, G3(5) and take V (h) such that

(7) C 0 = C 0 (c) = c 2π K 2 2 (g * ) 1 + g * 2 2 with c ≥ 16 max(1, K ∞ ). Then, under scheme S, E[|g(x 0 ) -ĝĥ (x 0 )| 2 ] ≤ C inf h∈H ess sup |g -E[ĝ h ]| 2 + V (h) + log(n∆) n∆ .
Thus our estimator ĝĥ has a risk as good as any of the collection (ĝ h ) h∈H , up to a logarithmic term. The pointwise control of the bias has been replaced by an uniform control. Actually, it is possible to keep the pointwise risk in the right term at the cost of a supplementary term sup h ∈H |K h b h (x 0 )| 2 . Although our estimator is not linear (we have an extra bias), it is exactly the same situation as in [START_REF] Goldenshluger | General selection rule from a family of linear estimators[END_REF], and we can conjecture it is in some sense unavoidable.

Note that the theorem is valid for c large enough, say c ≥ c 0 . In the proof, we obtain the upper bound 16 max(1, K ∞ ) for c 0 , unfortunately we can conjecture that this bound is not the optimal one. To obtain a sharper bound we have tuned c 0 in the simulation study.

The definition of the estimator uses (g * ) 1 and g * 2 2 , but these quantities can be estimated with a preliminar estimator of g * . More precisely, we set

K * 0 = 1 [-1,1] and (g * ) 1 = 1 n∆ n k=1 (Z ∆ k ) 2 K * 0 (uh 1 )e iuZ ∆ k du with h 1 = (n∆) -1/3 , g * 2 2 = ĝ * h 2 2 2 = 1 n∆ n k=1 Z ∆ k K * 0 (uh 2 )e iuZ ∆ k 2 du with h 2 = (n∆) -1/3 .
We introduce the following smoothness condition: a function ψ belongs to the Sobolev space Sob(1) if |ψ * (u)| 2 |u| 2 du < ∞ (this means that ψ has a derivative which is squareintegrable). Then, reinforcing the conditions on g, we obtain a similar theorem with an empirical C 0 .

Theorem 3.3. We use a kernel satisfying K1 and K2(1) and M = O((n∆) 1/3 ). Assume that g satisfies G2, G3(32), G4(1), G5. Assume also that g and xg(x) belong to Sob(1).

In the definition of ĥ, replace V (h) by V (h) = C 0 log(n∆)/(nh∆) where

C 0 = c 2π K 2 (g * ) 1 + g * 2 2 with c ≥ 32 max(1, K ∞ ). Then, under scheme S, E[|g(x 0 ) -ĝĥ (x 0 )| 2 ] ≤ C inf h∈H ess sup |g -E[ĝ h ]| 2 + V (h) + log(n∆) n∆ ,
where V (h) is defined by ( 6) and (7).

Let us now conclude with the consequence of this theorem in term of rate of convergence. As already explained, as we need assumption G5 to control the bias, we can assume β ≥ 1. Then h opt ∝ (log(n∆)/n∆) 1/(2β+1) ≥ (n∆) -1/3 belongs to H as soon as M is larger than a constant times (n∆) 1/3 . Hence we can state the following corollary.

Corollary 3.1. Assume that g satisfies G2, G3(5), G4(β) with β ≥ 1 and G5. We choose a kernel satisfying K1 and K2(β), and M = (n∆) 1/3 . Take C 0 as in Theorem 3.2 (or as in Theorem 3.3 with assumptions of this latter theorem). Then, under scheme S,

E[|g(x 0 ) -ĝĥ (x 0 )| 2 ] = O (log(n∆)/n∆) -2β 2β+1 .
Then the price to pay to adaptivity is a logarithmic loss in the rate. Nevertheless this phenomenon is known to be unavoidable in pointwise estimation (see [START_REF] Butucea | Exact adaptive pointwise estimation on Sobolev classes of densities[END_REF]). Thus ĝĥ (x 0 ) (resp. ĝĥ (x 0 )/x 0 ) is an adaptive estimator for g(x 0 ) (resp. N (x 0 )).

Examples and Simulations

We have implemented the estimation method for four different processes (listed in Examples 1-4 below). As usual in nonparemetric estimation, to obtain the rate of convergence, the kernel has to be of order larger than β, or, equivalently, the smoothness has to be smaller than the order of the kernel. In practice this does not play a big role, so we use the kernel described in (4) with l = 2 and u the Gaussian density.

The bandwidth set has been fixed to H = { j 2M , 1 ≤ j ≤ M } with M = 2(n∆) 1/3 . For the implementation, a difficulty is the proper calibration of the constant c in (7). This is usually done by a large number of preliminary simulations. We have chosen c = 0.1 as the adequate value for a variety of models and number of observations (as previously announced, this practical c is different from the theoretical one). The estimation and adaptation are done for 50 points x 0 on the abscissa interval. For clarity, we have computed the Mean Integrated Square Error (MISE) of the estimators. We also give the M SE(x 0 ) = M SE(x 0 , ĥ) in some points x 0 , by way of example. Illustratively, Figures 1 and2 plot ten estimated curves corresponding to our four examples with in the first column ∆ = 0.2, n = 5.10 3 , and in the second ∆ = 0.05, n = 5.10 4 .

Example 1. Let L t = Nt i=1 Y i , where (N t ) is a Poisson process with constant intensity λ and (Y i ) is a sequence of i.i.d random variables with density f independent of the process (N t ). Then, (L t ) is a Lévy process with characteristic function

(8) ψ t (u) = exp λt R (e iux -1)f (x)dx .
Its Lévy density is N (x) = λf (x) and thus g(x) = λxf (x). For our first example, we choose λ = 2 and f such that g(x) = xf (x) = (1/2) x/2 for 0 < x ≤ 2. Then assumption G4(1/2) holds (on (0, 2)), but not G4(β) for other β. Since β is small, the rate of convergence is slow. The discontinuity in 2 damages the estimation as it can be seen in Figure 1.

Example 2. Let α > 0, γ > 0. The Lévy-Gamma process (L t ) with parameters (γ, α) is such that, for all t > 0, L t has Gamma distribution with parameters (γt, α), i.e the density:

α γt Γ(γt) x γt-1 e -αx 1 x≥0 .
The Lévy density is N (x) = γx -1 e -αx 1 x>0 so that g(x) = γe -αx 1 x>0 satisfies assumptions G1, G2 and G3(p). Here we choose α = γ = 1. This example allows to study the role of the discontinuity in 0, which invalidates assumptions G4-G5. It is simulated in [START_REF] Ueltzhöfer | An oracle inequality for penalised projection estimation of lévy densities from high-frequency observations[END_REF] who obtain a better MISE (for N ) than ours because of this singularity. Nevertheless we can observe that the estimation become very good if we move away from 0. Example 3. For our third example, we also choose a compound Poisson process, but with f the Gaussian density with variance δ 2 . Thus g(x) = λxf (x) = λxe -x 2 /(2δ 2 ) /(δ √ 2π) and g * (u) = iλδue -δ 2 u 2 /2 . Assumptions G1, G2, G3(p),G5 hold for g. Moreover g belongs to a Hölder class of regularity β for all β > 0. Thus the rate is close to (n∆/ log(n∆)) -1 , and the good performance of our estimator is visible on Figure 2. Note that is the jump part of the so-called Merton model used for describing the log price in financial modeling.

Here we choose λ = 0.1 and δ = 0.05.

Example 4. Our last example is the Variance Gamma process, as described in [START_REF] Madan | The variance gamma process and option pricing[END_REF]. It is used for modeling the dynamics of the logarithm of stock prices. The process is obtained in evaluating a Brownian motion at a time given by a Lévy-Gamma process. Denoting (B t ) a standard Brownian motion, and (X t ) a Lévy-Gamma process with parameters (1/ν, 1, ν) independent of (B t ), we set L t = θX t + σB Xt . Then L t is a Lévy process, with

g(x) = x exp(θx/σ 2 ) ν|x| exp - 1 σ 2 ν + θ 2 σ 2 |x| .
As in example 2, there is a discontinuity in 0. Here we choose θ = 0.1, σ 2 = 0.1, ν = 0.5.

Irregular sampling

For high frequency data, it is frequent that the sampling is irregular, i.e. the interval ∆ is not necessarily the same at each time. In this section we consider the following framework. The observations are (L t k , k = 1, ..., n) where (L t ) is still a Lévy process with characteristic function (1). For each k ≥ 1, we denote ∆ k = t k -t k-1 the sampling intervals. Notice that it includes the previous case when for each k, ∆ k = ∆. The increments are denoted by

Z k = L t k -L t k-1 .
In this context of irregular sampling, they are still independent but with non-identical distribution: Z k has the same law than L ∆ k . To define an estimator, we observe that E Z k e iuZ k = ∆ k ψ ∆ k (u)g * (u), and then Thus, denoting ∆ =

E 1 n k=1 ∆ k n k=1 Z k e iuZ k = n k=1 ∆ k ψ ∆ k (u) n k=1 ∆ k g * (u).
1 n n k=1 ∆ k , we introduce (9) ĝ * h (u) = 1 n ∆ n k=1 Z k e iuZ k K * (hu), ĝh (x) = 1 n ∆ n k=1 Z k K h (x -Z k ).
Additionally, for all real δ, we denote ∆ δ = 1 n n k=1 ∆ δ k . We can bound the Mean Squared Error of this estimate: Proposition 5.1. Under G2, G3(2), G4(β), G5 and if K satifies K1 and K2(β), we have

M SE(x 0 , h) ≤ c 1 h 2β + c 2 1 nh ∆ + c 2 ∆ 2 nh ∆2 + c 1 ∆ 2 ∆ 2 (10) with c 1 = 2 L/ β ! |K(v)||v| β dv 2 , c 1 = 2(2 g ∞ g 1 K 1 ) 2 , c 2 = (g * ) 1 K 2 2 /(2π), c 2 = K 2 2 g 2 2 .
The proof is similar to the case of regular sampling, therefore it is omitted.

In this section, we are still interested in the high frequency context: the asymptotic framework is S': ∆ → 0 and n ∆ → ∞ when n → ∞. We shall also assume that (∆ 2 ) 2 / ∆ ≤ 1 and As already noticed in Comte and Genon-Catalot (2010a), other estimation strategies than (9) are possible. For each real δ, we obtain an estimator by setting ĝh (x) = 1

(∆ 2 ) 2 ∆ = O(n -1 ). (11) Condition (11) is verified for instance if ∆ k = Ck -α with α ∈ [1/3, 1].
n∆ δ+1 n k=1 ∆ δ k Z k K h (x -Z k ).
Under suitable conditions, this estimate has a MSE bounded by a constant times (n∆

δ+1 2 / ∆ 2δ+1 ) - 2β
2β+1 . But, for all δ, by the Schwarz inequality, ∆ δ+1 2 /∆ 2δ+1 ≤ ∆. That is why we prefer estimator (9).

To build an adaptive estimator, we use the same method of bandwidth selection. The set of bandwidth is still

H = { j M , 1 ≤ j ≤ M }. We also define ĝh,h (x 0 ) = K h ĝh (x 0 ) = 1 n ∆ n k=1 Z k K h K h (x 0 -Z k )
and we set as previously

A(h, x 0 ) = sup h ∈H |ĝ h,h (x 0 ) -ĝh (x 0 )| 2 -V (h ) + with V (h) = C 0 log(n ∆) nh ∆ .
Then the estimator is ĝĥ (x 0 ) with ĥ = ĥ(x 0 ) ∈ arg min h∈H {A(h, x 0 ) + V (h)}. We can state the following oracle inequality (the proof is very similar to the one of Theorem 3.2 and is therefore omitted).

Theorem 5.1. We use a kernel satisfying K1 and M = O((n ∆) 1/3 ). Assume that g satisfies G1, G2, G3(5) and take

(12) C 0 = c 2π K 2 2 (g * ) 1 + g * 2 2
with c ≥ 16 max(1, K ∞ ). Then, under scheme S',

E[|g(x 0 ) -ĝĥ (x 0 )| 2 ] ≤ C inf h∈H ess sup |g -E[ĝ h ]| 2 + V (h) + log(n ∆) n ∆ .
Moreover, if g satisfies G5, G4(β) with β ≥ 1 and the kernel satisfying K1 and K2(β) , and M = (n ∆) 1/3 , then

E[|g(x 0 ) -ĝĥ (x 0 )| 2 ] = O (log(n ∆)/n ∆) -2β 2β+1 .
Thus the rate of convergence in this case of irregular sampling is (log(n ∆)/n ∆)

-2β 2β+1
provided that (∆ 2 ) 2 / ∆ = O(n -1 ).

Proofs

Let us first state two classical propositions (see for instance Proposition 2.1 in Comte and Genon-Catalot (2009) for a proof). Proposition 6.1. Denote by P ∆ the distribution of Z ∆ 1 and define µ ∆ (dx

) = ∆ -1 xP ∆ (dx). If R |x|N (x) < ∞, the distribution µ ∆ has a density h ∆ given by h ∆ (x) = g(x -y)P ∆ (dy) = Eg(x -Z ∆ 1 ). Proposition 6.2. Let p ≥ 1 an integer such that R |x| p-1 |g(x)|dx < ∞. Then E(|Z ∆ 1 | p ) < ∞ and E[(Z ∆ 1 ) p ] = ∆ R x p-1 g(x)dx + o(∆). Moreover, if g is integrable, E(|Z ∆ 1 |) ≤ 2∆ g 1 .
6.1. Proof of Lemma 3.1. First, we study b 2 (x 0 ) using Proposition 6.1:

b 2 (x 0 ) = 1 h∆ E Z ∆ 1 K x 0 -Z ∆ 1 h - 1 h K x 0 -u h g(u)du = 1 h K x 0 -u h E[g(u -Z ∆ 1 ) -g(u)]du.
Now, applying the mean value theorem to g, we get

|b 2 (x 0 )| = 1 h K x 0 -u h E[-Z ∆ 1 g (u Z 1 )]du with u Z 1 ∈ [u -Z ∆ 1 , u ] ≤ g ∞ K 1 E Z ∆ 1 .
From the results of Proposition 6.2 we obtain

|b 2 (x 0 )| ≤ 2 g ∞ K 1 g 1 ∆. ( 13
)
To study b 1 (x 0 ) = K h g(x 0 ) -g(x 0 ), it is sufficient to use Taylor's theorem and G4(β) (this is a classic computation, see [START_REF] Tsybakov | Introduction to nonparametric estimation[END_REF] for details) and we obtain

|b 1 (x 0 )| ≤ h β L l! |K(v)||v| β dv. (14)
Gathering ( 13) and ( 14) completes the proof of Lemma 3.1. 6.2. Proof of Lemma 3.2. As the Z ∆ k are i.i.d., we have:

Var[ g(x 0 )] = Var 1 nh∆ n k=1 Z ∆ k K x 0 -Z ∆ k h = 1 n(h∆) 2 Var Z ∆ 1 K x 0 -Z ∆ 1 h . Thus, Var[ g(x 0 )] ≤ 1 n(h∆) 2 E (Z ∆ 1 ) 2 K 2 x 0 -Z ∆ 1 h . Writing K 2 x 0 -Z ∆ 1 h = 1 2π K * (u)e -i (x 0 -Z ∆ 1 )u h du 2 , we obtain with v = u/h Var[ g(x 0 )] ≤ 1 n∆ 2 E (Z ∆ 1 ) 2 1 2π K * (vh)e -i(x 0 -Z ∆ 1 )v dv 2 ≤ 1 n∆ 2 (2π) 2 E Z ∆ 1 e iZ ∆ 1 v K * (vh)e -ix 0 v Z ∆ 1 e iZ ∆ 1 u K * (uh)e -ix 0 u dvdu . Using Fubini and E[(Z ∆ 1 ) 2 e iZ ∆ 1 (v-u) ] = -ψ ∆ (v -u) we find Var[ g(x 0 )] ≤ 1 n∆ 2 (2π) 2 | -ψ ∆ (v -u)K * (vh)K * (uh)|dvdu.
Now the following formula

ψ ∆ = i∆ψ ∆ g * + i∆ψ ∆ g * = -∆ 2 ψ ∆ g * 2 + i∆ψ ∆ g * .
gives Var[ g(x 0 )] ≤ T 1 + T 2 with

T 1 = 1 n∆ 2 (2π) 2 |∆ 2 ψ ∆ (v -u)(g * ) 2 (v -u)K * (vh)K * (uh)|dvdu, T 2 = 1 n∆ 2 (2π) 2 |∆ψ ∆ (v -u)(g * ) (v -u)K * (vh)K * (uh)|dvdu.
We first bound T 2 :

T 2 ≤ 1 n∆(2π) 2 |ψ ∆ (v -u)||(g * ) (v -u)||K * (vh)| 2 dvdu × |ψ ∆ (v -u)||(g * ) (v -u)||K * (uh)| 2 dvdu ≤ 1 n∆(2π) 2 |K * (vh)| 2 dv |ψ ∆ (z)||(g * ) (z)|dz ≤ 1 nh∆(2π) 2 |K * (u)| 2 du |(g * ) (z)|dz, because |ψ ∆ (z)| ≤ 1 ≤ K 2 2 2πnh∆ |(g * ) (z)|dz
where (g * ) exists and is integrable by G2. Following the same line for the study of T 1 , we get

T 1 ≤ K 2 2 2πnh |(g * ) 2 (z)|dz ≤ K 2 2 g 2 2 nh .
This completes the proof of Lemma 3.2.

6.3. Proof of the lower bound. Here we prove Theorem 3.1 The essence of the proof is to build two functions g 0 and g 1 which are far in term of pointwise distance but with close associated distribution. Let

g 0 (x) = xf λ (x) = 1 π λx 1 + (λx) 2
where f λ is the density of the Cauchy distribution C(0, λ) with scale parameter λ. Here λ is a positive and small enough real (it will be made precise later). Now let K a infinitely differentiable and even function such that K = 0, K(0) = 0 and |K (k) (x)| ≤ |x| -2-k for |x| large enough (say for |x| > B) and for all k ≥ 0. We shall also use that the Fourier transform K * exists, is differentiable almost everywhere with K * and (K * ) ∈ L 1 ∩L 2 ∩L ∞ . Take for instance K equals to the difference between the Cauchy density and the normal density. Using this auxiliary function K, we can define

g 1 (x) = g 0 (x) + ch β n K x -x 0 h n x
where c is a constant to be specified later and

h n = (n∆) -1 2β+1 .
We denote N 0 (x) = g 0 (x)/x and N 1 (x) = g 1 (x)/x. Remark that if L 0,t = Nt i=1 Y i is a compound Poisson process with N t a Poisson process of intensity 1 and Y i Cauchy C(0, λ) variables, then its characteristic function is

ψ 0,t (u) = exp (t R (e iux -1)N 0 (x)dx) and Z 0,∆ k = L 0,k∆ -L 0,(k-1)∆ has distribution P 0 (dx) = e -∆ δ 0 (dx) + ϕ 0 (x)dx with ϕ 0 (x) = ∞ k=1 e -∆ ∆ k k! f k λ (x)
(where denotes the convolution). Moreover N 1 is a density if c small enough. Indeed the definition of K guarantees that

N 1 (x)dx = N 0 (x)dx + ch β n K x-x 0 hn dx = 1. And to ensure the positivity of N 1 , it is sufficient to prove that |N 1 -N 0 | ≤ N 0 . But, if |x| > |x 0 | + Bh n , N -1 0 (x)|N 1 (x) -N 0 (x)| ≤ Cch β+2 n x 2 |x -x 0 | -2 ≤ 1 for c small enough, and if |x| ≤ |x 0 | + Bh n , N -1 0 (x)|N 1 (x) -N 0 (x)| ≤ Cch β n (1 + (λ(|x 0 | + Bh n )) 2 ) K ∞ ≤ 1 for c small enough. Then, if L 1,t = Nt i=1 Y i with N t a
Poisson process of intensity 1 and Y i random variables with density N 1 , it is a Lévy process with Lévy measure N 1 (x)dx. We denote ψ 1,∆ the characteristic function of L 1,∆ with distribution P 1 , and ϕ 1 the function such that P 1 (dx) = e -∆ δ 0 (dx) + ϕ 1 (x)dx. Now let us denote for two probability measures P and Q, χ 2 (P, Q) = (dP/dQ -1) 2 dQ. We shall use the following result stated in [START_REF] Tsybakov | Introduction to nonparametric estimation[END_REF] (section 2.2 andTheorem 2.2): Theorem. Let Θ be a nonparametric class of functions containing the function θ to estimate, and {P θ , θ ∈ Θ} be a family of probability measures on a measurable space (X , A) associated with the data. Let d be a distance on Θ. Let θ 0 and θ 1 be two functions in

Θ such that d(θ, θ 1 ) ≥ 2ψ n . If χ 2 (P 1 , P 0 ) ≤ α < ∞ then inf θn sup θ∈Θ E θ ψ -2 n d 2 ( θn , θ) ≥ max e -α 4 , 1 -α/2 2
Then it is sufficient to show that 1) g 0 , g 1 belong to H(β, L),

2) |g 1 (x 0 ) -g 0 (x 0 )| ≥ C(n∆) -β 2β+1 , 3) χ 2 (P n 1 , P n 0 ) ≤ C < ∞ where P n 1 (resp. P n 0 ) is the distribution of a sample Z ∆ 1 , . . . , Z ∆ n s.t the associated Lévy process L 0 (resp. L 1 ) has Lévy measure N 0 (x)dx (resp. N 1 (x)dx).
In the following we denote all constants by C, even if it changes from line to line.

Proof of 1). Belonging to the Hölder space

To prove that our hypotheses belong to H(β, L), it is sufficient to show that, for i = 0, 1, g (k+1) i p ≤ L where k = β and p -1 = 1 + k -β. Indeed Hölder inequality gives

|g (k) i (x) -g (k) i (y)| = g (k+1) i (v)1 [x,y] (v)dv ≤ g (k+1) i p |x -y| β-k for all x, y.
When x goes to infinity, g

(k+1) 0 (x) = Cλ -1 x -k-2 + o(x -k-2 ) so it belongs to L p since p(k + 2) = (k + 2)/(k + 1 -β) > 1. Choosing λ small enough ensures g (k+1) 0 p ≤ L/2 ≤ L. Now to study g 1 , we can write (g 1 -g 0 ) (k+1) (x) = cxK (k+1) x -x 0 h n h β-k-1 n + c(k + 1)K (k) x -x 0 h n h β-k n .
Let us see if this two terms are in L p . Writing x = x -x 0 + x 0 and changing variables

xK (k+1) x -x 0 h n p dx ≤ 2 p-1 h p+1 n |vK (k+1) (v)| p dv + 2 p-1 |x 0 | p h n |K (k+1) (v)| p dv.
In the same way

K (k) x -x 0 h n p dx ≤ h n |K (k) (v)| p dv.
These integrals are finite since 2+k) for v large enough and p(k

|K (k) (v)| ≤ v -(2+k) so that |vK (k+1) (v)| ≤ v -(
+ 2) = (k + 2)/(k + 1 -β) > 1. Thus (g 1 -g 0 ) (k+1) p p ≤ Cc p (h n h p(β-k-1) n + h n h p(β-k) n ) ≤ Cc p h p(1/p+β-k-1) n ≤ Cc p ≤ (L/2) p
for suitable c. Then g 1 -g 0 belongs to H(β, L/2) and g 1 belongs to H(β, L).

Proof of 2). Rate

By assumption, x 0 = 0 and we can see that

|g 1 (x 0 )-g 0 (x 0 )| = ch β n |K(0)x 0 | with K(0) = 0. Since h n = (n∆) -1
2β+1 , this quantity has the announced order of the rate: (n∆)

-β 2β+1 .
Proof of 3). Chi-square divergence Since the observations are i.i.d., χ 2 (P n 1 , P n 0 ) = (1 + χ 2 (P 1 , P 0 )) n -1. Thus, it is sufficient to prove that χ 2 (P 1 , P 0 ) = O(n -1 ) where

χ 2 (P 1 , P 0 ) = x =0 ϕ 1 (x) ϕ 0 (x) -1 2 ϕ 0 (x)dx.
Indeed P 1 ({0}) = e -∆ = P 0 ({0}). Now let us remark that

ϕ 0 (x) = ∞ k=1 e -∆ ∆ k k! f k λ (x) ≥ e -∆ ∆f λ (x) ≥ ∆e -C λπ -1 /(1 + (λx) 2 ) since ∆ is bounded. Then ϕ 0 (x) ≥ C -1 ∆x -2 for |x| large enough, say |x| ≥ A and ϕ 0 (x) ≥ C -1 ∆ for |x| ≤ A. Next we write χ 2 (P 1 , P 0 ) = x =0 (ϕ 1 (x) -ϕ 0 (x)) 2 (ϕ 0 (x)) -1 dx = I 1 +I 2
where I 1 is the integral for |x| < A and I 2 for |x| ≥ A. We will bound these two terms separately. Since ϕ 0 (x) ≥ C -1 ∆ for |x| small

I 1 = |x|<A (ϕ 1 (x) -ϕ 0 (x)) 2 (ϕ 0 (x)) -1 dx ≤ C∆ -1 |x|<A (ϕ 1 (x) -ϕ 0 (x)) 2 dx.
For i = 0, 1, the Fourier tranform of ϕ i is ψ i,∆ (u) -P i ({0}). Thus Parseval equality gives

I 1 ≤ C∆ -1 |ψ 1,∆ (u) -ψ 0,∆ (u)| 2 du.
In order to get a bound on |ψ 1,∆ -ψ 0,∆ |, we apply the mean value theorem:

|ψ 1 (u) -ψ 0 (u)| ≤ sup z∈Iu |e z ||∆ (e iux -1)(N 1 (x) -N 0 (x))dx|
where

I u is the segment in C between a u = ∆ (e iux -1)N 0 (x)dx and b u = ∆ (e iux - 1)N 1 (x)dx. But (e iux -1)(N 1 (x) -N 0 (x))dx = ch β n (e iux -1)K x -x 0 h n dx = ch β+1 n e iux 0 K * (h n u).
Thus

|ψ 1 (u) -ψ 0 (u)| ≤ (sup z∈Iu e R(z) )∆ch β+1 n |K * (h n u)| where R(x) means the real part of x. We can compute R(a u ) = a u = ∆(N * 0 (u) -1) = ∆(exp(-|u/λ|) -1) ≤ 0 and R(b u ) = R(∆(N * 0 (u) -1 + (N 1 -N 0 ) * (u))) = ∆(N * 0 (u) -1 + ch β+1 n R(K * (h n u)e iux 0 )). Since K is even, R(b u ) = ∆(exp(-|u/λ|) -1 + ch β+1 n K * (h n u) cos(ux 0 )) ≤ c∆h β+1 n K * ∞ ≤ C so that |ψ 1 (u) -ψ 0 (u)| ≤ e C ∆ch β+1 n |K * (h n u)|. ( 15
)
Then

I 1 ≤ C∆ -1 ∆h β+1 n K * (h n u) 2 du ≤ C∆h 2β+1 n . (16)
Let us now bound the term I 2 , using that ϕ 0 (x) ≥ C -1 ∆x -2 for |x| large enough

I 2 = |x|≥A (ϕ 1 (x) -ϕ 0 (x)) 2 ϕ 0 (x) dx ≤ C∆ -1 (ϕ 1 (x) -ϕ 0 (x)) 2 x 2 dx.
But F = ϕ 1 -ϕ 0 has Fourier transform

F * = ψ 1,∆ -ψ 0,∆ = exp(∆(e -|u/λ| + ch β+1 n K * (h n u)e iux 0 -1)) -exp(∆(e -|u/λ| -1)
) and this function is differentiable everywhere except at u = 0, with derivative

F * = ∆γ 1 ψ 1,∆ -∆γ 0 ψ 0,∆ where γ 0 (u) = -sign(u).e -|u/λ| /λ, γ 1 (u) = γ 0 (u) + ch β+1 n e iux 0 (ix 0 K * (h n u) + h n K * (h n u)). Let us now prove that the Fourier transform of F * is -2πixF (-x). Let us write the factorization (17) ∆ -1 F * = γ 1 ψ 1,∆ -γ 0 ψ 0,∆ = (γ 1 -γ 0 )ψ 1,∆ + γ 0 (ψ 1,∆ -ψ 0,∆ )
with |ψ 1,∆ | ≤ 1. Since K * and K * are uniformly bounded, γ 1 -γ 0 is bounded as well.

In the same way, the inequality (15) entails that ψ 1,∆ -ψ 0,∆ ∞ < ∞, so that F * is bounded. Thus F * is Lipschitz and absolutely continuous. Moreover, using again (17), we can see that F * is integrable. Then, according to [START_REF] Rudin | Real and complex analysis[END_REF], the Fourier transform of F * is -ixF * * (x) (it is in fact a simple integration by parts). Since F * is integrable, F * * (x) = 2πF (-x) almost everywhere, and we have proved that (F * ) * (x) = -2πixF (-x) a.e.. Next, the Parseval equality provides |xF

(x)| 2 dx = (2π) -1 |F * (u)| 2 du. Thus I 2 ≤ C∆ -1 |xF (x)| 2 dx ≤ C∆(2π) -1 |γ 1 ψ 1,∆ -γ 0 ψ 0,∆ | 2 .
Hence, using the factorization (17) we can split

I 2 ≤ π -1 C∆(I 2,1 + I 2,2 ) with I 2,1 = |γ 1 -γ 0 | 2 , I 2,2 = |γ 0 (ψ 1,∆ -ψ 0,∆ )| 2 .
Using the definition of γ 1 , we compute

I 2,1 = c 2 h 2β+2 n |ix 0 K * (h n u) + h n K * (h n u)| 2 du ≤ 2c 2 h 2β+1 n x 2 0 |K * | 2 + h n |K * | 2 ≤ 4πc 2 h 2β+1 n x 2 0 |K| 2 + h n |xK(x)| 2 ≤ Ch 2β+1 n . (18)
Now, in order to deal with I 2,2 , we use the previous bound (15

) on |ψ 1,∆ -ψ 0,∆ | I 2,2 ≤ Cc 2 ∆ 2 h 2β+2 n |γ 0 (u)K * (h n u)| 2 du ≤ Cc 2 ∆ 2 h 2β+2 n K * ∞ γ 0 2 2 ≤ Ch 2β+1 n ( 19
)
since ∆ is bounded.

Finally, by gathering ( 16), ( 18) and ( 19), since h n = (n∆) 1/(2β+1) , we get

χ 2 (P 1 , P 0 ) ≤ C∆h 2β+1 n = O(n -1 ).
This ends the proof of Theorem 3.1.

6.4. Proof of Theorem 3.2. The goal is to bound E[|g(x 0 ) -ĝĥ (x 0 )| 2 ].
To do this, we fix h ∈ H. We write

|g(x 0 ) -ĝĥ (x 0 )| ≤ |ĝ ĥ(x 0 ) -ĝh, ĥ(x 0 )| + |ĝ h, ĥ(x 0 ) -ĝh (x 0 )| + |ĝ h (x 0 ) -g(x 0 )|.
So we have

|g(x 0 ) -ĝĥ (x 0 )| 2 ≤ 3R 2 1 + 3R 2 2 + 3R 2 3 with R 1 = |ĝ ĥ(x 0 ) -ĝh, ĥ(x 0 )|, R 2 = |ĝ h, ĥ(x 0 ) -ĝh (x 0 )|, R 3 = |ĝ h (x 0 ) -g(x 0 )|.

According to the definition of A(h):

A(h) ≥ |ĝ ĥ(x 0 ) -ĝh, ĥ(x 0 )| 2 -V ( ĥ) = R 2 1 -V ( ĥ).

So R 2 1 ≤ A(h) + V ( ĥ). In the same way, A( ĥ) ≥ |ĝ h, ĥ(x 0

) -ĝh (x 0 )| 2 -V (h) = R 2 2 -V (h). So R 2 2 ≤ A( ĥ) + V (h). Therefore, |g(x 0 ) -ĝĥ (x 0 )| 2 ≤ 3(A(h) + V ( ĥ)) + 3(A( ĥ) + V (h)) + 3|ĝ h (x 0 ) -g(x 0 )| 2 .
Now, by definition of ĥ, A( ĥ) + V ( ĥ) ≤ A(h) + V (h). This allows us to write

|g(x 0 ) -ĝĥ (x 0 )| 2 ≤ 6A(h) + 6V (h) + 3|ĝ h (x 0 ) -g(x 0 )| 2 . Let us denote b h (x 0 ) = E[ĝ h (x 0 )] -g(x 0 ) and b h,2 (x 0 ) = E[ĝ h (x 0 )] -K h g(x 0
) (these are the same notation as in Lemma 3.1, but with subscript h). Thus

E[|g(x 0 ) -ĝĥ (x 0 )| 2 ] ≤ 6E[A(h)] + 6V (h) + 3b 2 h (x 0 ) + 3Var(ĝ h (x 0 )) ≤ 6E[A(h)] + 3b 2 h (x 0 ) + (6 + 3/c)V (h). It remains to bound E[A(h)]. Let us denote by g h,h = E[ĝ h,h ] and g h = E[ĝ h ]. We write (20) ĝh,h -ĝh = ĝh,h -g h,h -ĝh + g h + g h,h -g h ,
and we study the last term of the above decomposition. We have

g h,h (x 0 ) -g h (x 0 ) = E[ĝ h,h (x 0 ) -ĝh (x 0 )] = E[K h ĝh (x 0 ) -ĝh (x 0 )] = K h E[ĝ h (x 0 ) -g(x 0 )] + K h g(x 0 ) -E[ĝ h (x 0 )].
This can be written:

g h,h (x 0 ) -g h (x 0 ) = K h b h (x 0 ) -b h ,2 (x 0 ). Now, using inequality (13), |b h ,2 (x 0 )| ≤ 2 g ∞ K 1 g 1 ∆, so that |g h,h (x 0 ) -g h (x 0 )| 2 ≤ 2|K h b h (x 0 )| 2 + O(∆ 2 ) ≤ 2 K 2 1 ess sup |b h | 2 + O(∆ 2 ). ( 21
)
Then by inserting (21) in decomposition (20), we find:

A(h) = sup h {|ĝ h,h (x 0 ) -ĝh (x 0 )| 2 -V (h )} + ≤ 3 sup h {|ĝ h,h (x 0 ) -g h,h (x 0 )| 2 -V (h )/6} + +3 sup h {|ĝ h (x 0 ) -g h (x 0 )| 2 -V (h )/6} + + 6 K 2 1 ess sup |b h | 2 + O(∆ 2 ). ( 22
)
We can prove the following concentration result: Proposition 6.3. Assume that g satisfies G1, G2, G3(5) , K satisfies K1, M = O((n∆) 1/3 ), ∆ ≤ 1 and take c in (7) such that c ≥ 16 max(1, K ∞ ). Then

E sup h {|ĝ h (x 0 ) -g h (x 0 )| 2 -V (h )/6} + = O log(n∆) n∆ , (23) E sup h {|ĝ h,h (x 0 ) -g h,h (x 0 )| 2 -V (h )/6} + = O log(n∆) n∆ . ( 24
)
This proposition is proved in Section 6.6 page 23. Inequalities ( 23) et (24) together with ( 22) imply

E[|g(x 0 ) -ĝĥ (x 0 )| 2 ] ≤ C 1 ess sup |b h | 2 + C 2 V (h) + C 3 log(n∆) n∆ .
This completes the proof of Theorem 3.2.

6.5. Proof of Theorem 3.3. In all this proof, we shall use the following notation:

θ∆ (u) = 1 n n k=1 Z ∆ k e iZ ∆ k u , η∆ (u) = 1 n n k=1 (Z ∆ k ) 2 e iZ ∆ k u ,
and θ ∆ (u) = E θ∆ (u), η ∆ (u) = Eη ∆ (u). We also denote f (x) = xg(x), so that f * (u) = -i(g * ) (u) is estimated by f * h 1 = η∆ (u)K * (uh 1 )/∆. We shall use the following Lemma. Lemma 6.1 (Proposition 2.3 in [START_REF] Comte | Nonparametric estimation for pure jump Lévy processes based on high frequency data[END_REF]). Assume that g is integrable, then we have:

(25) |ψ ∆ (u) -1| ≤ |u|∆ g 1 . Moreover under G3(2p), for p ≥ 1, (26) ∆ -2p E θ∆ (v) -θ ∆ (v) 2p ≤ C(n∆) -p . Now, let Ω = { g * -ĝ * h 2 2 ≤ g * 2 (1 -1/ √ 2) and f * -f * h 1 1 ≤ f * 1 /2},
where h 1 = (n∆) -1/3 = h 2 , as defined page 7. The proof is decomposed in three steps.

First we shall prove that the inequality is true on Ω, i.e.

E[|g(x 0 ) -ĝĥ (x 0 )| 2 1 Ω ] ≤ C inf h∈H ess sup |g -E[ĝ h ]| 2 + V (h) + log(n∆) n∆ .
The second step is to show the rough upper bound

E[|g(x 0 ) -ĝĥ (x 0 )| 4 ] ≤ C(n∆) 2/3 .
Finally we will show that P(Ω c ) ≤ C(n∆) -8/3 . Consequently

E[|g(x 0 ) -ĝĥ (x 0 )| 2 1 Ω c ] ≤ E[|g(x 0 ) -ĝĥ (x 0 )| 4 ]P(Ω c ) ≤ C(n∆) -1
and the theorem is proved.

• First step:

Following the proof of Theorem 3.2, we can obtain

|g(x 0 ) -ĝĥ (x 0 )| 2 1 Ω ≤ 6A(h)1 Ω + 6 V (h)1 Ω + 3|g(x 0 ) -ĝh (x 0 )| 2 . Now, let us remark that on Ω 1 2 g * 2 2 ≤ ĝ * h 2 2 2 ≤ (2 -1/ √ 2) 2 g * 2 2 and 1 2 f * 1 ≤ f * h 1 1 ≤ 3 2 f * 1 with f * 1 = (g * ) 1 , so that on Ω, 1 2 C 0 ≤ C 0 ≤ 2C 0 and (27) 1 2 V (h)1 Ω ≤ V (h)1 Ω ≤ 2V (h)
We thus get

E |g(x 0 ) -ĝĥ (x 0 )| 2 1 Ω ≤ 6E [A(h)1 Ω ] + 12V (h) + 3E|g(x 0 ) -ĝh (x 0 )| 2 , which leads to E |g(x 0 ) -ĝĥ (x 0 )| 2 1 Ω ≤ 6E[A(h)1 Ω ] + 3b 2 h (x 0 ) + (12 + 3/c) V (h). Using the definition of A(h), it is then sufficient to prove E sup h {|ĝ h (x 0 ) -g h (x 0 )| 2 -V (h )/6} + 1 Ω = O log(n∆) n∆ , (28) E sup h {|ĝ h,h (x 0 ) -g h,h (x 0 )| 2 -V (h )/6} + 1 Ω = O log(n∆) n∆ (29)
to obtain the result. Using ( 27) and Proposition 6.

3, since c/2 ≥ 16 max(1, K ∞ ), E sup h {|ĝ h (x 0 ) -g h (x 0 )| 2 -V (h )/6} + 1 Ω ≤ E sup h {|ĝ h (x 0 ) -g h (x 0 )| 2 - 1 6 c/2 2π K 2 (g * ) 1 + g * 2 2 log(n∆) n∆ } + = O log(n∆) n∆
and we prove (29) in the same way.

• Second step: First, using Lemma 3.1, |g ĥ

(x 0 ) -g(x 0 )| 2 ≤ sup h∈H c 1 h 2 + c 1 ∆ 2 ≤ C.
Then the bias term is uniformly bounded. Let us now study the "variance" term. We can write ĝh (

x 0 ) = 1 2π e -ix 0 u K * (uh) 1 ∆ θ∆ (u)du
and, since all h ∈ H is larger than 1/M , |ĝ ĥ(x 0 ) -g ĥ

(x 0 )| ≤ 1 2π sup h∈H |K * (uh)| θ∆ (u) -θ ∆ (u) ∆ du ≤ M 2π h∈H |K * (u)| θ∆ (u/h) -θ ∆ (u/h) ∆ du.
With a convex inequality |ĝ ĥ(x 0 ) -g ĥ

(x 0 )| 4 ≤ M 7 (2π) 4 h∈H |K * (u)| θ∆ (u/h) -θ ∆ (u/h) ∆ du 4
Next, we use the following inequality (obtained with two uses of the Schwarz inequality):

E ( φ(u)du) 4 = E [φ(u 1 ) . . . φ(u 4 )] du 1 . . . du 4 ≤ E 1/4 φ(u 1 ) 4 . . . E 1/4 φ(u 4 ) 4 du 1 . . . du 4 = E 1/4 φ(u) 4 du 4 with φ(u) = |K * (u)| θ∆ (u/h)-θ ∆ (u/h) ∆ . Thus, E |ĝ ĥ(x 0 ) -g ĥ(x 0 )| 4 ≤ M 7 (2π) 4 h∈H   |K * (u)|E 1/4   θ∆ (u/h) -θ ∆ (u/h) ∆ 4   du   4 .
But, according to (26) in Lemma 6.1, under G3(4),

∆ -4 E θ∆ (v) -θ ∆ (v) 4 ≤ C(n∆) -2 . Hence, under G3(4), E|ĝ ĥ(x 0 ) -g ĥ(x 0 )| 4 ≤ CM 7 h∈H |K * (u)|(n∆) -1/2 du 4 ≤ C K * 4 1 M 8 (n∆) -2 ≤ C K * 4 1 (n∆) 2/3 . • Third step: P(Ω c ) = P( g * -ĝ * h 2 2 > g * 2 (1 -1/ √ 2) or f * -f * h 1 1 > f * 1 /2) ≤ ( g * 2 (1 -1/ √ 2)) -8 E ĝ * h 2 -g * 8 2 + ( f * 1 /2) -16 E f * h 1 -f * 16 1 ≤ C E ĝ * h 2 -g * h 2 8 2 + E g * h 2 -g * 8 2 + E f * h 1 -f * h 1 16 1 + E f * h 1 -f * 16 1 .
Thus we have four terms to upperbound.

First term: Since ĝ * h 2 (u) = K * 0 (uh 2 ) θ∆ (u)/∆, ĝ * h 2 -g * h 2 2 2 = 1 h 2 |K * 0 (u)| 2 θ∆ (u/h 2 ) -θ ∆ (u/h 2 ) ∆ 2 du.
Then, according to (26) in Lemma 6.1, under G3( 8),

E ĝ * h 2 -g * h 2 8 2 ≤ 1 h 4 2   E 1/4   |K * 0 (u)| 8 θ∆ (u/h 2 ) -θ ∆ (u/h 2 ) ∆ 8   du   4 ≤ 1 h 4 2 |K * 0 (u)| 2 (n∆) -1 du 4 ≤ K * 0 8 2 M 4 (n∆) -4 ≤ 16(n∆) -8/3 . Second term: Since g * h 2 = K * 0 (uh 2 )g * (u)ψ ∆ (u), we can decompose the bias into g * (u) -g * h 2 (u) = g * (u)(1 -K * 0 (uh 2 )) + g * (u)K * 0 (uh 2 )(1 -ψ ∆ (u)) = b 1 + b 2 . Using that g ∈ Sob(1) ( |g * (u)| 2 u 2 du < ∞), b 1 2 2 = |g * (u)(1 -K * 0 (uh 2 ))| 2 du = |g * (u)| 2 1 |uh 2 |>1 du ≤ |g * (u)| 2 |uh 2 | 2 du ≤ Ch 2 2 .
On the other hand, using (25) in Lemma 6.1,

b 2 2 2 = |g * (u)K * 0 (uh 2 )(1 -ψ ∆ (u))| 2 du ≤ C∆ 2 |g * (u)u| 2 du ≤ C∆ 2 ≤ C(n∆) -1 . Thus, taking h 2 = (n∆) -1/3 gives g * -g * h 2 8 ≤ Ch 8 2 + C(n∆) -4 ≤ C(n∆) -8/3 . Third term: Since f * h 1 (u) = K * 0 (uh 1 )η ∆ (u)/∆, f * h 1 -f * h 1 1 ≤ 1 h 1 |K * 0 (u)| η∆ (u/h 1 ) -η ∆ (u/h 1 ) ∆ du.
Next, we use the following inequality

E ( φ(u)du) 16 ≤ E 1/16 φ(u) 16 du 16 .
Exactly as Lemma 6.1, using the Rosenthal inequality, we can prove under G3(4p),

for p ≥ 1, ∆ -2p E |η ∆ (v) -η ∆ (v)| 2p ≤ C(n∆) -p . Then, under G3(32), E f * h 1 -f * h 1 16 1 ≤ 1 h 16 1 E 1/16 |K * 0 (u)| 16 η∆ (u/h 1 ) -η ∆ (u/h 1 ) ∆ 16 du 16 ≤ 1 h 16 1 |K * 0 (u)|(n∆) -1/2 du 16 ≤ C K * 1 (n∆) -8/3 since h 1 = (n∆) -1/3 . Fourth term: Since η ∆ = -ψ ∆ = ∆f * ψ ∆ + ∆ 2 (g * ) 2 ψ ∆ , we can decompose the bias into f * (u) -f * h 1 (u) = f * (u) -K * 0 (uh 1 )f * (u)ψ ∆ (u) -∆K * 0 (uh 1 )(g * (u)) 2 ψ ∆ (u) = f * (u)(1 -K * 0 (uh 1 )) + f * (u)K * 0 (uh 1 )(1 -ψ ∆ (u)) -∆K * 0 (uh 1 )(g * (u)) 2 ψ ∆ (u) = b 1 + b 2 + b 3 . Since xg(x) ∈ Sob(1) ( |f * (u)| 2 u 2 du < ∞), b 1 1 ≤ |f * (u)(1 -K * 0 (uh 1 ))|du = |f * (u)|1 |uh 1 |>1 du ≤ |f * (u)| 2 |uh 1 | 2 du |uh 1 | -2 1 |uh 1 |>1 du 1/2 ≤ Ch 1/2 1 .
On the other hand, using ( 25

Z k ∆ ∆ K h x 0 -Z k ∆ -E Z k ∆ ∆ K h x 0 -Z k ∆ . ( 30
)
In order to apply a Bernstein inequality, since the Z ∆ k 's are not bounded, we truncate these variables and consider the following decomposition:

{|Z k ∆ | ≤ µ n } and {|Z k ∆ | > µ n }
where (31)

µ n = µ n (h ) = K 2 2 ( (g * ) 1 + g * 2 2 ) 2π K ∞ V (h )/6
.

We then decompose (30) as follows ĝh (x 0 ) -g h (x 0 ) = 1 n

n k=1 W k (h ) + T k (h ) -E W k (h ) + T k (h ) = S n (W (h )) + S n (T (h ))
where S n (X) means (1/n) n i=1 [X i -E(X i )] and

W k (h) = Z k ∆ ∆ K h x 0 -Z k ∆ 1 {|Z k ∆ |≤µn(h)} , (32)
T k (h) = Z k ∆ ∆ K h x 0 -Z k ∆ 1 {|Z k ∆ |>µn(h)} . (33) Thus E sup h {|ĝ h (x 0 ) -g h (x 0 )| 2 -V (h )/6} + ≤ 2 h ∈H E S n (W (h )) 2 -V (h )/12 + + 2 h ∈H E S n (T (h )) 2 .
Then we use the two following lemmas Lemma 6.2. Assume that g satisfies G1, G2, K satisfies K1, and c ≥ 16, M = O((n∆) 1/3 ), ∆ ≤ 1. Then there exists C > 0 only depending on K and g such that Lemmas 6.2 and 6.3 yield

E sup h {|ĝ h (x 0 ) -g h (x 0 )| 2 -V (h )/6} + ≤ C 1 n∆ + log(n∆) n∆ .
Inegality ( 24) is obtained by following the same lines as for inequality (23) with K h replaced by K h K h . This ends the proof of Proposition 6.3. We apply this form of Bernstein inequality to W i (h) defined by ( 32) and η = (1/12 + y)V (h). Using Lemma 3.2 and ∆ ≤ 1, it is easy to see that

Var(W i ) ≤ ν 2 := K 2 2 ( (g * ) 1 + g * 2 2 ) 2π∆h
and

|W i | ≤ b := K ∞ µ n (h) ∆h .
We find exp -nη 2 4ν 2 = exp -π(1/12)V (h)n∆h 2 K 2 2 ( (g * ) 1 + g * 2 2 )

× exp -πyV (h)n∆h 2 K 2 2 ( (g * ) 1 + g * 2 2 ) = (n∆) -c/48 × (n∆) -cy/4 as soon as c ≥ 16. This completes the proof of lemma 6.2.

Figure 1 .

 1 Figure 1. Function g (solid line) and estimators ĝĥ (dotted lines).

Figure 2 .

 2 Figure 2. Function g (solid line) and estimators ĝĥ (dotted lines).

  Then we find the same rate of convergence replacing ∆ by ∆: Proposition 5.2. Under the assumptions of Proposition 5.1 and under condition (11), the choice h opt ∝ ((n ∆) -1 2β+1 ) minimizes the risk bound (10) and gives M SE(x 0 , h opt ) =

  ) in Lemma 6.1, b 2 1 ≤ |f * (u)K * 0 (uh 1 )(1 -ψ ∆ (u))|du ≤ C∆ |f * (u)uK * 0 (uh 1 )|du ≤ C∆ |f * (u)u| 2 du |K * 0 (uh 1 )| 2 du ≤ ∆ |K * 0 (uh 1 )(g * (u)) 2 ψ ∆ (u)|du ≤ ∆ |(g * (u)) 2 |du ≤ C∆ ≤ C(n∆) -1/2 . Thus f * -f * h 1 16 1 ≤ Ch 8 1 + C(h 1 n∆) -8 + C(n∆) -8 ≤ C(n∆) -8/3. This completes the proof of Theorem 3.3. 6.6. Proof of Proposition 6.3. Note that ĝh (x 0 ) -g h (x 0 ) = 1 n n k=1

  Under assumptions K1, G3(5) and ifM = O((n∆) 1/3 ), h∈H E S 2 n (T (h)) ≤ C 1 n∆ .

6. 7 . 0 V

 70 Proof of lemma 6.2. First, note thatE S 2 n (W (h)) -V (h)/12 + ≤ ∞ 0 P(S 2 n (W (h)) ≥ V (h)/12 + x)dx ≤ ∞ (h)P |S n (W (h))| ≥ V (h)(1/12 + y) dy.Next, we recall the classical Bernstein inequality (see e.g.[START_REF] Birgé | Minimum contrast estimators on sieves: exponential bounds and rates of convergence[END_REF] for a proof):Lemma 6.4. Let W 1 , ..., W n n independent and identically distributed random variables andS n (W ) = (1/n) n i=1 [W i -E(W i )].Then, for η > 0,P(|S n (W )| ≥ η) (W 1 ) ≤ ν 2 and |W 1 | ≤ b.

V

  (h)(n∆) -c/48 max (n∆) -cy/4 , (n∆) y/λ = λ and ∞ 0 e -√ y/λ = 2λ 2 . Replacing V (h) by its value, it givesh∈H E S 2 n (W (h)) -V (h)/12 + ≤ -1 log(n∆) + 96 c

Acknowledgement

The authors thank Fabienne Comte and Valentine Genon-Catalot for enlightening discussions and helpful advices.

6.8. Proof of lemma 6.3. For a fixed bandwidth h in H, we can establish the following bound:

for any w > 0. Recall that, according to Proposition 6.2,

The following equalities hold up to constants:

Finally, as M = O((n∆) 1/3 ), we have

We need that (2 + w/2) × 1/3 -w/2 < 0, so we need the Z i admit a moment of order w + 2 ≥ 5. This completes the proof of lemma 6.3.