Ëlle Gwenna

Mabon

ADAPTIVE DECONVOLUTION OF LINEAR FUNCTIONALS ON THE NONNEGATIVE REAL LINE

Keywords: Deconvolution, Nonparametric density estimation, Adaptive estimation, Linear functionals, Laguerre basis, Mean squared risk AMS Subject Classification 2010: Primary 62G07; secondary 62G99, 62J99

.

Introduction

In many experiments statisticians do not observe directly the variable of interest X ; instead they have at hand observations of Z, equal to the sum of X and another random variable Y . In various situations, Y can modelize a measurement error, and as such, is assumed to be symmetric or centered. But we can also, in reliability fields, observe the sum of the lifetimes of two components, the second one being well known. In survival analysis, X can be the time of infection of a disease and Y the incubation time, and this happens in the so called back calculation problems in AIDS research. In these last two cases, distributions of X and Y are R + -supported. In this situation, Y is not considered as a noise but as an additional nuisance process. Indeed a noise distribution is assumed to be centered, which is not the case anymore. Thus we consider the following model

Z i = X i + Y i , i = 1, . . . , n, (1)
where the X i 's are independent identically distributed (i.i.d.) nonnegative random variables (r.v.) with unknown density f . The Y i 's are also i.i.d. nonnegative variables with known density g. We denote by h the density of the Z i 's. Moreover the X i 's and the Y i 's are assumed to be independent, they are not observed. Our goal is to estimate linear functionals of f defined by

ϑ(f ) = ψ, f = E[ψ(X 1 )
] with ψ a known function, from observations Z 1 , . . . , Z n .

Those assumptions imply that, in Model (1), h(x) = (f g)(x) where (ϕ ψ)(x) = ϕ(xu)ψ(u) du denotes the convolution product. This setting matches convolution models which is a classical topic in nonparametric statistics. The problem of recovering the signal distribution f when it is observed with an additive noise with known error distribution on the real line, has been extensively studied, see for rates of convergence and their optimality for kernel estimators [START_REF] Carroll | Optimal rates of convergence for deconvolving a density[END_REF] and [START_REF] Fan | On the optimal rates of convergence for nonparametric deconvolution problems[END_REF], for wavelets strategy [START_REF] Pensky | Adaptive wavelet estimator for nonparametric density deconvolution[END_REF], for projection strategies with penalization [START_REF] Comte | Penalized contrast estimator for adaptive density deconvolution[END_REF] and Butucea and Tsybakov (2008a,b) for the study of sharp asymptotic optimality ; with an unknown error density, see [START_REF] Neumann | On the effect of estimating the error density in nonparametric deconvolution[END_REF] for kernel estimator and minimax optimality, [START_REF] Johannes | Deconvolution with unknown error distribution[END_REF] for minimax optimality under various regularity conditions on f , [START_REF] Comte | Data-driven density estimation in the presence of additive noise with unknown distribution[END_REF] and [START_REF] Kappus | Adaptive density estimation in deconvolution problems with unknown error distribution[END_REF] for projection strategies with penalization.

The problem of one-sided error in the convolution model has been first introduced by Groeneboom and Wellner (1992) under a constraint of monotonicity of the cumulative distribution function (c.d.f.). They concentrate on deriving nonparametric maximum likelihood estimators (NPMLE) of the c.d.f. Some particular cases have been tackled as uniform or exponential deconvolution by [START_REF] Groeneboom | Density estimation in the uniform deconvolution model[END_REF] and [START_REF] Jongbloed | Exponential deconvolution: two asymptotically equivalent estimators[END_REF] who propose NPMLE of the c.d.f. of the X i 's, which have explicit expressions. For other cases [START_REF] Van Es | Isotonic inverse estimators for nonparametric deconvolution[END_REF] circumvent the lack of explicit expression for the NPMLE by proposing an isotonic inverse estimator. In this paper, we do not use the approach of the NPLME.

Moreover Model (1) is also related to the field of mixture models, see [START_REF] Roueff | Nonparametric estimation of mixing densities for discrete distributions[END_REF] and [START_REF] Rebafka | Nonparametric estimation of the mixing density using polynomials[END_REF] who study in particular mixtures of Exponential and Gamma distributions, which are contained in our framework. These models play a major role in natural sciences phenomena of discharge or disexcitation as in radioactive decays, the electric discharge of a capacitor or the temperature difference between two objects.

On one hand the problem of estimating linear functionals in linear models has been widely studied especially in the setting of the white noise model, see Cai andLow (2003, 2005) and [START_REF] Laurent | Adaptive estimation of linear functionals by model selection[END_REF] for instance. On the other hand adaptive estimation of linear functionals has not been much studied in the convolution model. [START_REF] Butucea | Adaptive estimation of linear functionals in the convolution model and applications[END_REF], in the context of Model (1) when the variables X i 's and Y i 's are R-supported, propose a general estimator of ϑ(f ) using a spectal cut-off in the Fourier domain when the random variables are distributed on the real line. They apply it to the pointwise estimation of the density on the real line and prove that their losses in their adaptive procedure is optimal in the minimax sense. They do not prove it for their general estimator. They also apply their adaptive procedure to the pointwise Laplace transform estimation and the stochastic volatility model. Recently [START_REF] Pensky | Minimax theory of estimation of linear functionals of the deconvolution density with or without sparsity[END_REF] has improved their results by deriving minimax lower bounds for estimators of a general linear functional of the deconvolution density. The author even extends the techniques when ψ is not integrable or square integrable and considers the possibility that the vector of observations is sparse (i.e. has a lot of zeros).

In this paper, our goal is to establish a specific method for the estimation of linear functional when we know that the random variables are R + -supported. In that way we can cite the work of [START_REF] Mabon | Adaptive deconvolution on the nonnegative real line[END_REF] who proposes a specific estimation for nonnegative variables in Model (1). The methodology, based on a penalized projection strategy in a Laguerre basis, allows to estimate the density and survival function of X. Moreover this work has showed that for certain classes of density, the Laguerre estimation in the convolution model (1) gives faster rates of convergence of the estimators than with the classical method based on Fourier inversion as obtained in [START_REF] Comte | Penalized contrast estimator for adaptive density deconvolution[END_REF] for instance. In particular, it is verified for mixed Gamma distributions. The contribution of this paper is to extend the particular methodology developed for nonnegative variables to estimate linear functionals of f . Thanks to this new methodology, we can derive the pointwise estimation of the probability density function (p.d.f.) f , of the c.d.f of X and also its Laplace transform.

In Section 2, we explain how projection coefficient estimators in the Laguerre basis can be used to define estimators of linear functionals. Next we lead a theoretical study of estimators of ϑ(f ) and derive upper bound on the quadratic risk and rates of convergence in function of the smoothness of f, g and ψ when these three functions belong to L 2 (R + ). We show that under some assumptions on g or ψ, the parametric rate of estimation can be achieved. In Section 3 we propose a nonparametric data driven strategy, following [START_REF] Goldenshluger | Bandwidth selection in kernel density estimation: Oracle inequalities and adaptive minimax optimality[END_REF] method, for selecting a relevant projection space. In Section 4, when f and g belong to L 2 (R + ) we adapt the procedure to pointwise estimation of f in the setting of Model (1) in this case ψ is specified and do not belong to L 2 (R + ) anymore. The method is then illustrated through simulations. Following [START_REF] Lacour | Minimal penalty for Goldenshluger-Lepski method[END_REF], we apply a new procedure to choose the tuning parameter appearing in the penalization term of the data-driven estimator. This procedure is promising and shows good results.

To sum up the paper is organized as follows. In Section 2, we give the notations, specify the statistical model and estimation procedures for projection estimators of f and ϑ(f ), upper bound on the pointwise mean squared error. In Section 3, we propose a new data-driven procedure for choosing the tuning parameter linear functionals of f which allows in particular to estimate the cumulative distribution function of X. In Section 4, we derive an adaptive procedure for the pointwise estimation of f and provide an empirical study on simulations. All the proofs are postponed to Section 5. For two functions ϕ, ψ : R → R belonging to L 2 (R), we denote ϕ the L 2 norm of ϕ defined by ϕ 2 = R |ϕ(x)| 2 dx, ϕ, ψ the scalar product between ϕ and ψ defined by ϕ, ψ = R ϕ(x)ψ(x)dx. Let d be an integer, for two vectors u and v belonging to R d , we denote u 2 the Euclidean norm defined by u 2 2 = t u u where t u is the transpose of u. The scalar product between u and v is u, v 2 = t u v = t v u. We introduce the square of the spectral norm of a matrix A :

2 (A) = λ max t AA where λ max t AA is the largest eigenvalue of t AA in absolute value. We also introduce the following operator norm:

A 1 = max 1≤j≤d d i=1 |a ij |. 2.
2. Laguerre basis. We define the Laguerre basis as

∀k ∈ N, ∀x ≥ 0, ϕ k (x) = √ 2L k (2x)e -x with L k (x) = k j=0 (-1) j k j x j j! . (2)
The Laguerre polynomials L k defined by Equation ( 2) are orthonormal with respect to the weight function x → e -x on R + . In other words, R + L k (x)L k (x)e -x dx = δ k,k where δ k,k is the Kronecker symbol. Thus (ϕ k ) k≥0 is an orthonormal basis of L 2 (R + ). We can also notice that the Laguerre basis verifies the following inequality for any integer k

sup x∈R + |ϕ k (x)| = ϕ k ∞ ≤ √ 2. ( 3
)
We also introduce the space S m = Span{ϕ 0 , . . . , ϕ m-1 }. For a function p in L 2 (R + ), we note

p(x) = k≥0 a k (p)ϕ k (x) where a k (p) = R + p(u)ϕ k (u) du.
According to formula 22.13.14 in [START_REF] Abramowitz | Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF], what makes the Laguerre basis relevant in our deconvolution setting is the relation

ϕ k ϕ j (x) = x 0 ϕ k (u)ϕ j (x -u) du = 2 -1/2 (ϕ k+j (x) -ϕ k+j+1 (x)) (4)
where stands for the convolution product.

2.3. Projection estimator of the linear functional.

2.3.1. Statistical procedure. The goal is to estimate ϑ(f ) = ψ, f = E[ψ(X 1 )] when only the Z i 's are observed for i = 1, . . . , n. If f and ψ ∈ L 2 (R + ) then they can be decomposed on the Laguerre basis, it yields that

ϑ(f ) = ψ, f = ∞ k=0 a k (ψ)a k (f ).
Since ψ is known and the ϕ k 's are known, coefficients a k (ψ) can be explicitly computed. Therefore we have to define estimators âk of a k (f ) but an estimator of ϑ(f ) would not be ϑ(f ) = ∞ k=0 a k (ψ)â k . Indeed the last sum is not necessarily convergent nor computable in practice, so we need to truncate it. Let us consider ϑ m (f ) defined by

ϑ m = ϑ m (f ) = m-1 k=0 a k (ψ)a k (f ).
Let us define ψ m and f m respectively the projection of ψ and f on the space S m . It yields that

ψ m , f m = ϑ m = ψ, f m = ψ m , f . Then we can estimate ϑ m by θm = ϑ( fm ) = ψ m , fm = m-1 k=0 a k (ψ)â k , (5)
with fm = m-1 k=0 âk ϕ k . The coefficients âk are computed as follows

t (â 0 . . . âm-1 ) := ˆ f m = G -1 m ˆ h m , with ˆ h m = t (â 0 (Z) . . . âm-1 (Z)) (6)
where âk (

Z) = (1/n) n i=1 ϕ k (Z i ) for k = 0, . . . , m -1 are estimators of a k (h) and [G m ] := [G m ] i,j 0≤i,j≤m-1 =      2 -1/2 a 0 (g) if i = j, 2 -1/2 (a i-j (g) -a i-j-1 (g)) if j < i, 0 otherwise. ( 7
)
We know from [START_REF] Mabon | Adaptive deconvolution on the nonnegative real line[END_REF] that formula (4) implies that θm is an unbiased estimator of ϑ m , see details therein, i.e. E θm = ϑ m . Before deriving the statistical properties of our estimator let us give some examples to show that the method is feasible. Since the vector ˆ h m is computed directly from the data, let us concentrate on the matrix G m and derive, if possible, an explicit expression of its coefficients.

Example 1. If g is an Exponential distribution of parameter λ > 0, then for all integer k we have

a k (g) = R + g(u)ϕ k (u) du = √ 2λ k j=0 k j (-1) j j! R + (2u) j e -(1+λ)u du = √ 2λ λ + 1 k j=0 k j (-2) j (1 + λ) j = √ 2λ λ + 1 (λ -1) k (λ + 1) k .
We can compute the coefficients of the matrix as

a 0 (g) = √ 2λ 1 + λ and a i-j (g) -a i-j-1 (g) = -2 √ 2λ (λ -1) i-j-1 (λ + 1) i-j+1 if j < i.
In that case, we can also inverse the matrix as follows

G -1 m i,j = λ + 1 λ        1 if i = j, 2 λ + 1 if j < i, 0 otherwise. ( 8
)
Thus we can explicitly compute the coefficients âk with formula (6) and θm is easy to obtain by (5). Extension to the case of a Gamma distribution of parameters p an integer, p ≥ 1 and λ > 0, can be obtained by using the more general formula

a k (g) = R + λ p u p-1 e -λu √ 2e -u L k (2u) du = √ 2λ p k j=0 k j (-2) j j! R + u p+j-1 e -(1+λ)u du = √ 2λ p (1 + λ) p k j=0 k j (-2) j (1 + λ) j (p + j -1)! j! = √ 2λ p (1 + λ) p S p-1,k 2 1 + λ . ( 9
)
with

S p-1,k (x) = 1 (p -1)! d p-1 dx p-1 x p-1 (1 -x) k .
2.3.2. Discussion: noticeable linear functionals. We propose to point out some interesting linear functionals. Some can be directly obtained with the previous methodology since they can be written as an inner product of f with a square integrable function. Others do not fit this particular assumption but can nonetheless be estimated. As said in the introduction, [START_REF] Mabon | Adaptive deconvolution on the nonnegative real line[END_REF] studies the problem of global estimation in Model (1) and proposes adaptive estimators of the probability density and survival functions. Thanks to the particular procedure for linear functionals, we can derive the pointwise estimation of the c.d.f., p.d.f and Laplace transform. a) Cumulative distribution function. Let us define ψ(x) = 1 [0,c] (x) for c > 0. The function ψ belongs to L 2 (R + ). This definition of ψ enables us to consider the estimation of the c.d.f. of f denoted by F . Indeed we can write the c.d.f. as

F (c) = c 0 f (u) du = ∞ 0 1 [0,c] (u)f (u) du = ψ, f .
The methodology of Subsection 2.3.1 can therefore be applied. b) Probability density function. Let x 0 be a nonnegative real number, we have

f (x 0 ) = ∞ 0 f (u)δ {x 0 } (u) du = δ {x 0 } (.), f = ψ, f ,
with ψ(x) = δ {x 0 } (x) and δ stands for the Dirac measure. Obviously in that case ψ is not square integrable. This case is studied in details in Section 4. c) Laplace transform. Let t be a postive real number and consider the function ψ t (x) = e -tx which is square integrable, we can write the Laplace transform of X as follows

Ee -tX = ∞ 0 e -tu f (u) du = ∞ 0 e -tu k≥0 a k (f )ϕ k (u) du = k≥0 a k (f ) ∞ 0 e -tu ϕ k (u) du = k≥0 a k (f )a k (ψ t ).
We can apply the procedure of Subsection 2.3.1. Moreover we explain in Subsection 2.4.4 how to estimate the Laplace transform with a parametric rate.

2.4. Upper bound on the mean squared error and rates of convergence.

2.4.1. Risk bound. Let us notice that if f, g ∈ L 2 (R + ) then h is bounded. Indeed h ∞ = sup x≥0 |f g(x)| ≤ f g .
Then we can state the following Proposition: Proposition 2.1. For f, g and ψ ∈ L 2 (R + ), for G m defined by Equation (7) and θm defined by Equation (5), the following result holds

E (ϑ -θm ) 2 ≤ (ϑ -ϑ m ) 2 + h ∞ t ψ m G -1 m 2 2 n . ( 10
)
The terms of the right-hand side of Equation ( 10) correspond to a squared bias variance decomposition. We can notice by applying the Cauchy-Schwarz inequality that

(ϑ -ϑ m ) 2 =   k≥m a k (ψ)a k (f )   2 ≤ ψ -ψ m 2 f -f m 2 = k≥m a 2 k (ψ) k≥m a 2 k (f ). ( 11
)
Therefore the first term gets smaller when m increases. On the contrary noticing that

t ψ m G -1 m 2 2 ≤ ψ 2 2 (G -1 m )
, we obtain a bound on the variance which is nondecreasing with m (see Lemma 3.4 in [START_REF] Mabon | Adaptive deconvolution on the nonnegative real line[END_REF]). In that case we have to tackle the usual bias variance trade-off.

We can note that if there exist some function ψ and/or some density function g such that

t ψ m G -1 m 2
2 is bounded by a constant not depending on m, then we can estimate the resulting linear functionals with a parametric rate. First let us consider the general rates of convergence.

2.4.2. General rates of convergence. In order to derive general rates of convergence of estimators θm defined by Equation ( 5), we need to evaluate the smoothness of the signal along with the order of 2 G -1 m . In the first place, we assume that f belongs to a Laguerre-Sobolev space defined as

W s (R + , L) =    p : R + → R, p ∈ L 2 (R + ), k≥0 k s a 2 k (p) ≤ L < +∞    with s ≥ 0 ( 12
)
where we recall that a k (p) = p, ϕ k . [START_REF] Bongioanni | What is a Sobolev space for the Laguerre function systems?[END_REF] have introduced Laguerre-Sobolev spaces but the link with the coefficients of a function on a Laguerre basis was done by [START_REF] Comte | Adaptive Laguerre density estimation for mixed Poisson models[END_REF] when s is an integer. Indeed, let s be an integer, f ∈ W s (R + , L) is equivalent to the fact that f admits derivatives up to order s -1 with f (s-1) absolutely continuous and for 0

≤ k ≤ s -1, x (k+1)/2 k+1 j=0 k+1 j f (j) (x) ∈ L 2 (R + ).
For more details we refer to section 7 of [START_REF] Comte | Adaptive Laguerre density estimation for mixed Poisson models[END_REF].

Thanks to those spaces we can evaluate the bias order. Let

f ∈ W s (R + , L) and ψ ∈ W α (R + , L ) with α, s > 1, starting from Equation (11) we have (ϑ -ϑ m ) 2 ≤ k≥m a 2 k (ψ)k -α k α k≥m a 2 k (f )k -s k s ≤ LL m -(s+α) .
Then we must evaluate the variance term of Equation ( 10) which means assess the order of 2 G -1 m . [START_REF] Comte | Laplace deconvolution and its application to dynamic contrast enhanced imaging[END_REF] show that under the following conditions on the density g, we can recover the order of the spectral norm of G -1 m . First we define an integer r ≥ 1 such that

d j dx j g(x) | x=0 = 0 if j = 0, 1, . . . , r -2 B r = 0 if j = r -1.
And we make the two following assumptions:

(C1) g ∈ L 1 (R + ) is r times differentiable and g (r) ∈ L 1 (R + ). (C2)
The Laplace transform of g, defined by G(z) = E[e -zY ], has no zero with non negative real parts except for the zeros of the form ∞ + ib.

For instance if Y follows a Gamma distribution of parameter p and λ, then its density g verifies these three conditions for r = p. Especially an Exponential distribution satisfies those assumptions for r = 1. According to [START_REF] Comte | Laplace deconvolution and its application to dynamic contrast enhanced imaging[END_REF], under Assumptions (C1)-(C2) there exist some positive constants C and C such that

C m 2r ≤ 2 G -1 m ≤ C m 2r .
Then the order of the variance can be derived as follows s+α+2r) , then there exists a positive constant c 1 depending on L, L , s, r, α, ψ , C such that s+α+2r) .

t ψ m G -1 m 2 2 ≤ t ψ m 2 2 2 (G -1 m ) ≤ ψ 2 2 (G -1 m ) ≤ ψ 2 C m 2r . ( 13
) Proposition 2.2. Assume that ψ ∈ W α (R + , L ) and f ∈ W s (R + , L) with α, s > 1, that g satis- fies Assumptions (C1)-(C2) and let m opt ∝ n 1/(
sup f ∈W s (R + ,L) E (ϑ -θmopt ) 2 ≤ c 1 n -(s+α)/(
Thus the choice m = m opt enables us to compute the rate of convergence of the estimator. However this choice depends on the regularity of the unknown function f and cannot be used in practice. This is why we will look for another way of performing the compromise between the squared bias and the variance.

2.4.3. Almost parametric rate: mixed Gamma densities. We want to point out that the Laguerre method enables us to obtain fast rates of convergence on the class of mixed Gamma densities noted MΓ(q, α, p, λ) and defined by

MΓ(q, α, p, λ) = f = q i=1 α i γ i ≥ 0, γ i ∼ Γ(p i , λ i ), λ i > 0, p i ∈ N, q ∈ N ∪ {∞}, α i ∈ R, q i=1 |α i | < ∞, q i=1 α i = 1 , (14)
Let us assume that f belongs to MΓ(q, α, p, λ). We obtain from (9) (see details in [START_REF] Mabon | Adaptive deconvolution on the nonnegative real line[END_REF]):

(ϑ -ϑ m ) 2 =   k≥m a k (ψ) q i=1 α i a k (γ i )   2 ≤ k≥m a 2 k (ψ) k≥m q i=1 |α i |a 2 k (γ i ) ≤ ψ 2 q i=1 |α i |C(p i , λ i ) λ i -1 λ i + 1 2m m 2(p i -1) ≤ C ψ 2 m 2(p * -1) ρ 2m . with ρ = max i |(λ i -1)/(λ i + 1)| ∈ (0, 1
) and p * = max i p i . Thus the squared bias decays exponentially. And if we assume that g verifies (C1)-(C2) according to Equation ( 13), the order of the variance is less than C ψ 2 m 2r . Therefore we obtain the following result Proposition 2.3. Assume that f ∈ MΓ(q, α, p, λ), that g satisfies Assumptions (C1)-(C2) and let m opt = c log n/| log ρ| with c ≥ 1, then there exists a positive constant c 2 depending on q, α, p, λ, α, ψ , C such that

sup f ∈MΓ(q, α, p, λ) E (ϑ -θmopt ) 2 ≤ c 2 (log n) 2r n .
So for the classes of mixed Gamma densities, the Laguerre procedure enables us to obtain almost parametric rates of convergence, which is remarkable in the deconvolution model. As previously, the choice m opt cannot be performed in practice, since f is unknown.

2.4.4. Parametric rate. In this section, we propose to find conditions ensuring that the parametric rate can be achieved.

Proposition 2.4. If ψ(x) = αe -x with α ∈ R, then E (ϑ -θ1 ) 2 = α 2 a 2 0 (g) Var[ϕ 0 (Z 1 )] n . ( 15
)
This result is interesting in the sense that it does not require any additional assumptions on the density g. It allows us to recover the Laplace transform of X at the point t = 1 with a parametric rate. Note that, as Model (1) implies that E[e

-Z i ] = E[e -X i ]E[e -Y i
] and the distribution of Y is assumed to be known, we can estimate the Laplace transform with a parametric rate with

(1/n) n i=1 e -X i /E[e -Y 1 ],
without knowing anything about Laguerre basis. This formula shows that the Laplace transform can be estimated at a parametric rate at any point t > 0. It can also be achieved with the Laguerre basis but the procedure is a little bit more complicated than with a plug-in estimator.

In fact, the Laguerre basis can be parametrized. In the present work for sake of clarity we consider that this scale parameter is equal to 1. More generally we can write the basis as follows

∀k ∈ N, ∀x ≥ 0, ϕ τ k (x) = √ 2τ L k (2τ x)e -τ x with τ > 0.
For τ > 0, if we take the Laguerre basis (ϕ τ k ) k≥0 and proceed exactly as in the proof of Proposition 2.4, we find that the functions proportional to exp(-τ x) satisfy Equation ( 15) for the corresponding scale parameter τ of the basis. The extension is straightforward. So for ψ(x) = α exp(-τ x), the parametric rate is still achieved. We can use this methodology to estimate the Laplace transform of X at any point.

The following two results are more elaborate.

Proposition 2.5. If j≥2 |a j-1 (g) -a j-2 (g)| /a 0 (g) < 1 then G -1 m
is upper bounded by a constant independent of m and the parametric rate is achieved.

We can notice that Exponential distributions with parameter λ = 1 satisfies this Proposition. From Example 1, we have the coefficients of g ∼ E(λ), which yields for λ = 1

j≥2 |a j-1 (g) -a j-2 (g)| a 0 (g) = 0,
and for λ = 1

j≥2 |a j-1 (g) -a j-2 (g)| a 0 (g) = j≥2 2 |λ -1| j-2 |λ + 1| j-1 = 2 λ + 1 j≥2 |λ -1| j-2 |λ + 1| j-2 = 2 λ + 1 1 1 -(λ -1)/(λ + 1) = 1. Proposition 2.6. If g ∼ E(λ) with λ > 0 and m-1 k=0 ( m-1 j=k |a j (ψ)|) 2 < ∞, then there exists a positive constant C depending on λ such that E (ϑ -θm ) 2 ≤ (ϑ -ϑ m ) 2 + C n .
Propositions 2.5 and 2.6 give conditions ensuring that the variance can be upper bounded by C/n whatever the value of m. Then to derive a parametric rate in this case we need to choose m n large enough such that (ϑ -ϑ mn ) 2 = O(n -1 ).

Let us give some examples of functions ψ to illustrate Proposition 2.6.

Example 2. Let us consider here that ψ is the density Γ(p, µ). According to Equation ( 9) we have

a j (ψ) = √ 2µ p (1 + µ) p S p-1,j 2 1 + µ
The term S p-1,j (x) has the following order for in x ∈ [0, 1]

|S p-1,j (x)| ≤ 2 p |1 -x| j-p+1 , for j ≥ p -1 (see details in Appendix B), it yields m-1 k=0   m-1 j=k |a j (ψ)|   2 ≤ 2 2p+1 µ 2p (µ + 1) 2p m-1 k=p-1   m-1 j=k µ -1 µ + 1 j-p+1   2 ≤ 2 2p+1 µ 2p (µ + 1) 2p m-1 k=p-1 µ -1 µ + 1 2k m-1-k =0 µ -1 µ + 1 2 ≤ 2 2p+1 µ 2p (µ + 1) 2p (µ + 1) 4 2µ(µ + 1 -|µ -1|) 2 Thus condition m-1 k=0 ( m-1 j=k |a j (ψ)|) 2 < ∞ of Proposition 2.
6 is fulfilled. Therefore we conclude that if g is E(λ) and ψ a finite mixture of Gamma distributions, then the parametric rate is achieved.

Example 3. Let α an integer such that α > 1 and assume ψ ∈ W α (R + , L ), it yields   m-1 j=k |a j (ψ)|   2 =   m-1 j=k j α/2 |a j (ψ)|j -α/2   2 ≤ m-1 j=k a 2 j (ψ)j α m-1 j=k j -α ≤ L αk -α+1 α -1 , then m-1 k=0   m-1 j=k |a j (ψ)|   2 ≤ L α α -1 m-1 k=0 k -(α-1) < ∞ if α > 1.
In conclusion any function ψ such that k≥0 k α a 2 k (ψ) ≤ L < +∞ with α > 1 satisfies the assumption of Proposition 2.6.

For instance ψ(x) = (1 + x) -β 1 x≥0 with β > (α + 1)/2 belongs to W α (R + , L ) (see details in Appendix C).

Model selection and adaptive estimation

To build data-driven estimators of linear functionals of the density f , we propose a new selection strategy in the spirit of [START_REF] Goldenshluger | Bandwidth selection in kernel density estimation: Oracle inequalities and adaptive minimax optimality[END_REF]. First we add the following assumption on the model collection:

(A) M = m ∈ {1, . . . , n}, m ≤ n/ log n and max 1≤k≤m t ψ k G -1 k 2 2 ≤ n .
The model selection is done by minimizing the following quantity

m = argmin m∈M {A(m) + V (m)} ( 16
)
where V (m) has the order of the variance in the right-hand side of Equation ( 10) and is defined by

V (m) = κ( h ∞ ∨ 1) max 1≤k≤m t ψ k G -1 k 2 2 log n n ,
with κ a numerical constant which will be chosen later (see comment below). The term of variance is slightly modified to ensure that V is nondecreasing with m. The term A(m) has the order of the bias and is based on the comparison of the estimators built in the previous section as follows

A(m) = sup m ∈M ( θm -θm ∧m ) 2 -V (m ) + .
We can now state the following oracle inequality:

Theorem 3.1. For f, g and ψ ∈ L 2 (R + ), assume that Assumption (A) is true. Let θ m be defined by Equations ( 5) and (16). Then there exists a constant κ 0 such that for κ ≥ κ 0 such that

E (ϑ -θ m ) 2 ≤ C ad inf m∈M      sup k>m k∈N ∪ {+∞} (ϑ k -ϑ m ) 2 + V (m)      + C n . ( 17
)
where C ad is a positive numerical constant and C depends only on h ∞ .

The oracle inequality (17) establishes a non asymptotic oracle bound. It shows that the squared bias variance tradeoff is automatically made up to a loss of logarithmic order and a multiplicative constant. Let us notice that C ad = 7 and κ 0 = 384 would suit. Let us comment the specific squared bias term which differs from the squared bias established in Proposition 2.1. The term sup k>m (ϑ k -ϑ m ) 2 is defined such that it is non-decreasing in m and we take for convention ϑ ∞ = ϑ. Usually in Lepski's method, it is assumed that the p.d.f. belongs to some regularity space indexed by β and the squared bias is expressed as a function of β and m which is non-decreasing in m. Nevertheless it a strong assumption since we do not know the coefficient β depending on f . Our approach enables us to circumvent this assumption. Theorem 3.1 is derived under mild assumptions. Indeed we do not need to assume that ψ or f belong to some semi-parametric space to derive the theorem. Concerning the model selection procedure, it is simpler in its formulation than those of [START_REF] Laurent | Adaptive estimation of linear functionals by model selection[END_REF] and [START_REF] Butucea | Adaptive estimation of linear functionals in the convolution model and applications[END_REF] thanks to the Goldenshluger and Lepski (2011) method. [START_REF] Butucea | Adaptive estimation of linear functionals in the convolution model and applications[END_REF] obtain as a consequence of their general adaptive procedure the pointwise estimation of the density on the real line, the stochastic volatility model and the pointwise Laplace transform estimation when X is a positive random variable, even when the Laplace transform of the noise is infinite. In the next section, we also show that a procedure for the pointwise estimation on the nonnegative real line is possible. A direct application of Theorem 3.1 is the estimation of the c.d.f. of X as pointed out in Subsection 2.4, which is a novelty compared to [START_REF] Butucea | Adaptive estimation of linear functionals in the convolution model and applications[END_REF]. [START_REF] Pensky | Minimax theory of estimation of linear functionals of the deconvolution density with or without sparsity[END_REF] also develops an adaptive procedure based on Lepski's method. This paper clearly improves previous papers and covers a lot of cases (c.d.f, pointwise, moments) for which lower bounds are provided. Some comments for practical use are in order. Indeed in the variance term V (m), there are two quantities which deserve some explanations: κ and h ∞ . First note that when κ gets larger, so do V (m) and thus the bound (17). It follows from the proof that κ = 384 would suit. But in practice, values obtained from the theory are generally too large and the constant is calibrated by simulations. Once chosen, it remains fixed for all simulation experiments. There is still an unknown term in the penalty, h ∞ , that needs to be estimated. We have to check that we can derive an oracle inequality when this term is estimated, which is done in the following Corollary.

Beforehand let

ĥD (x) = D-1 k=0 âk (Z)ϕ k (x), (18)
with âk (Z) = (1/n) n i=1 ϕ k (Z i ). We can see that ĥD is an unbiased estimator of

h D (x) = D-1 k=0 a k (h)ϕ k (x).
Corollary 3.2. For f, g and ψ ∈ L 2 (R + ), assume that Assumption (A) is true. Let θ m be defined by (5) and

m = argmin m∈M A(m) + V (m) (19) with A(m) = sup m ∈M ( θm -θm ∧m ) 2 -V (m ) + and V (m) = κ(2 ĥD ∞ ∨1) max 1≤k≤m t ψ k G -1 k 2 2 log n n
where ĥD is given by (18) and

D satisfies log n ≤ D ≤ h ∞ 128 √ 2 n (log n) 3 .
Then there exists a constant κ 0 such that for κ ≥ κ 0 such that

E (ϑ -θ m ) 2 ≤ C ad inf m∈M      sup k>m k∈N ∪ {+∞} (ϑ k -ϑ m ) 2 + V (m)      + C n .
where C ad is a positive numerical constant and C depends only on h ∞ .

Note that the constraint on D is fulfilled for n large enough as soon as D √ n for instance. In this sense Corollary 3.2 has rather an asymptotic flavor.

4. Particular case of pointwise estimation 4.1. Theoretical results. For pointwise estimation of f , we take ψ(x) = δ {x 0 } (x) for any nonnegative x 0 , where δ stands for the Dirac measure. It yields that ϑ = f (x 0 ). In this case Equation ( 5) becomes

fm (x 0 ) = m-1 k=0 âk ϕ k (x 0 ), ( 20
) since a k (ψ) = δ {x 0 } , ϕ k = ϕ k (x 0 )
. We can notice that ψ is not square integrable.

Proposition 4.1. For f, g ∈ L 2 (R + ), for G m defined by Equation (7) and fm (x 0 ) defined by Equation (20) for any nonnegative x 0 , the following result holds

E (f (x 0 ) -fm (x 0 )) 2 ≤ (f (x 0 ) -f m (x 0 )) 2 + h ∞ t ϕ m (x 0 )G -1 m 2 2 n ,
where t ϕ m (x 0 ) = (ϕ 0 (x 0 ) . . . ϕ m-1 (x 0 )).

We can notice that the squared bias and the variance have not the same order as in the case where ψ is square integrable. Let us assume as in Section 2.4.2 that f belongs to W s (R + , L), applying Equation (3) and the Cauchy-Schwarz inequality we get the following order for the squared bias

(f (x 0 ) -f m (x 0 )) 2 =   k≥m a k (f )ϕ k (x 0 )   2 ≤ 2   k≥m |a k (f )|   2 = 2   k≥m |a k (f )|k s/2 k -s/2   2 ≤ 2 k≥m |a k (f )| 2 k s k≥m k -s ≤ 2sL s -1 m -s+1
for s > 1. Moreover applying Equation (3), we get

t ϕ m (x 0 )G -1 m 2 2 ≤ t ϕ m (x 0 ) 2 2 2 (G -1 m ) ≤ m-1 k=0 |ϕ k (x 0 )| 2 2 (G -1 m ) ≤ 2m 2 (G -1 m ). ( 21
)
Proposition 4.2. Assume that f ∈ W s (R + , L) with s > 1, that Assumptions (C1)-(C2) are true and let m opt ∝ n 1/(s+2r) , then there exists a positive constant c 2 depending on L, s, r, C such that s+2r) .

sup f ∈W s (R + ,L) E (f (x 0 ) -f mopt (x 0 )) 2 ≤ c 2 n -(s-1)/(
Note that this rate is different from the global one: according to [START_REF] Mabon | Adaptive deconvolution on the nonnegative real line[END_REF] for the global estimation of the density f with the L 2 -risk, the rate of convergence is of order n -s/(s+2r+1) . As in Section 2.4.2, we can point out that faster rates of convergence of order log n/n can be obtained for functions of Gamma type defined by Equation ( 14).

The data-driven strategy is done according to the one developed in Section 3 for the particular case of pointwise estimation, with the specific variance term according to this model. It yields that

A(m, x 0 ) = sup m ∈M ( fm (x 0 ) -fm ∧m (x 0 )) 2 -V (m , x 0 ) + with V (m, x 0 ) = κ(2 ĥD ∞ ∨ 1) max 1≤k≤m t ϕ k (x 0 )G -1 k 2 2 log n n ( 22
)
where κ is a numerical constant, ĥD is the estimator of the projection of h on S D given by ( 18)

and

D satisfies log n ≤ D ≤ h ∞ 128 √ 2 n (log n) 2 and m = argmin m∈M A(m, x 0 ) + V (m, x 0 ) . ( 23
)
Theorem 4.3. For f, g ∈ L 2 (R + ), assume that Assumption (A) is true. Let f m (x 0 ) be defined by Equations ( 20)-( 23). Then there exists a constant κ 0 such that for κ ≥ κ 0 such that

E (f (x 0 ) -f m (x 0 )) 2 ≤ C ad inf m∈M      sup k>m k∈N ∪ {+∞} (f k (x 0 ) -f m (x 0 )) 2 + V (m, x 0 )      + C n .
where C ad is a positive numerical constant, C depends only on h ∞ and V (m, [START_REF] Mabon | Adaptive deconvolution on the nonnegative real line[END_REF]). These two bounds are independent of x 0 which proves that the risk, may tend uniformly to 0 for an adequate choice m = m(n), when n tends to infinity. 4.2. Illustrations. The whole implementation is conducted using R software. The mean squared error (MSE) E(f (x 0 ) -f m (x 0 )) 2 is computed as the empirical mean over 500 simulation samples. 4.2.1. Simulation setting. The performance of the procedure is studied for the three following distributions for X. Moreover all the densities are normalized with unit variance. Each density is computed over an interval I. Thus the first Gamma distribution has a variance 1/10 and the second 1/4. We refer to Example 1 for the computation of the matrix G m . We also compute estimators where there is no additional noise, i.e. σ 2 = 0. The procedure is applied with G m = I m . 4.2.2. Implementation of the estimation procedure. We consider the following model collection M n = {m ∈ N, 0 ≤ m ≤ n/ log n }. The pointwise estimation at a point x 0 is computed according to the following adaptive procedure:

x 0 ) = κ( h ∞ ∨ 1) max 1≤k≤m t ϕ k (x 0 )G -1 k 2 2 log n/n. Applying Equation (3) note that sup k>m |f k (x 0 ) -f m (x 0 )| 2 = sup k>m k-1 j=m a j (f )ϕ k (x 0 ) 2 ≤ 2   j≥m |a j (f )|   2 and using Equation (21) yields V (m, x 0 ) ≤ 2κ h ∞ m 2 (G -1 m ) log n n , since m → 2 (G -1 m ) is nondecreasing with m (see Lemma 3.4 in
For m ∈ M n , compute m = argmin m∈Mn A(m, x 0 ) + V (m, x 0 ) , with • A(m, x 0 ) = sup m ∈Mn ( fm (x 0 ) -fm ∧m (x 0 )) 2 -V (m , x 0 ) + • V (m, x 0 ) = κ(2 ĥD ∞ ∨ 1) max 0≤k≤m-1 t ϕ k (x 0 )G -1 k 2 2 log n/n with ĥD defined by (18). Compute f m (x 0 ) = m-1 k=0 âk ϕ k (x 0 ).
In addition we compute the integrated squared risk over an interval I of 300 equidistant points, by computing length(I) 300

300 k=1 (f (x k ) -f m k (x k )) 2
and (x k ) k a subdivision of I, which gives an approximation of the integrated risk for our method.

Thus we can measure the performances of the pointwise estimation by comparing our method with the method of [START_REF] Mabon | Adaptive deconvolution on the nonnegative real line[END_REF] who proposes adaptive estimators of the density f associated with the L 2 -risk.

One of the major issues in data-driven strategies is the choice of the constant κ. To detect this value in the model selection paradigm, we can use the slope heuristics for instance based on the works of [START_REF] Birgé | Minimal penalties for gaussian model selection[END_REF]. [START_REF] Baudry | Slope heuristics: overview and implementation[END_REF] developed a method based on slope heuristics or on dimension jump. We cannot use those methods in the present work since we use Goldenshluger-Lepski methodology for the automatic selection of an estimator among a collection.

The results of [START_REF] Goldenshluger | Bandwidth selection in kernel density estimation: Oracle inequalities and adaptive minimax optimality[END_REF] ensure that if the variance term is large enough, the data-driven estimator is almost as efficient as the best one in the collection. In the proof we derive a theoretical value of κ defined by Equation ( 22). Nonetheless it is well known that theoretical constants in Ṽ (m, x 0 ) are often too large in practice. Thus for practical implementation, the good behavior of the estimator is based on the calibration of the constant κ. Few studies have been conducted on the constant calibration in Goldenshluger-Lepski methodology to see if there exists a critical value such that, if V (m) > V 0 the risk is quasi optimal.

To find this value in the setting of Goldenshluger-Lepski method, we follow the recent work of [START_REF] Lacour | Minimal penalty for Goldenshluger-Lepski method[END_REF] who show that, for density estimation based on kernel estimators with Goldenshluger-Lepski type bandwith selection, if the variance term is chosen too small then the procedure fails. They are the first to give a minimal penalty for Goldenshluger-Lepski methodology. In our case, we do not necessarily expect the same behavior of the variance term since we are in the particular case of the deconvolution model but we expect to find a critical value of κ which characterizes a failure of the procedure.

To detect such a value, we propose to observe the evolution of the choice m for different values of κ. It is clear that to challenge the global estimation, our pointwise procedure must be able to adapt to the point and to choose different m for different points.

To avoid over fitting, we choose two distributions that we will not estimate in the following: β(4, 5) × 5 and χ 2 (10)/ √ 20. The first one has variance 50/81 and has a compact support [0,5]. The other one is normalized with unit variance and its support is R + . Only here in our procedure, we replace V (m) by its uniform bound 2κm ĥD ∞ 2 (G -1 m ) log n/n. Indeed with this approach it is easier to detect stable ranges where the choice of m does not change. Otherwise we observe stronger oscillations of the choice m and it is difficult to isolate the good values of κ from the bad ones.

In Figure 1, we apply this methodology for a wide range of κ and in Figure 2 we concentrate on smaller range of κ which corresponds to the neighborhood of the κ of the black thin line with circles. In Figure 1, we also draw the data driven choice for κ = 0 and κ = 20 which respectively corresponds to the bold black and the bold red lines. For κ = 0, the selected dimension is systematically the largest as possible while for κ = 20 it is mainly the smallest. We choose to concentrate on the black thin line with circles because it is the curve which moves the most and is the farthest from the two curves κ = 0 and κ = 20 which represent extreme situations. In Figure 2, we see that for both distributions the curves have the same behavior. Finally according to this methodology we choose a small value of κ = 3 • 10 -4 . In a second step we also propose to test another calibration suggested by [START_REF] Lacour | Minimal penalty for Goldenshluger-Lepski method[END_REF] in their concluding remark section. The idea is to consider

A(m, x 0 ) = sup m ∈M ( θm -θm ∧m ) 2 -κ(2 ĥD ∞ ∨ 1) max 1≤k≤m t ϕ k (x 0 )G -1 k 2 log n/n + m = argmin m∈M A(m, x 0 ) + κ (2 ĥD ∞ ∨ 1) max 1≤k≤m t ϕ k (x 0 )G -1 k 2 2 log n/n .
Then, it is easy to prove that Theorem 3.1 holds for ϑ m as soon as κ ≥ κ. The authors suggest to find an adequate value of κ for κ = κ and then to put κ = 2κ. This method is illustrated in the next section.

Simulation results.

In Table 1 we report the results of the simulation (averaged values of the integrated squared errors multiplied by 100) for the global estimator computed according to [START_REF] Mabon | Adaptive deconvolution on the nonnegative real line[END_REF]. The results of the pointwise estimation are presented in Table 2. It reports averaged values of the approximation of the integrated risk for our method of pointwise estimation. First remarks: increasing the sample size improves the quality of the estimation and increasing the variance degrades the estimation but in an acceptable way for both methods.

If we compare the methods, we can see for the mixed Gamma distribution that the pointwise estimation is better than the global estimation. The risk is approximately divided by 2. For the Gamma and the log-normal, the estimation is clearly better with the global method where there is no additional noise. Otherwise the two methods are equivalent. We can note for the pointwise estimation that when the sample size increases, the results are very close to those without noise (σ 2 = 0).

In Table 2 we compute the risk associated to the oracles. We see that the oracles are very good, approximately divided by 30 compared to data driven estimators. It shows that the pointwise estimation could be excellent if the data driven strategy could perform adequately. Nonetheless in Section 3, we have seen that the bias variance compromise was made up to a logarithmic loss. It may also explain the gap between oracles and data driven procedure results. We also compare the quality of the estimation when κ = κ or κ = 2κ. The results are always and slightly better when κ = κ. Our examples do not permit us to highlight the idea of dissociation of the constant presented in [START_REF] Lacour | Minimal penalty for Goldenshluger-Lepski method[END_REF] for the Goldenshluger-Lepski strategy. Nonetheless this detection method of the constant κ leads to very good results.

We also illustrate the results with some figures. Figures 345display the results of the data driven estimation respectively for the Gamma and the mixed Gamma. For each Figure we present the Goldenshluger-Lepksi estimator with the calibration proposed by [START_REF] Lacour | Minimal penalty for Goldenshluger-Lepski method[END_REF] for κ = κ. First we see that the risk decreases when the sample size increases. For the Gamma, we see that data driven estimators are very close to oracle estimators which is not really the case for the mixed Gamma especially for n = 2000. We can observe some oscillations near the origin. Thus the pointwise estimation does not seem to radically improve the estimation near the origin. [START_REF] Mabon | Adaptive deconvolution on the nonnegative real line[END_REF]. MISE E( f -f m 2 )×100 averaged over 500 samples. σ 2 denotes the level of variance of the noise. σ 2 = 0 corresponds to the model without noise (Y = 0). The noise is

σ 2 = 0 σ 2 = 1 10 σ 2 =
Γ(2, 1 √ 20 ) for σ 2 = 1 10 and Γ(2, 1 √ 8 ) for σ 2 = 1 4 . σ 2 = 0 σ 2 = 1 10 σ 2 =
E(ϑ -θm ) 2 = (ϑ -ϑ m ) 2 + E(ϑ m -θm ) 2 .
The first term corresponds to the squared bias term of Equation ( 10). Let us study the second term, by definition we have Then we get

(ϑ m -θm ) 2 = ψ, f m -fm 2 = ψ m , f m - ˆ f m 2 2 = ψ m , G -1 m ( h m - ˆ h m ) 2 2 .
E(ϑ m -θm ) 2 = Var[ θm ] = Var m-1 k=0 m-1 l=0 a k (ψ) G -1 m k,l âl (Z) = Var 1 n n i=1 m-1 k=0 m-1 l=0 a k (ψ) G -1 m k,l ϕ l (Z i ) = 1 n Var m-1 k=0 m-1 l=0 a k (ψ) G -1 m k,l ϕ l (Z 1 ) ≤ 1 n E   m-1 k=0 m-1 l=0 a k (ψ) G -1 m k,l ϕ l (Z 1 ) 2   ≤ 1 n R + m-1 k=0 m-1 l=0 a k (ψ) G -1 m k,l ϕ l (u) 2 h(u) du ≤ h ∞ n t ψ m G -1 m 2 2 .
5.1.2. Proof of Proposition 2.4. According to Equation ( 10), we just have to find the eigenvectors of the matrix G -1 m . Indeed if t ψ m is an eigenvector of G -1 m associated with eigenvalue λ m , it yields that t ψ m G -1

m 2 2 = λ 2 m t ψ m 2
2 and since we assumed that ψ ∈ L 2 (R + ), the variance is upper bounded λ 2 m ψ 2 . First note that

• For any vector u, t G -1 m u 2 2 = t uG -1 m 2 2 . • If we have t G m u = α u with α = 0, then (1/α) u = t G -1
m u. Therefore let us look for the eigenvalues of t G m .

Since t G m is a triangular matrix, its eigenvalues are its diagonal elements. Thus the only eigenvalue of t G m is a 0 (g)/ √ 2 associated with the eigenvectors belonging to S 0 = Span{ϕ 0 }. Then the matrix t G -1 m has for eigenvalue √ 2/a 0 (g) associated with the same space of eigenvectors. Let ψ ∈ S 0 , i.e. ψ(x) = αe -x 1 R + (x) with α ∈ R, it yields that t ψ m = (α/ √ 2 0 . . . 0) and

t ψ m G -1 m = (α/a 0 (g) 0 . . . 0) which implies t G -1 m ψ m 2 2 = t ψ m G -1 m 2 2 = α 2 /a 2 0 (g) = ϑ 1 .
Thus the variance term does not depend on the dimension m of the model as announced.

Now for the bias term, let us notice that ϑ = ψ, f = k≥0 a k (ψ)a k (f ) = αa 0 (f )/ √ 2 which implies that E[ θ1 ] = ϑ 1 = ϑ. So the estimator θ1 of ϑ, in that case, is unbiased.

Finally, we have that

E(ϑ -θ1 ) 2 = E(ϑ 1 -θ1 ) 2 = Var[ θ1 ] = 1 n Var a 0 (ψ)[G -1 m ] 1,1 ϕ 0 (Z 1 ) = α 2 a 2 0 (g)
Var[ϕ 0 (Z 1 )] n .

5.1.3. Proof of Proposition 2.5. The matrix G m admits only one eigenvalue a 0 (g)/ √ 2 with multiplicity m, it yields that

G m - a 0 (g) √ 2 I m m = 0
which implies that

I m - √ 2 a 0 (g) G m m = 0.
Writing that

I m = I m -I m - √ 2 a 0 (g) G m m = I m -I m - √ 2 a 0 (g) G m m-1 k=0 I m - √ 2 a 0 (g) G m k , therefore G -1 m = √ 2 a 0 (g) m-1 k=0 I m - √ 2 a 0 (g) G m k .
It yields that

G -1 m ≤ √ 2 a 0 (g) m-1 k=0 I m - √ 2 a 0 (g) G m k .
Yet using the fact that for any matrix A ∈ R m×m (see Chapter 5 in [START_REF] Horn | Matrix Analysis[END_REF]),

(A) ≤ A 1 A ∞
where . 1 and . ∞ are respectively the maximum column sum matrix and the maximum row sum matrix. Besides noticing that since G -1 m is a triangular Toeplitz matrix we have that

G -1 m 1 = t G -1 m ∞ . So we get I m - √ 2 a 0 (g) G m ≤ I m - √ 2 a 0 (g) G m 1
Moreover let us notice that

I m - √ 2 a 0 (g) G m 1 = m j≥2 a j-1 (g) -a j-2 (g) a 0 (g) ≤ j≥2 a j-1 (g) -a j-2 (g) a 0 (g) . Thus if q = j≥2 a j-1 (g) -a j-2 (g) a 0 (g) < 1, it yields that G -1 m ≤ √ 2 a 0 (g) m-1 k=0 q k ≤ √ 2 a 0 (g) ∞ k=0 q k = √ 2 a 0 (g) 1 1 -q then G -1 m
is upper bounded by a constant independent of m.

5.1.4. Proof of Proposition 2.6. Let the coefficients of G -1 m be defined by Equation ( 8), it yields

t ψ m G -1 m 2 2 = λ + 1 λ 2 m-1 k=0   a k (ψ) + 2 λ + 1 m-1 j=k+1 a j (ψ)   2 ≤ λ + 1 λ 2 1 ∨ 2 λ + 1 2 m-1 k=0   m-1 j=k |a j (ψ)|   2 . Then if m-1 k=0 ( m-1 j=k |a j (ψ)|) 2 < ∞, we get that t ψ m G -1 m 2
2 ≤ C where C does not depend on m.

Proofs of

(ϑ -θ m ) 2 ≤ 3(ϑ -θm ) 2 + 3( θm -θm∧ m ) 2 + 3( θm∧ m -θ m ) 2 ≤ 3(ϑ -θm ) 2 + 3(A( m) + V (m) + A(m) + V ( m)) ≤ 3(ϑ -θm ) 2 + 6(A(m) + V (m))
Taking expectation, we get

E(ϑ -θ m ) 2 ≤ 3E(ϑ -θm ) 2 + 6E[A(m)] + 6V (m) (24)
with

A(m) = sup m ∈M ( θm -θm ∧m ) 2 -V (m ) + (25)
Proposition 5.1. Under Assumptions of Theorem 3.1, it holds true that

E[A(m)] ≤ sup k>m k∈M (ϑ k -ϑ m ) 2 + C n .
Starting from Equation ( 24), applying Proposition 2.1 and Proposition 5.1 we get

E(ϑ -θ m ) 2 ≤ 3(ϑ -ϑ m ) 2 + h ∞ t ψ m G -1 m 2 2 n + 6 sup k>m k∈M (ϑ k -ϑ m ) 2 + C n + 6V (m) ≤ 3(ϑ -ϑ m ) 2 + 6 sup k>m k∈M (ϑ k -ϑ m ) 2 + 1 κ + 6 V (m) + C n ≤ 6 sup k>m k∈N ∪ {+∞} (ϑ k -ϑ m ) 2 + 7V (m) + C n , since κ ≥ 1 and given that ϑ = ϑ ∞ = lim m→∞ ϑ m if f, g ∈ L 2 (R + )
Finally by taking the infimum over all m ∈ M we get the desired result

E(ϑ -θ m ) 2 ≤ C ad inf m∈M      sup k>m k∈N ∪ {+∞} (ϑ k -ϑ m ) 2 + V (m)      + C n .
Proof of Proposition 5.1. First let us notice the following upper bound

A(m) = sup m ∈M ( θm -θm ∧m ) 2 -V (m ) + ≤ 3(D 1 + D 2 + D 3 )
where

D 1 = sup m ∈M (ϑ m -ϑ m∧m ) 2 D 2 = sup m ∈M ( θm -ϑ m ) 2 - V (m ) 6 + D 3 = sup m ∈M ( θm∧m -ϑ m∧m ) 2 - V (m ) 6 + .
• First consider D 1 , we have

D 1 = sup m ∈M (ϑ m -ϑ m∧m ) 2 = sup k>m k∈M (ϑ k -ϑ m ) 2 . ( 26
) • Now consider D 2 E[D 2 ] ≤ m ∈M E ( θm -ϑ m ) 2 - V (m ) 6 + ≤ m ∈M R + P ( θm -ϑ m ) 2 - V (m ) 6 + ≥ u du ≤ m ∈M R + P | θm -ϑ m | ≥ V (m ) 6 + u du. ( 27
)
To handle this probability, we will apply Bernstein's inequality. First we need to identify the empirical process, by definition we have

θm = m -1 k=0 a k (ψ)â k = m -1 k=0 a k (ψ)   m -1 i=0 G -1 m k,i 1 n n j=1 ϕ i (Z j )   = 1 n n j=1 m -1 k=0 m -1 i=0 a k (ψ) G -1 m k,i ϕ i (Z j ) = 1 n n j=1 ω m (Z j ), which yields θm -ϑ m = 1 n n j=1 (ω m (Z j ) -E[ω m (Z j )]
). We need to ensure that there exist b and v such that |ω m (Z

j )| ≤ b a.s. and Var[ω m (Z j )] ≤ v. First notice |ω m (Z j )| = 0≤i,k≤m -1 a k (ψ) G -1 m k,i ϕ i (Z j ) ≤ m -1 i=0 ϕ i (Z j ) m -1 k=0 a k (ψ) G -1 m k,i ≤ √ 2 m -1 i=0 m -1 k=0 a k (ψ) G -1 m k,i ≤ √ 2 t ψ m G -1 m 1 := b.
Then we have

Var[ω m (Z j )] ≤ E[ω 2 m (Z j )] ≤ E   m -1 k=0 m -1 i=0 a k (ψ) G -1 m k,i ϕ i (Z j ) 2   ≤ R + m -1 k=0 m -1 i=0 a k (ψ) G -1 m k,i ϕ i (u) 2 h(u) du ≤ h ∞ t ψ m G -1 m 2 2 := v.
At last applying Bernstein inequality, we get

P | θm -ϑ m | ≥ V (m ) 6 + u ≤ 2 exp - n 4v V (m ) 6 + u ∨ exp - n 4b V (m ) 6 + u ≤ 2 exp - n 4v V (m ) 6 + u ∨ exp - nα 4b V (m ) 6 + u - n(1 -α) 4b V (m ) 6 + u ≤ 2 exp - n 4v V (m ) 6 + u ∨ exp - nα 4b V (m ) 6 exp - n(1 -α) 4b √ u , with α ∈ [0, 1]. Yet we get n 4v V (m ) 6 ≥ κ log n 24 ≥ p log n
as soon as κ ≥ 24p.

By definition we have nα 4b

V (m ) 6 = nα 4 √ 2 t ψ m G -1 m 1 κ( h ∞ ∨ 1) max 1≤k≤m t ψ k G -1 k 2 2 log n 6n = α κ( h ∞ ∨ 1) 96 max 1≤k≤m t ψ k G -1 k 2 2 n log n t ψ m G -1 m 2 1
And as m ∈ M, m ≤ n/ log n and according to the following inequalites for any u

∈ R k u 2 ≤ u 1 ≤ √ k u 2 , it yields using (A) nα 4b V (m ) 6 ≥ α κ( h ∞ ∨ 1) 96 n log n m ≥ α ( h ∞ ∨ 1) 96 log n ≥ α h ∞ 96 log n. Thus nα 4b V (m ) 6 ≥ p log n as soon as κ ≥ 96p 2 /α 2 .
Now putting these results into Equation ( 27), we get

E[D 2 ] ≤ 2 m ∈M R + exp - nu 4v -p log n ∨ exp - n(1 -α) 4b √ u -p log n du ≤ 2 m ∈M n -p R + exp - nu 4v ∨ exp - n(1 -α) 4b √ u du ≤ 8n -p m ∈M max h ∞ t ψ m G -1 m 2 2 n , 8 t ψ m G -1 m 2 1 n 2 (1 -α) 2 ≤ C(α, h ∞ )n -p m ∈M max t ψ m G -1 m 2 2 n , t ψ m G -1 m 2 1 n 2 .
Since m ∈ M, we get

E[D 2 ] ≤ C(α, h ∞ )n -p m ∈M t ψ m G -1 m 2 2 n ≤ C(α, h ∞ )n -p Card(M).
Thus for p = 2, we have

E[D 2 ] ≤ C n . (28)
• Finally noticing that

D 3 = max sup m >m ( θm -ϑ m ) 2 - V (m ) 6 + , sup m ≤m ( θm -ϑ m ) 2 - V (m ) 6 + ≤ max sup m >m ( θm -ϑ m ) 2 - V (m ) 6 + , D 2 ≤ max sup m >m ( θm -ϑ m ) 2 - V (m) 6 + , D 2 ≤ ( θm -ϑ m ) 2 - V (m) 6 + + D 2 ≤ 2D 2 ,
where we used the fact that m → V (m) is nondecreasing. Now applying the same reasoning as for D 2 it yields that

E[D 3 ] ≤ C n . (29)
Now gathering Equations ( 26), ( 28) and (29) into Equation ( 25), we get To bound the first two terms on the r.h.s. of the previous inequality we apply Proposition 2.1 and Proposition 5.1 we get

E[A(m)] ≤ sup k>m k∈M (ϑ k -ϑ m ) 2 + C n .
E(ϑ -θ m ) 2 ≤ 3 sup k>m k∈N ∪ {+∞} (ϑ k -ϑ m ) 2 + 48V (m) + C n + 12E    sup m ≥m m,m ∈M V (m ) -V (m )    . ( 30
)
To control the fluctuation of the random penalty V (m ) around the true one, we need to introduce the following set

Λ = ĥD ∞ -h ∞ < h ∞ 2 .
On one hand, on the set Λ we have that ĥD ∞ ≤ (3/2) h ∞ , it yields that

E    sup m >m m,m ∈M V (m ) -V (m ) 1 Λ    = E ( h ∞ ∨ 1) -(2 ĥD ∞ ∨ 1) 1 Λ κ max 1≤k≤m t ψ k G -1 k 2 2 log n n ≤ 4κ( h ∞ ∨ 1) max 1≤k≤m t ψ k G -1 k 2 2
log n n = 4V (m).

On the set Λ c , on the other hand using that m ∈ M and Assumption (A)

E    sup m >m m,m ∈M V (m ) -V (m ) 1 Λ c    = E ( h ∞ ∨ 1) -(2 ĥD ∞ ∨ 1) 1 Λ c κ max 1≤k≤m t ψ k G -1 k 2 2 log n n ≤ κ log n E ( h ∞ ∨ 1) -(2 ĥD ∞ ∨ 1) 1 Λ c ≤ κ log n E [( h ∞ ∨ 1)1 Λ c ] + 2κ log n E ( ĥD ∞ ∨ 1)1 Λ c . Noticing that ĥD ∞ ≤ D-1 k=0 ϕ k ∞ ≤ 2D ≤ 2n, we get E    sup m >m m,m ∈M V (m ) -V (m ) 1 Λ c    ≤ κ log n( h ∞ ∨ 1)E [1 Λ c ] + 2κn log n E [1 Λ c ] .
We now apply the following Lemma proved just after Lemma 5.2. For p > 0, log n ≤ D ≤ h ∞ 128 √ 2 n (log n) p and n large enough

P ĥD ∞ -h ∞ ≥ h ∞ 2 ≤ 2D n p .
It yields for p = 2

log n E [1 Λ c ] ≤ 2 log n D n 2 ≤ 2 log n h ∞ 128 √ 2 n (log n) 2 1 n 2 ≤ √ 2 128 h ∞ n
and similarly for p = 3

n log n E [1 Λ c ] ≤ √ 2 128 h ∞ n .
Finally, we have

E    sup m ≥m m,m ∈M V (m ) -V (m )    ≤ 4V (m) + C n . (31)
In conclusion, gathering Equations ( 30) and ( 31) gives the desired result.

Proof of Lemma 5.2. Let us notice that

P | ĥD ∞ -h ∞ | ≥ h ∞ /2 ≤ P ĥD -h ∞ ≥ h ∞ /2 ≤ P ĥD -h D ∞ ≥ h ∞ /4 + P [ h D -h ∞ ≥ h ∞ /4] := P 1 + P 2 . ( 32
)
• First consider P 1 , the following inequalities hold true.

P 1 = P sup x∈R + D-1 k=0 (â k (Z) -a k (h))ϕ k (x) ≥ h ∞ /4 ≤ P D-1 k=0 |â k (Z) -a k (h)| ≥ h ∞ /(4 √ 2) ≤ D-1 k=0 P |â k (Z) -a k (h)| ≥ h ∞ /(4 √ 2D) ≤ D-1 k=0 P 1 n n i=1 (ϕ k (Z i ) -E[ϕ k (Z i )]) ≥ h ∞ /(4 √ 2D)
In order to apply Bernstein's inequality, let us notice that |ϕ k (Z 1 )| ≤ √ 2 and Var[ϕ k (Z 1 )] ≤ 2 which yields that

P 1 ≤ 2D exp - n 8 h ∞ 4 √ 2D ∨ exp - n 4 √ 2 h ∞ 4 √ 2D .
Taking D ≤ h ∞ 128 √ 2 n (log n) p , we get that

P 1 ≤ 2D n p .
• Now let us prove that for n large enough P 2 = 0. According to [START_REF] Mabon | Adaptive deconvolution on the nonnegative real line[END_REF] we can write the following series of equalities, Then there exists a certain n 0 such that f D -f g + f g -g D ≤ ε. Besides D ≥ log n and n ≥ n 0 ensure that f D -f g + f g -g D ≤ ε. Then starting from Equation (32), we get

P 2 ≤ P ε ≥ h ∞ 4 = 0
In the end

P ĥD ∞ -h ∞ ≥ h ∞ 2 ≤ 2D n p .
Appendix A.

2.

  Statistical model and estimation procedure 2.1. Notations. For two real numbers a and b, we denote a ∨ b = max(a, b) and a ∧ b = min(a, b).

  with W ∼ 0.4Γ(5, 1) + 0.6Γ(13, 1), I = [0, 13]. Log-normal distribution : LN (1, 1), I = [0, 15]. We then choose a Gamma distribution for the error distribution which verifies (C1)-(C2) for r = 2. Gamma noise: Γ(2, 1 √ 20 ) and Γ(2, 1 √

Figure 1 .

 1 Figure 1. m in function of κ ∈ [10 -4 , 10] (15 equidistant calibration constants, some are superposed) plus κ = 0 (top) and κ = 20 (bottom), n = 5000 and σ 2 = 1/10 and 15 equidistant x 0 in the interval [0, 5] for the β and [0, 10] for the χ 2 .

Figure 3 .

 3 Figure 3. Estimation of the Gamma density: left for n = 200 and right n = 2000; σ 2 = 1/10. In bold black line the true density, in orange data-driven estimators computed on various samples and in green dashed line the oracle (the best in the collection) estimator.

Figure 4 .Figure 5 .

 45 Figure 4. Estimation of the Mixed Gamma density: left for n = 200 and right n = 2000, σ 2 = 1/10. In bold black line the true density, in orange data-driven estimators computed on various samples and in green dashed line the oracle (the best in the collection) estimator.

  Section 3. 5.2.1. Proof of Theorem 3.1. By definition of m, the following inequalities hold

5. 2 . 2 .

 22 Proof of Corollary 3.2. By definition of m, the following inequalities hold(ϑ -θ m ) 2 ≤ 3(ϑ -θm ) 2 + 3( θm -θm∧ m ) 2 + 3( θm∧ m -θ m ) 2 ≤ 3(ϑ -θm ) 2 + 3( A( m) + V (m) + A(m) + V ( m)) ≤ 3(ϑ -θm ) 2 + 6( A(m) + V (m)) ≤ 3(ϑ -θm ) 2 + 6(A(m) + V (m)) + 6( V (m) -V (m)) + 6 sup m >m m,m ∈M V (m ) -V (m ) + ≤ 3(ϑ -θm ) 2 + 6(A(m) + V (m)) + 12 sup m ≥m m,m ∈M V (m ) -V (m ) .Taking expectation, we get E(ϑ -θ m ) 2 ≤ 3E(ϑ -θm ) 2 + 6 (E[A(m)] + V (m)

0 f 0 f

 00 2 (a k-l (g) -a k-l-1 (g)) a l (f a k (f )a j (g) x 0 ϕ k (u)ϕ j (x -u) du = x 0 f D (x -u)g D (u) du.Thus we have|h D (x) -h(x)| = x 0 f D (x -u)g D (u) du --f )(x -u)g D (u) du -x (x -u)(g -g D )(u) du ≤ x 0 (f D -f )(x -u)g D (u) du + x (x -u)r D (u) du ≤ f D -f g + f g -g D .Since f, g ∈ L 2 (R + ), we get g -

  Lemma A.1. (Bernstein's inequality) Let X 1 , . . . , X n be i.i.d.random variables such that Var(X 1 ) ≤v 2 and |X 1 | ≤ b a.s. Let S n = n i=1 (X i -E[X i ]), then for ε > 0 P [|S n -E[S n ]| ≥ nε] ≤ 2 max expk (1 -x) p-k ≤ |1 -x| j-p max 0≤k≤p j k (|x| + |1 -x|) p ≤ 2 p |1 -x| j-p .

Table 1 .

 1 Results of simulation for the global estimation based on

	1
	4

Table 2 .

 2 Results of simulation for the pointwise estimation. Approximation of the integrated risk multiplied by 100 averaged over 500 samples. σ 2 denotes the level of variance of the noise.

	1
	4

Appendix C.

According to [START_REF] Comte | Adaptive Laguerre density estimation for mixed Poisson models[END_REF], proving that a function ψ belongs to a Laguerre-Sobolev defined by ( 12), with α an integer, is equivalent to prove that

And we have for ψ defined as in Example 3 that

acknowledgements The author would like to thank Fabienne Comte for initiating this work as well as for her good advice and suggestions all along the writing of this paper.


Antoine Lemenant

Rectifiability of non Euclidean planar self-contracted curves

.

Introduction

The concept of self-contracted curve was first introduced by Daniilidis is non-increasing on [a, b]. Sometimes to emphasis on the used distance we will also say d-self-contracted curve or • -self-contracted curve when the distance comes from a norm.

The origin of this definition comes from the fact that, in the Euclidean space (R n , |•|), any solution of the gradient descent of a proper convex fonction f : R n → R, i.e. a solution γ of the gradient system γ ′ (t) = -∇f (γ(t))

t > 0,

γ(0) = x 0 ∈ R n (1.1)
is a self-contracted curve (actually it is enough for f to be quasiconvex, that is, its sublevel sets to be convex).

One of the main question about solutions of (1.1), is whether or not bounded solutions are of finite length. If f is analytic (and not necessarily convex) then it follows from the famous Lojasiewicz inequality [START_REF] Lojasiewicz | Une propriété topologique des sous-ensembles analytiques réels[END_REF], while it fails for general C ∞ functions [8, p. 12]. Now for a general convex function, the Lojasiewicz inequality does not need to hold (see [START_REF] Bolte | Characterizations of Lojasiewicz inequalities: subgradient flows, talweg, convexity[END_REF]Section 4.3]). However, bounded solutions have finite length.

The latter follows from the main result of [START_REF] Daniilidis | Asymptotic behaviour of selfcontracted planar curves and gradient orbits of convex functions[END_REF] (see also [START_REF] Manselli | Maximum length of steepest descent curves for quasiconvex functions[END_REF]), which focuses on the following more general question, which is purely metric: Does any bounded self-contracted curve have finite length?

(1.2)

In [START_REF] Daniilidis | Asymptotic behaviour of selfcontracted planar curves and gradient orbits of convex functions[END_REF] it is proved that the answer is yes in a two dimensional Euclidean space. It was then established in higher dimensions in [START_REF] Daniilidis | Lemenant Rectifiability of Self-contracted curves in the euclidean space and applications[END_REF], still in the Euclidean setting. However, it is no more true in a general infinite dimensional Hilbert space.

A natural question is whether the answer to (1.2) remains yes in a finite dimensional space, when the metric space is not Euclidean anymore. In the recent paper [START_REF] Daniilidis | Rifford Self contracted curves in Riemannian manifolds Preprint[END_REF], it is proved that it holds true on a Riemannian manifold.

In this paper we prove that the answer to (1.2) is yes in R 2 endowed with a strictly convex C 2 norm.

Theorem 1.1. Let • be a C 2 and strictly convex norm on R 2 , let I ⊂ R be an interval and let γ :

I → R 2 be a • -self-contracted curve. Then there exists a constant C > 0 depending only on • such that ℓ(γ) ≤ Cdiam(K(γ)) (1.3)
where ℓ(γ) is the length of the curve, K(γ) is the closed convex hull of the support of the curve γ, and diam(K) is the diameter of K. In particular, any bounded • -self-contracted curve has finite length.

Notice that to establish the inequality (1.3), the norm which is used to compute ℓ(γ) or diam(K) does not really matter, since all the norms are equivalent on R 2 . Actually, we will establish (1.3) with ℓ(γ) and diam(K) computed using the Euclidean norm on R 2 , denoted | • |. More precisely, our strategy is to reproduce the Euclidean argument used in [START_REF] Daniilidis | Lemenant Rectifiability of Self-contracted curves in the euclidean space and applications[END_REF], in the simple case of dimension 2, and prove that if γ is no more | • |-self-contracted but merely • -self-contracted, then the proof still works almost by the same way.

However, there are some notable differences. The main one concerns the starting point of the proof, which consists, for any t 0 < t 1 , of localizing the curve "after" t 1 on one side of the perpendicular bisector of the segment [γ(t 0 ), γ(t 1 )]. This follows directly from the definition of being self-contracted. Indeed, for t 0 < t 1 < t, since the function s → d(γ(s), γ(t)) is non-decreasing on [t 0 , t] we have

d(γ(t 0 ), γ(t)) ≤ d(γ(t 1 ), γ(t))
which means that γ(t) is situated on one side of the bisector

M(γ(t 0 ), γ(t 1 )) := {x ∈ R 2 : d(γ(t 0 ), x) = d(γ(t 1 ), x)}.
For the Euclidean distance this yields a "separating line" for the curve after t 0 , whose direction is orthogonal to γ(t 0 )γ(t 1 ). If the distance is not Euclidean anymore, the line segment bisector is no longer perpendicular, and not even a line anymore. However, if the distance is coming from a C 2 and strictly convex norm, we prove that the bisector is a curve which is asymptotic to a line at infinity, whose direction is well identified: in a certain sense it is a direction dual to γ(t 0 )γ(t 1 ). Moreover, the bisector stays close enough to the middle line having the same direction. After noticing those facts, we are able to adapt the proof of [START_REF] Daniilidis | Lemenant Rectifiability of Self-contracted curves in the euclidean space and applications[END_REF] and this is how we prove Theorem 1.1.

Acknowledgements:

The author wishes to thank Aris Daniilidis, Estibalitz Durand-Cartagena, Michael Goldman and Vincent Munnier for useful discussions on the subject of this paper. This work was partially supported by the project ANR-12-BS01-0014-01 GEOMETRYA financed by the French Agence Nationale de la Recherche (ANR).

Notation and terminology

In this paper we will work on R 2 endowed with the Euclidean norm |x| with scalar product x, y . The Euclidean ball with center x and radius R will be denoted by B(x, R) and S N -1 is the Euclidean unit sphere. For v ∈ R 2 we will denote by v ⊥ the image of v the rotation of angle π 2 in the anticlockwise direction. Eventually, on this Euclidean space R 2 we shall also consider another non Euclidean norm that will be denoted by • . The associated ball will be denoted by B • (x, r) and the sphere for the norm • will then be denoted by ∂B • (x, r).

A curve is a mapping γ : I → R 2 , not necessarily continuous, from some interval I ⊂ R. The length of a curve is the quantity

ℓ(γ) := sup m-1 i=0 |γ(t i ) -γ(t i+1 )| ,
where the supremum is taken over all finite increasing sequences t 0 < t 1 < • • • < t m that lie in the interval I.

The mean width of a convex K ⊂ R 2 is the quantity

W (K) = 1 2π ˆS1 H 1 (P u (K))du
where P u is the orthogonal projection onto the real line in R 2 directed by the vector u ∈ S 1 , and H 1 denotes the 1-dimensional Hausdorff measure. Although we shall not use it in this paper, let us mention the following nice identity valid for any compact and convex set K ⊂ R 2 ,

W (K) = H 1 (∂K) π .
It is also clear from the definition that

W (K) ≤ diam(K).
If x, y ∈ R 2 we will denote by [x, y] ⊂ R 2 the segment between x and y.

A norm • is said to be C 2 if x → x is of class C 2 on R 2 \ {0}.
This implies that the sphere ∂B • (x, r) is a C 2 manifold. A norm is said to be C 2 and strictly convex if, for any x ∈ R 2 \ {0} we have D 2 x > 0 (in the sense of quadratic forms). This implies that the ball B • (0, 1) is strictly convex in the geometrical sense, i.e. for any couple of points x, y ∈ ∂B • (0, 1) we have [x, y] ∩ ∂B • (0, 1) = {x, y}. In other words the segment [x, y] without its extremities x and y lies in the interior of the ball B • (0, 1). This means that the sphere ∂B • (0, 1) = {x, y} contains no "flat-parts".

Preliminaries about the curve bisector

Let • be a C 2 and strictly convex norm on R 2 , and a, b ∈ R 2 be any given points. Then we focus on the curve bisector defined by

M(a, b) := {z ∈ R 2 : a -z = b -z }.
If the norm is Euclidean, it is well known that M(a, b) is the line passing through the middle point a+b 2 and perpendicular to ba. But for a general norm, the curve bisector may not be perpendicular and may even not be a line. However, it is not difficult to see that M(a, b) is always asymptotically converging to a line at infinity, and the direction of that line is given by the dual vector to ba (i.e. the vector on the sphere B • (0,1) at which the tangent to the sphere has direction ba). Figure 1 represents the curve bisector of two points for the norm

• 4 in R 2 .
To prove this, we shall need to consider the direction defined in the following definition.

Definition 2.1 (Definition of L x ). Let • be a C 2 and strictly convex norm on R 2 . For any x ∈ ∂B • (0, 1) we define the vectorial line "dual to x", denoted L x , in the following way: there exists exactly two points y 1 , y 2 ∈ ∂B • (0, 1) such that the tangent line to ∂B • (0, 1) at point y i is directed by x. Then L x is the line passing through y 1 and y 2 . If x = 0 and x = 1, we denote by L x the line L x/ x . Figure 1: The curve bisector of the segment [A, D] for the norm • 4 is a curve which is asymptotic to a line at infinity, whose direction is given by the point on the sphere ∂B • 4 (0, 1) at which the tangent to the sphere has the same direction as the segment [A,D].

Remark 2.2. A geometrical way to find the y i is as follows. Let H x ⊂ R 2 \B • (0, 1) be a line directed by x. Then translate this line in the direction x ⊥ until it touches the sphere ∂B • (0, 1). The touching point is then one of the y i . Since the norm is C 2 and strictly convex, there exists only two of those points, which from the symmetry of the ball with respect to the origin must satisfy y 1 = -y 2 . For example, in Figure 1, the line passing through the origin is

L D-A = L A-D . Remark 2.3. For all x ∈ ∂B • (0, 1), let α(x) ∈ [0, π
2 ] be the smallest angle between the two lines L x and Rx. Then it is easy to see that π 2 ≥ inf x∈∂B • (0,1) α(x) =: α 0 > 0. Indeed, the infimum is actually a minimum, and it cannot be zero because the origin lies in the interior of the ball. Roughly speaking, the angle α 0 quantifies how far the ball for • is from being the Euclidean ball, in the sense of how the tangent line to the sphere can be far from making an angle of π 2 with the radius. Another way to define α 0 is as follows: for every x ∈ ∂B • (0, 1) let ν x be outer unit normal vector to ∂B • (0, 1) at point x. Then α 0 is the unique real number in [0, π 2 ] solution to

sin(α 0 ) = inf x∈∂B • (0,1) ν x , x |x| . (2.1)
The following proposition says that M(a, b) converges to the line

L(a, b) := b + a 2 + L b-a (2.2)
at infinity. converging on the compact set ∂B • (0, 1) to some w and w ′ . We actually have that w = w ′ because

w n -w ′ n = b -a /R n = 1/R n → 0.
Now we claim that w is directing the line L x , where x = ba. To see this, it is enough showing that the tangent line to ∂B • (0, 1) at point w is directed by x. This follows from the fact that, since ∂B • (0, 1) is a C 2 manifold, the rescaled secants (w nw ′ n )/ w nw ′ n , which is nothing but ab, converges to the tangent line to ∂B • (0, 1) at the point w.

Now to prove the proposition, we need to go one order further, namely use a C 2 argument. For this purpose, we adopt the following point of view: let a, b ∈ R 2 be such that ab = 1 and let v = ba the direction of the segment [a, b]. For all t ∈ R, we denote by H t := (0, t) + Rv the line directed by v in R 2 , of height t. Let t 0 > 0 be the maximum t such that H t ∩ B • (0,1) = ∅. Let also a t and b t be the two points of H t ∩ ∂B • (0, 1), for t ∈ (-t 0 , t 0 ). Then for any t ∈ (-t 0 , t 0 ), it is clear that the origin lies on the curve bisector M(a t , b t ) (see Figure 2). Actually, by using the rescaling (b ta t )/ b ta t we get this way a complete parametrization on (-t 0 , t 0 ) of M(0, v), the curve bisector of a the segment of unit length directed by v starting from the origin (which is the same as M(a, b), after translation to the origin).

Moreover, when t converges to -t 0 or t 0 , the points a t and b t converge to some points -y and y on the sphere ∂B • (0, 1), the ones at which the tangent line to ∂B • (0, 1) is directed by v. The line passing through -y and y is exactly L v .

Next, for any R ≥ 1 2 , there exists a unique t R ∈ (-t 0 , 0) such that

1 at R -bt R = R. Therefore, if z R ∈ M(0, v) is such that z R = R, dist(z R , L(0, v)) ≤ R|m t R -l t R |, (2.3)
where m t = (a t + b t )/2 is the middle point of the segment [a t , b t ], and l t is the intersecting point of the two segments [0, -y] and [a t , b t ] (see Figure 2). 0

H t a t b t m t -y 1 1 B • (0, 1) v l t
|m t -l t | ≤ C|a t -b t | 2 + o(|a t -b t | 2 ), (2.4)
for some C > 0 (depending only on • ). This is enough to conclude because returning to (2.3) we get, using that

a t R -b t R = 1 R , dist(z R , L(0, v)) ≤ R|m t R -l t R | ≤ C ′ 1 R -→ R→+∞ 0.
Now to prove (2.4), we work locally around the point -y ∈ ∂B • (0, 1). Let T -y be the tangent line at point -y. Since the norm is C 2 and strictly convex, in a certain well chosen coordinate system, the sphere ∂B • (0, 1) almost concides with the graph of s → As 2 for some A > 0, up to some error of second order. Consequently, in that coordinate system, the segments [a tb t ] which have the same direction as T -y , almost concides with the segment [-x s , x s ] where x s = (s, As 2 ), again up to some error of second order. On the other hand the angle θ between the radius [0, -y] of the ball B • (0, 1) and the vertical line orthogonal to T -y , is at most π 2α 0 , where α 0 is defined in Remark 2.3 (see Figure 3).

-y We deduce that |m t -

[0, -y] T -y θ ≤ π 2 -α 0 As 2 -x s x s s ∂B • (0, 1)
a t b t
l t | ≤ |A sin(θ)s 2 | + o(s 2 ) = Cs 2 + o(s 2
), as desired. This achieves the proof of (2.4), and of the proposition. Definition 2.5. For any x ∈ R 2 and v ∈ R 2 with v = 1 we denote by Q v (x) the projection of x ∈ R 2 on Rv parallely to L v (i.e. the v component of x written in the basis (v, v ′ ) where v ′ is a vector director for L v ).

The next proposition quantifies the maximum distance between M(a, b) and the line L(a, b). Proposition 2.6. Let • be a C 1 and strictly convex norm on R 2 . There exists 0 < κ < 1/2 depending on the norm • , such that for any given a, b ∈ R 2 , the curve bisector M(a, b) is contained in the strip S κ (a, b) defined as

S κ (a, b) := {x ∈ R 2 : |Q b-a b-a (x - a + b 2 )| ≤ κ b -a }.
Proof. We first prove that for all a, b in R 2 , M(a, b) ⊆ S 1/2 (a, b). Let a, b be given. We can assume that ba = 1 and we denote by v = ba. Adopting the same notation a t , b t , m t as in the proof of Proposition 2.4, we see that

sup{|Q v (x -v/2)| : x ∈ M(0, v)} = sup t∈[-t 0 ,t 0 ] |Q v (m t )| b t -a t .
By convexity of the ball B • (0,1) we have

L v ∩ [a t , b t ] = ∅ for all |t| ≤ t 0 , from which we find that sup t∈[-t 0 ,t 0 ] |Q v (m t )| b t -a t ≤ 1/2,
as claimed. Now to pass from 1/2 to some κ < 1/2 we argue by contradiction. Assuming the proposition is false, we can find a n := a tn , b n := b tn , such that

|Q v (m n )| b n -a n -→ n→+∞ 1/2, (2.5)
where m n := (a n + b n )/2. According to Proposition 2.4, we can assume that

inf n a n -b n ≥ δ, (2.6)
for some δ > 0. Indeed, let R > 0 be given by Proposition 2.4 for ε := 1/4 and let δ := 1/R. Then, for all a t and b t satifying r := a tb t ≤ δ, we have that a t /r = 1/r ≥ 1/δ = R thus (0, 0) ∈ M(a t /r, b t /r) \ B(a t /r, R) and therefore the conclusion of Proposition 2.4 says

dist((0, 0), L(a t /r, b t /r)) ≤ ε = 1 4 .
In particular this implies

|Q v ( at+bt 2 )|/ b t -a t ≤ 1/4.

Rectifiability of • -SC-curves

In the sequel if γ is a • -self-contracted curve, we denote by Γ := γ(I) the image of the curve, and for all x ∈ Γ we introduce the "piece of curve after x" namely, Γ(x) := {y ∈ Γ : x y},

where denotes the order on the curve given by its parameterization. We will also denote by Ω(x) the convexe hull of Γ(x).

We start with a first lemma about the maximum aperture of the angle between y ′x 0 and yx 0 when x 0 , y, y ′ are all lying on Γ with y and y ′ after x 0 . Namely, for an Euclidean self-contracted-curve, it is easy to see that whenever x 0 , y, y ′ ∈ Γ are satisfying x 0 y y ′ we have

y -x 0 , y ′ -x 0 ≥ 0.
The way this is proved in [START_REF] Daniilidis | Lemenant Rectifiability of Self-contracted curves in the euclidean space and applications[END_REF] is as follows. From the self-contracted property we infer that |y ′x 0 | ≥ |y ′ -y|.

Thus writing y ′y = (y ′x 0 ) + x 0y and squaring the estimates we get

|y ′ -x 0 | 2 ≥ |y ′ -x 0 | 2 + |y -x 0 | 2 -2 y ′ -x 0 , y -x 0 which yields y ′ -x 0 , y -x 0 ≥ 0.
For a general norm which is no more Euclidean, we get a similar estimate from a different argument. Lemma 3.1. Let • be a C 2 and strictly convex norm on R 2 . Let α 0 be the constant of Remark 2.3 and let γ : I → R 2 be a • -self-contracted curve. Then for every x 0 , y, y ′ ∈ Γ satisfying x 0 y y ′ we have

y -x 0 |y -x 0 | , y ′ -x 0 |y ′ -x 0 | ≥ -cos(α 0 ). (3.1)
Proof. The argument is purely geometric. Let x 0 , y, y ′ ∈ Γ satisfying x 0 y y ′ . The self contracting property yields y ′x 0 ≥ y ′y which means that x 0 ∈ B • (y ′ , yy ′ ). Hence, assuming y ′ = 0 and y = 1, to find a lower bound for the left-hand side of (3.1) we can consider the following problem, denoting B := B • (0, 1), and fixing y ∈ ∂B,

inf x∈R 2 \B y -x |y -x| , -x |x| . (3.2)
To solve this problem, we first prove that it is equivalent to take the infimum in x ∈ ∂B. Indeed, let us define

ϕ(x) := x -y |y -x| , x |x|
and consider the function f of the real variable t > 0 defined by

f (t) = ϕ(tx). A simple computation shows that d dt |y -tx| = tx-y,x |y-tx| and d dt tx -y, x = |x| 2 thus f ′ (t) = |x| 2 |y -tx| 2 -tx -y, x 2 |x||y -tx| 3 ≥ 0,
due to the Cauchy-Schwarz inequality. This implies that t → ϕ(tx) is non decreasing in t and the infimum in (3.2) is the same as:

inf x∈∂B x -y |y -x| , x |x| . (3.3)
But now it is easy to conclude using the convexity of B. Indeed, for a given x ∈ ∂B, let ν x be the unit inner normal vector to ∂B at point x. Then by convexity of B, since y ∈ ∂B we must have yx, ν x ≥ 0, and by definition of α 0 we deduce that

x -y |y -x| , x |x| ≥ -cos(α 0 ).
We are now ready to prove the main result of this paper.

Proof of Theorem 1.1. We use the same notation ≺, Γ, Γ(x), Ω(x) as before. The begining of the proof follows essentially the proof of [START_REF] Daniilidis | Lemenant Rectifiability of Self-contracted curves in the euclidean space and applications[END_REF]Theorem 3.3] which is divided into several steps. Although the first two steps are very close to the Euclidean situation, we write here the full detail. The first step consists in first noticing that to prove the theorem, it is enough finding some c 0 > 0 such that for any pair of points x, x ′ ∈ Γ with x ′ x it holds

W (Ω(x)) + c 0 |x -x ′ | ≤ W (Ω(x ′ )) (3.4) (this is Claim 1 of [2, Theorem 3.3]).
Indeed, letting t 0 < t 1 . . . < t m be any increasing sequence in I, and set

x i := γ(t i ). If (3.4) holds, then m-1 i=0 |γ(t i+1 ) -γ(t i )| = m-1 i=0 |x i+1 -x i | ≤ 1 c 0 m-1 i=0 W (Ω(x i ) -W (Ω(x i+1 )) = 1 c 0 (W (Ω(x 0 )) -W (Ω(x m ))) ≤ 1 c 0 W (Ω(x 0 )),
since the mean width W (H) is a nondecreasing function of H (the variable H is ordered via the set inclusion). Taking the supremum over all choices of t 0 < t 1 . . . < t m in I we obtain (1.3) for C = 1/c 0 .

Therefore, the theorem will be proved, if we show that (3.4) holds for some constant c 0 > 0. Before we proceed, we introduce some extra notation similar to the ones of [START_REF] Daniilidis | Lemenant Rectifiability of Self-contracted curves in the euclidean space and applications[END_REF] but modified with the constant κ. Indeed, let κ be the constant given by Proposition 2.6 depending only on • and let λ := 1 2κ > 0. Let x, x ′ be fixed in Γ with x ′ ≺ x. We set (see Figure 4)

v 0 := x ′ -x |x ′ -x| and x 0 := x ′ -v 0 λ 2 |x -x ′ |. (3.5)
Let us also set

ξ 0 (y) = y -x 0 |y -x 0 | ∈ S 1 , for any y ∈ Γ(x). (3.6)
Notice that x 0 , v 0 and ξ 0 (y) depend on the points x, x ′ , while the desired constant c 0 does not (may depend only on • ). To determine c 0 , we shall again transform the problem into another one (similar to Claim 2 of [2, Theorem 3.3]).

Let us assume that there exists some constants δ > 0 and τ > 0 (depending only on • ), such that for all x, x ′ in Γ with x ′ ≺ x (and for x 0 , v 0 defined by (3.5)), there exists v ∈ S 1 such that the following two properties hold:

v, v 0 ≥ τ (3.7)
v, ξ 0 (y) ≤ -δ for all y ∈ Γ(x).

(3.8) Then (3.4) holds true (and consequently (1.3) follows). This is a slight modification of Claim 2 of [2, Theorem 3.3]). To prove the latter, assume that such constants δ, τ , and a vector v exist, so that (3.7) and (3.8) holds. Up to change δ into τ /2 if necessay, we may assume that δ ≤ τ /2. Set

V := v ∈ S 1 ; |v -v| ≤ δ .
From (3.8) we get v, yx 0 ≤ 0, for all v ∈ V and y ∈ Ω(x).

(3.9)

Let us now assume that x 0 is the origin. Recall that, for v ∈ R 2 , P v denotes the orthogonal projection onto the line Rv. Observe that Ω(x ′ ) contains the convex hull of Ω(x) ∪ [x, x ′ ], whence P v (Ω(x)) ⊂ P v (Ω(x ′ )).

In particular,

H 1 (P v (Ω(x))) ≤ H 1 (P v (Ω(x ′ ))) for v ∈ S n-1 , (3.10)
Then (3.9) says that for all directions v in V we have

sup P v (Ω(x)) ≤ 0 < P v (x ′ ) ≤ sup P v (Ω(x ′ )). Now for every v ∈ V we have v, v 0 = v, v 0 + v -v, v 0 ≥ τ -|v 0 ||v -v| ≥ τ /2 (3.11)
because δ ≤ τ /2. This gives a lower bound for the length of the projected segment [x 0 , x ′ ] onto Rv. Precisely, recalling that

|x 0 -x ′ | = λ 2 |x -x ′ |,
we get

P v (x ′ ) ≥ λτ 4 |x -x ′ |.
This yields

H 1 (P v (Ω(x))) + λτ 4 |x -x ′ | ≤ H 1 (P v (Ω(x ′ ))) for v ∈ V . ( 3
.12) Integrating (3.12) for v ∈ V and (3.10) for v ∈ S 1 \ V , and summing up the resulting inequalities we obtain (3.4).

Consequently, our next goal is to determine δ > 0 and τ > 0 so that (3.7) and (3.8) hold.

Here comes the point where the proof slightly differs from [2, Theorem 3.3] and where the preliminary section about the curve bisector plays a role. First we notice that from the self-contracted property of the curve, the set Γ(x) lies only on one side of the curve bisector M(x, x ′ ) (the one containing x). On the other hand by applying Proposition 2.6, we know that M(x, x ′ ) ⊂ S κ (x, x ′ ) (see Proposition 2.6 for the definition of S κ (x, x ′ )). Let w be a vector parallel to L v 0 and let ν ∈ S 1 the vector orthogonal to it pointing in the opposite direction with respect to Γ(x). Then it holds, thanks to our definition of x 0 in correspondance to the width of S κ , yx 0 , ν ≤ -λ 2 ≤ 0 ∀y ∈ Γ(x). (3.13) Moreover, the angle between ν and v 0 can be estimated by the constant α 0 defined in Remark 2.3, which yields ν, v 0 ≥ cos( π 2 α 0 ) =: τ 1 > 0.

The vector ν is very close to satisfy the required conditions (3.7) and (3.8). Actually, as already used in [START_REF] Daniilidis | Lemenant Rectifiability of Self-contracted curves in the euclidean space and applications[END_REF], the desired vector v will be constructed as a little perturbation of ν, precisely, of the form

ν ε := ν + εν ⊥ |ν + εν ⊥ |
for some ε > 0 small, to be chosen later.

Let us check first that for any ε ≤ τ 1 /6, it holds ν ε , v 0 ≥ τ 1 /2 so that condition (3.7) would be satisfied for any of those ν ε , with τ = τ 1 /2. Indeed, using the inequality √ 1 + t ≤ 1 + t 2 , we obtain Consequently,

|ν -ν ε | 2 = 1 1 + ε 2 [( √ 1 + ε 2 -1) 2 + ε 2 ] ≤ 1 2 ε 4 + ε 2
ν ε , v 0 = ν, v 0 + ν ε -ν, v 0 ≥ τ 1 -|ν -ν ε | ≥ τ 1 /2
when 3ε ≤ τ 1 /2. Now it remains to find ε ≤ τ 1 /6 that would moreover make ν ε satisfying an inequality like (3.8). To do so, we decompose the half space H := {y : P ν (y) ≤ 0} in two different regions. Let

v 0 -ε 0 v ⊥ 0 |v 0 -ε 0 v ⊥ 0 | = - ε 0 √ 1 + ε 0 ≤ - ε 0 2 , provided that ε 0 ≤ 1.
Gathering all the estimates together, we have found some ν satisfying sup y∈Γ(x) ξ 0 (y), ν ≤ -τ, with τ = min(µ, λ 4 , ε 0 2 ), and this finishes the proof of the Theorem.

  , Ley and Sabourau in [5, Definition 1.2]. For a given metric space (E, d) and a possibly unbounded interval I of R, a map γ : I → E is called a self-contracted curve, if for every [a, b] ⊂ I, the real-valued function t → d(γ(t), γ(b))

Proposition 2 . 4 .

 24 Let • be a C 2 and strictly convex norm on R 2 . Then for every ε > 0 there exists R > 0 such that for all couple of points a, b ∈ R 2 satisfying a = b and ba = 1, sup{dist(z, L(a, b)) : z ∈ M(a, b) \ B(a, R)} ≤ ε, where L(a, b) is defined in (2.2), and dist is the Euclidean distance. Proof. Let a, b ∈ R 2 be satisfying a = b and ba = 1, and let z R ∈ M(a, b) be such that z Ra = R → +∞. Assume for simplicity that z R stays "above" the line directed by ba. Then it is easy to see that z R -a R and z R -b R both converge to the same direction w ∈ ∂B • (0, 1), the one directing the line L b-a . To see this, we let R n be a subsequence making w n := z Rn -a Rn and w ′ n := z Rn -b Rn

Figure 2 :

 2 Figure 2: The distance from M(a t , b t ) to L(a, b) is the same as the distance from m t to L(ba) (the line passing through 0 and -y), which is less than the distance between m t and l t .

Figure 3 :

 3 Figure 3: Situation in the proof of (2.4).

  Hence, our sequence of a n and b n satisfying (2.5) must satisfy a nb n ≥ δ for n large enough. Now by compactness of ∂B • (0, 1) we can assume that a n → a = a t 0 and b n → b = b t 0 satisfying ab ≥ δ. By (2.5) we also have that |Q v (m)| ba = 1/2, where m = (a + b)/2. But this implies that L v ∩ [a, b] is either the point a or the point b, which is a contradiction because, by strict convexity of the ball B, the line L v must meet the segment [a, b] only by its interior. This finishes the proof of the Proposition.

Figure 4 :

 4 Figure 4: Γ(x) lies on one side of the curve bisector M(x, x ′ ) which is contained in the strip S κ (x, x ′ ).

which yields, for ε ≤ 1 ,

 1 |νν ε | ≤ 3ε.(3.14)

H 1 : 1 θFigure 5 :

 115 Figure 5: The minimum angle between yx 0 and ν -ε 0 for y ∈ H + µ is achieved when yx 0 ∈ ν ⊥ .

as desired, and we only need to take care of H 2 . For this purpose, we furthermore decompose H 2 itself in the two regions,

Therefore, if ε ≤ min(α 0 /6, µ/3, λ/12) then (3.15) holds and moreover,

, for all y ∈ H µ .

It remains finally to consider y ∈ H ′ µ . For this we may assume that 2µ ≤ 1 2 so that H ′ µ is divided in two connected components, one denoted H + µ lying in the upper half-space {x : x, v ⊥ 0 ≥ 0} and another one denoted H - µ lying in the other part {x : x, v ⊥ 0 ≤ 0}. Now for µ ∈ (0, 1 2 ) let us define the quantity

It is clear that β µ → -1 when µ goes to 0. Therefore, since α 0 > 0, there exists µ > 0 small enough (depending only on α 0 ), such that β µ <cos(α 0 ). By applying Lemma 3.1, we infer that the following alternative holds

Next, we set ε 0 = min(α 0 /6, µ/3, λ/12).

In the first case (i.e. when Γ(x) ∩ H ′ µ ⊂ H + µ ) we define ν := ν -ε 0 and in the second case (i.e. when Γ(x)

We finally check that this choice suit our purposes. Indeed, it is not difficult to see that

where θ is the angle between ν and ν -ε 0 (see Figure 5).