Christophe Chesneau
email: chesneau@math.unicaen.fr

A general result on the mean integrated squared error of the wavelet hard thresholding estimator under α-mixing dependence

Keywords: AMS 2000 subject classifications: Primary 62G07, 62G20; secondary 60K35 Nonparametric statistics, α-mixing dependence, Wavelets, Besov balls, Hard thresholding mixing2.tex

In this note, we consider the estimation of an unknown function f for weakly dependent data (α-mixing) in a general setting. Our contribution is theoretical: we prove that a wavelet hard thresholding estimator attains a sharp rate of convergence under the mean integrated squared error (MISE) over Besov balls without imposing too restrictive assumptions on the model. Applications are given for two types of inverse problems: the deconvolution density estimation and the density estimation in a GARCH-type model, both improve existing results in this dependent context. Another application concerns the regression model with random design.

Introduction

A general nonparametric problem is adopted: we aim to estimate an unknown function f via n random variables V 1 , . . . , V n from a strictly stationary stochastic process (V t ) t∈Z . We suppose that (V t ) t∈Z has a weak dependence structure; the α-mixing case is considered. This kind of dependence naturally appears in numerous models as Markov chains, GARCH-type models and discretely observed diffusions (see, e.g., [START_REF] Doukhan | Mixing. Properties and Examples[END_REF], [START_REF] Carrasco | Mixing and moment properties of various GARCH and stochastic volatility models[END_REF] and [START_REF] Bradley | Introduction to strong mixing conditions[END_REF]). The problems where f is the density of V 1 or a regression function have received a lot of attention. A partial list of related works includes [START_REF] Robinson | Nonparametric estimators for time series[END_REF], [START_REF] Roussas | Nonparametric estimation in mixing sequences of random variables[END_REF][START_REF] Roussas | Nonparametric regression estimation under mixing conditions[END_REF], Truong and Stone (1992), [START_REF] Tran | Nonparametric function estimation for time series by local imsart-generic ver[END_REF], Masry (1996a,b), [START_REF] Masry | Local polynomial estimation of regression functions for mixing processes[END_REF], [START_REF] Bosq | Nonparametric statistics for stochastic processes[END_REF] and [START_REF] Liebscher | Estimation of the density and the regression function under mixing conditions[END_REF].

For an efficient estimation of f , many methods can be considered. The most popular of them are based on kernels, splines and wavelets. In this note we deal with wavelet methods that have been introduced in i.i.d.setting by Donoho andJohnstone (1994, 1995) and Donoho et al. (1995[START_REF] Donoho | Density estimation by wavelet thresholding[END_REF]. These methods enjoy remarkable local adaptivity against discontinuities and spatially varying degree of oscillations. Complete reviews and discussions on wavelets in statistics can be found in, e.g., [START_REF] Antoniadis | Wavelets in statistics: a review (with discussion)[END_REF] and [START_REF] Härdle | Wavelet, Approximation and Statistical Applications[END_REF]. In the context of α-mixing dependence, various wavelet methods have been elaborated for a wide variety of nonparametric problems. Recent developments can be found in, e.g., [START_REF] Leblanc | Wavelet linear density estimator for a discrete time stochastic process: L p -losses[END_REF], Tribouley and Viennet (1998), [START_REF] Masry | Wavelet-Based estimation of multivariate regression functions in besov spaces[END_REF], [START_REF] Patil | Asymptotics for wavelet based estimates of piecewise smooth regression for stationary time series[END_REF], [START_REF] Doosti | Wavelets for nonparametric stochastic regression with mixing stochastic process[END_REF], [START_REF] Doosti | Multivariate Stochastic Regression Estimation by Wavelets for Stationary Time Series[END_REF], [START_REF] Doosti | Two dimensional wavelets for nonlinear autoregressive models with an application in dynamical system[END_REF], [START_REF] Cai | Nonlinear Wavelet Density Estimation for Truncated and Dependent Observations[END_REF], [START_REF] Niu | Nonlinear Wavelet Estimation of Conditional Density under Left-Truncated and α-Mixing Assumptions[END_REF], [START_REF] Benatia | Nonlinear wavelet regression function estimator for censored dependent data[END_REF], [START_REF] Chesneau | On the adaptive wavelet deconvolution of a density for strong mixing sequences[END_REF]Chesneau ( , 2013a,b),b), [START_REF] Chaubey | On MISE of a Nonlinear Wavelet Estimator of the Regression Function Based on Biased Data under Strong Mixing[END_REF] and [START_REF] Abbaszadeh | Wavelet Density Estimation and Statistical Evidences Role for a GARCH Model in the Weighted Distribution[END_REF].

In the general dependent setting described above, we provide a theoretical contribution to the performance of a wavelet estimator based on a hard thresholding. This nonlinear wavelet procedure has the features to be fully adaptive and efficient over a large class of functions f (see, e.g., Donoho andJohnstone (1994, 1995), Donoho et al. (1995[START_REF] Donoho | Density estimation by wavelet thresholding[END_REF] and [START_REF] Delyon | On minimax wavelet estimators[END_REF]). Following the spirit of [START_REF] Kerkyacharian | Thresholding algorithms, maxisets and well-concentrated bases[END_REF], we determine necessary assumptions on (V t ) t∈Z and the wavelet basis to ensure that the considered estimator attains a fast rate of convergence under the MISE over Besov balls. The obtained rate of convergence often corresponds to the near optimal one in the minimax sense for the standard i.i.d. case. The originality of our result is to be general and sharp; it can be applied for nonparametric models of different natures and improves some existing results. This fact is illustrated by the consideration of the density deconvolution estimation problem and the density estimation problem in a GARCH-type model, improving (Chesneau, 2012, Proposition 5.1) and (Chesneau, 2013a, Theorem 2) respectively. A last part is devoted to the regression model with random design. The obtained result completes the one of [START_REF] Patil | Asymptotics for wavelet based estimates of piecewise smooth regression for stationary time series[END_REF].

The organization of this note is as follows. In the next section we describe the considered wavelet setting. The hard thresholding estimator and its rate of convergence under the MISE over Besov balls are presented in Section 3. Applications of our general result are given in Section 4. The proofs are carried out in Section 5.

Wavelets and Besov balls

In this section we introduce some notations corresponding to wavelets and Besov balls.

Wavelet basis

We consider the wavelet basis on [0, 1] constructs from the Daubechies wavelets db2N with N ≥ 1 (see, e.g., [START_REF] Daubechies | Ten lectures on wavelets[END_REF]). A brief description of this basis is given below. Let φ and ψ be the initial wavelet functions of the family db2N. These functions have the particularity to be compactly supported and to belong to the class C a for N > 5a. For any j ≥ 0, we set Λ j = {0, . . . , 2 j -1} and, for k ∈ Λ j , With appropriated treatments at the boundaries, there exists an integer τ such that, for any integer

φ j,k (x) = 2 j/2 φ(2 j x -k), ψ j,k (x) = 2 j/2 ψ(2 j x -k).
ℓ ≥ τ , B = {φ ℓ,k , k ∈ Λ ℓ ; ψ j,k ; j ∈ N -{0, . . . , ℓ - 1}, k ∈ Λ j } is an orthonormal basis of L 2 ([0, 1]), where L 2 ([0, 1]) = f : [0, 1] → R; ||f || 2 = 1 0 |f (x)| 2 dx 1/2 < ∞ .
For any integer ℓ ≥ τ and f ∈ L 2 ([0, 1]), we have the following wavelet expansion:

f (x) = k∈Λ ℓ c ℓ,k φ ℓ,k (x) + ∞ j=ℓ k∈Λj d j,k ψ j,k (x), x ∈ [0, 1],
where c j,k and d j,k denotes the wavelet coefficients of f defined by

c j,k = 1 0 f (x)φ j,k (x)dx, d j,k = 1 0 f (x)ψ j,k (x)dx. (2.1)
Technical details can be found in, e.g., [START_REF] Cohen | Wavelets on the interval and fast wavelet transforms[END_REF] and [START_REF] Mallat | A wavelet tour of signal processing[END_REF].

In the main result of this paper, we will investigate the MISE rate of the proposed estimator by assuming that the unknown function of interest f belongs to a wide class of functions: the Besov class. Its definition in terms of wavelet coefficients is presented below.

Besov balls

We say that f ∈ B s p,r (M ) with s > 0, p, r ≥ 1 and M > 0 if and only if there exists a constant C > 0 such that the wavelet coefficients of f given by (2.1) satisfy

2 τ (1/2-1/p) k∈Λτ |c τ,k | p 1/p +    ∞ j=τ   2 j(s+1/2-1/p)   k∈Λj |d j,k | p   1/p    r    1/r ≤ C,
with the usual modifications if p = ∞ or r = ∞. Note that, for particular choices of s, p and r, B s p,r (M ) contains the classical Hölder and Sobolev balls (see, e.g., [START_REF] Meyer | Ondelettes et Opérateurs[END_REF] and [START_REF] Härdle | Wavelet, Approximation and Statistical Applications[END_REF]).

Remark 2.1. We have chosen a wavelet basis on [0,1] to fix the notations; wavelet basis on another interval can be considered in the rest of the study without affecting the results.

Statistical framework, estimator and result

Statistical framework

As mentioned in Section 1, a nonparametric estimation setting as general as possible is adopted: we aim to estimate an unknown function f ∈ L 2 ([0, 1]) via imsart-generic ver. 2009/12/15 file: mixing2.tex date: January 4, 2014 n random variables (or vectors) V 1 , . . . , V n from a strictly stationary stochastic process (V t ) t∈Z defined on a probability space (Ω, A, P). We suppose that (V t ) t∈Z has a α-mixing dependence structure with exponential decay rate, i.e., there exist two constants γ > 0 and θ > 0 such that sup m≥1 e θm α m ≤ γ,

where α m = sup (A,B)∈F V -∞,0 ×F V m,∞ |P(A ∩ B) -P(A)P(B)|, F V
-∞,0 is the σalgebra generated by the random variables (or vectors) . . . , V -1 , V 0 and F V m,∞ is the σ-algebra generated by the random variables (or vectors) V m , V m+1 , . . ..

The α-mixing dependence is reasonably weak; it is satisfied by a wide variety of models including Markov chains, GARCH-type models and discretely observed diffusions (see, for instance, [START_REF] Bradley | Introduction to strong mixing conditions[END_REF], [START_REF] Carrasco | Mixing and moment properties of various GARCH and stochastic volatility models[END_REF], [START_REF] Doukhan | Mixing. Properties and Examples[END_REF] and [START_REF] Genon-Catalot | Stochastic volatility models as hidden Markov models and statistical applications[END_REF]).

The considered estimator for f is presented below.

Estimator

We define the wavelet hard thresholding estimator f by

f (x) = k∈Λj 0 ĉj0,k φ j0,k (x) + j1 j=j0 k∈Λj dj,k 1 {| dj,k |≥κλj } ψ j,k (x), (3.1)
where

ĉj,k = 1 n n i=1 q(φ j,k , V i ), dj,k = 1 n n i=1 q(ψ j,k , V i ), (3.2)
1 is the indicator function, κ > 0 is a large enough constant, j 0 is the integer satisfying

2 j0 = [τ ln n], (3.3)
where [a] denotes the integer part of a, j 1 is the integer satisfying

2 j1 = n (ln n) 3 1/(2ρ+1) (3.4) and λ j = 2 ρj ln n n . (3.5)
Here it is supposed that there exists a function q (H1) for γ ∈ {φ, ψ}, any integer j ≥ j 0 and k ∈ Λ j ,

: L 2 ([0, 1]) × V 1 (Ω) → C
E (q(γ j,k , V 1 )) = 1 0 f (x)γ j,k (x)dx,
where E denotes the expectation, (H2) there exist two constants, C > 0 and ρ ≥ 0, satisfying, for γ ∈ {φ, ψ}, for any integer j ≥ j 0 and k ∈ Λ j ,

(i) sup x∈V1(Ω) |q(γ j,k , x)| ≤ C2 ρj 2 j/2 , (ii) E |q(γ j,k , V 1 )| 2 ≤ C2 2ρj , (iii) for any m ∈ {1, . . . , n -1}, |C ov (q(γ j,k , V m+1 ), q(γ j,k , V 1 )) | ≤ C2 2ρj 2 -j ,
where C ov denotes the covariance, i.e., C ov (X, Y ) = E(XY )-E(X)E(Y ), Y denotes the complex conjugate of Y .

For well-known nonparametric models in the i.i.d. setting, wavelet hard thresholding estimators and important results can be found in, e.g., Donoho andJohnstone (1994, 1995), Donoho et al. (1995[START_REF] Donoho | Density estimation by wavelet thresholding[END_REF], [START_REF] Delyon | On minimax wavelet estimators[END_REF], [START_REF] Kerkyacharian | Thresholding algorithms, maxisets and well-concentrated bases[END_REF] and [START_REF] Fan | Wavelet deconvolution[END_REF]. In the α-mixing context, f defined by (3.1) is a general and improved version of the estimator considered in [START_REF] Chesneau | On the adaptive wavelet deconvolution of a density for strong mixing sequences[END_REF]Chesneau ( , 2013a)). The main differences are the presence of the tuning parameter ρ, and the global definition of the function q offering numerous possibilities of applications. Three of them are explored in Section 4.

Comments on the assumptions. The assumption (H1) ensures that (3.2) are unbiased estimators for c j,k and d j,k given by (2.1), whereas (H2) is related to their good performance. See Proposition 5.1 below. These assumptions are not too restrictive. For instance, if we consider the standard density estimation problem where (V t ) t∈Z are i.i.d. random variables with bounded density f , the function q(γ, x) = γ(x) satisfies (H1) and (H2) with ρ = 0 (note that, thanks to the independence of (V t ) t∈Z , the covariance term in (H2)-(iii) is zero). The technical details are given in [START_REF] Donoho | Density estimation by wavelet thresholding[END_REF].

Lemma 3.1 below describes a simple situation in which the assumption (H2)-(iii) is satisfied.

Lemma 3.1. We make the following assumptions.

(F1) Let u be the density of V 1 and, u (V1,Vm+1) be the density of (V 1 , V m+1 ) for any m ∈ Z. We suppose that there exists a constant C > 0 such that

sup m∈{1,...,n-1} sup (x,y)∈V1(Ω)×Vm+1(Ω) |u (V1,Vm+1) (x, y) -u(x)u(y)| ≤ C.
(F2) There exist two constants, C > 0 and ρ ≥ 0, satisfying, for γ ∈ {φ, ψ}, for any integer j ≥ j 0 and k ∈ Λ j ,

V1(Ω) |q(γ j,k , x)|dx ≤ C2 ρj 2 -j/2 .
imsart-generic ver. 2009/12/15 file: mixing2.tex date: January 4, 2014

Then, under (F1) and (F2), (H2)-(iii) is satisfied.

Result

Theorem 3.1 below determines the rate of convergence attained by f under the MISE over Besov balls.

Theorem 3.1. We consider the general statistical setting described in Subsection 3.1. Let f be (3.1) under (H1) and (H2). Suppose that

f ∈ B s p,r (M ) with r ≥ 1, {p ≥ 2 and s ∈ (0, N )} or {p ∈ [1, 2) and s ∈ ((2ρ + 1)/p, N )}. Then there exists a constant C > 0 such that E f -f 2 2 ≤ C ln n n 2s/(2s+2ρ+1)
.

The rate of convergence "((ln n)/n) 2s/(2s+2ρ+1) " is often the near optimal one in the minimax sense for numerous statistical problems in a i.i.d. setting (see, e.g., [START_REF] Härdle | Wavelet, Approximation and Statistical Applications[END_REF] and Tsybakov ( 2004)). Moreover, note that Theorem 3.1 is flexible; the assumptions on (V t ) t∈Z , related to the definition of q in (H1) and (H2), are mild. In the next section, this flexibility is illustrated for three sophisticated nonparametric estimation problems: the density deconvolution estimation problem, the density estimation problem in a GARCH-type model and the regression function estimation in the regression model with random design.

Applications

Density deconvolution

Let (V t ) t∈Z be a strictly stationary stochastic process such that

V t = X t + ǫ t , t ∈ Z, (4.1)
where (X t ) t∈Z is a strictly stationary stochastic process with unknown density f and (ǫ t ) t∈Z is a strictly stationary stochastic process with known density g. It is supposed that ǫ t and X t are independent for any t ∈ Z and (V t ) t∈Z is a α-mixing process with exponential decay rate (see Subsection 3.1 for a precise definition). Our aim is to estimate f via V 1 , . . . , V n from (V t ) t∈Z . Some related works are [START_REF] Masry | Strong consistency and rates for deconvolution of multivariate densities of stationary processes[END_REF], [START_REF] Kulik | Nonparametric deconvolution problem for dependent sequences[END_REF], [START_REF] Comte | Adaptive density deconvolution for dependent inputs with measurement errors[END_REF] and van Zanten and Zareba (2008).

We formulate the following assumptions: (G4) For any m ∈ Z, let u (V1,Vm+1) be the density of (V 1 , V m+1 ). We suppose that there exists a constant C > 0 such that

(G1) The support of f is [0, 1]. (G2) There exists a constant C > 0 such that sup x∈R f (x) ≤ C < ∞.
sup m∈Z sup (x,y)∈R 2 u (V1,Vm+1) (x, y) ≤ C.
(G5) For any integrable function γ, we define its Fourier transform by

F(γ)(x) = ∞ -∞ γ(y)e -ixy dy, x ∈ R.
We suppose that there exist three known constants C > 0, c > 0 and δ > 1 such that, for any x ∈ R,

• the Fourier transform of g satisfies

| F(g)(x)| ≥ c (1 + x 2 ) δ/2 , • for any ℓ ∈ {0, 1, 2}, the ℓ-th derivative of the Fourier transform of g satisfies |(F(g)(x)) (ℓ) | ≤ C (1 + |x|) δ+ℓ .
We are now in the position to present the result.

Theorem 4.1. We consider the model (4.1). Suppose that (G1)-(G5) are satisfied. Let f be defined as in (3.1) with

q(γ, x) = 1 2π ∞ -∞ F (γ)(y) F(g)(y) e -iyx dy, (4.2)
where F (γ)(y) denotes the complex conjugate of F (γ) (y) and ρ = δ (appearing in (G5)).

Suppose that f ∈ B s p,r (M ) with r ≥ 1, {p ≥ 2 and s ∈ (0, N )} or {p ∈ [1, 2) and s ∈ ((2δ + 1)/p, N )}. Then there exists a constant C > 0 such that • exactly the rate of convergence attained by the wavelet hard thresholding estimator, • the near optimal rate of convergence in the minimax sense.

E f -f 2 2 ≤ C ln n n 2s/(2s+2δ+1
The details can be found in [START_REF] Fan | Wavelet deconvolution[END_REF]. Thus Theorem 4.1 can be viewed as an extension of this existing result to the weak dependent case.

GARCH-type model

We consider the strictly stationary stochastic process (V t ) t∈Z where, for any t ∈ Z,

V t = σ 2 t Z t , (4.3) (σ 2 t )
t∈Z is a strictly stationary stochastic process with unknown density f and (Z t ) t∈Z is a strictly stationary stochastic process with known density g. It is supposed that σ 2 t and Z t are independent for any t ∈ Z and (V t ) t∈Z is a αmixing process with exponential decay rate (see Subsection 3.1 for a precise definition). Our aim is to estimate f via V 1 , . . . , V n from (V t ) t∈Z . Some related works are [START_REF] Comte | Adaptive density deconvolution for dependent inputs with measurement errors[END_REF] and Chesneau (2013a).

We formulate the following assumptions:

(J1) There exists a positive integer δ such that

g(x) = 1 (δ -1)! (-ln x) δ-1 , x ∈ [0, 1].
Let us remark that g is the density of (J4) For any m ∈ Z, let u (V1,Vm+1) be the density of (V 1 , V m+1 ). We suppose that there exists a constant C > 0 such that

sup m∈Z sup (x,y)∈R 2 u (V1,Vm+1) (x, y) ≤ C.
We are now in the position to present the result.

Theorem 4.2. We consider the model (4.3). Suppose that (J1)-(J4) are satisfied. Let f be defined as in (3.1) with

q(γ, x) = T δ (γ)(x), (4.4)
where, for any positive integer ℓ, T (γ

)(x) = (xγ(x)) ′ and T ℓ (γ)(x) = T (T ℓ-1 (γ))(x)
and ρ = δ (appearing in (J1)).

imsart-generic ver. 2009/12/15 file: mixing2.tex date: January 4, 2014

Suppose that f ∈ B s p,r (M ) with r ≥ 1, {p ≥ 2 and s ∈ (0, N )} or {p ∈ [1, 2) and s ∈ ((2δ + 1)/p, N )}. Then there exists a constant C > 0 such that

E f -f 2 2 ≤ C ln n n 2s/(2s+2δ+1)
.

Theorem 4.2 significantly improves (Chesneau, 2013a, Theorem 2) in terms of rate of convergence; we gain an exponent 1/2.

Nonparametric regression model

We consider the strictly stationary stochastic process (V t ) t∈Z where, for any

t ∈ Z, V t = (Y t , X t ), Y t = f (X t ) + ξ t , (4.5)
(X t ) t∈Z is a strictly stationary stochastic process with unknown density g, (ξ t ) t∈Z is a strictly stationary centered stochastic process and f is the unknown regression function. It is supposed that X t and ξ t are independent for any t ∈ Z and (V t ) t∈Z is a α-mixing process with exponential decay rate (see Subsection 3.1 for a precise definition). Our aim is to estimate

f via V 1 , . . . , V n from (V t ) t∈Z .
Applications of this problem can be foud in [START_REF] Härdle | Applied Nonparametric Regression[END_REF]. Wavelet methods can be found in [START_REF] Patil | Asymptotics for wavelet based estimates of piecewise smooth regression for stationary time series[END_REF], [START_REF] Doosti | Wavelets for nonparametric stochastic regression with mixing stochastic process[END_REF], [START_REF] Doosti | Two dimensional wavelets for nonlinear autoregressive models with an application in dynamical system[END_REF] and [START_REF] Doosti | Multivariate Stochastic Regression Estimation by Wavelets for Stationary Time Series[END_REF].

We formulate the following assumptions:

(K1) The support of f and g are [0, 1] and f and g

∈ L 2 ([0, 1]). (K2) ξ 1 (Ω) is bounded. ( K3
c * ≤ inf x∈[0,1] g(x), sup x∈[0,1] g(x) ≤ C.
(K5) Let u be the density of V 1 . We suppose that there exists a constant C > 0 such that sup

x∈R×[0,1] u(x) ≤ C.
(K6) For any m ∈ Z, let u (V1,Vm+1) be the density of (V 1 , V m+1 ). We suppose that there exists a constant C > 0 such that

sup m∈Z sup (x,y)∈(R×[0,1])×(R×[0,1]) u (V1,Vm+1) (x, y) ≤ C.
We are now in the position to present the result.

imsart-generic ver. 2009/12/15 file: mixing2.tex date: January 4, 2014

Theorem 4.3. We consider the model (4.5). Suppose that (K1)-(K6) are satisfied. Let f be the truncated ratio estimator:

f (x) = v(x) ĝ(x) 1 {|ĝ(x)|≥c * /2} , (4.6)
where

• v is defined as in (3.1) with q(γ, (x, x * )) = xγ(x * ) (4.7) and ρ = 0, • ĝ is defined as in (3.1) with X t instead of V t , q(γ, x) = γ(x) (4.8)
and ρ = 0, • c * is the constant defined in (K4).

Suppose that f g ∈ B s p,r (M ) and g ∈ B s p,r (M ) with r ≥ 1, {p ≥ 2 and s ∈ (0, N )} or {p ∈ [1, 2) and s ∈ (1/p, N )}. Then there exists a constant C > 0 such that

E f -f 2 2 ≤ C ln n n 2s/(2s+1)
.

The estimator (4.6) is derived by combining the procedure of [START_REF] Patil | Asymptotics for wavelet based estimates of piecewise smooth regression for stationary time series[END_REF] with the truncated approach of Vasiliev (2012).

Theorem 4.3 completes [START_REF] Patil | Asymptotics for wavelet based estimates of piecewise smooth regression for stationary time series[END_REF] in terms of rates of convergence under the MISE over Besov balls.

Remark 4.1. The assumption (K2) can be relaxed with another strategy to the one developed in Theorem 4.3. Some technical elements are given in Chesneau (2013b).

Conclusion.

Considering the weak dependent case on the observations, we prove a general result on the rate of convergence attains by a hard wavelet thresholding estimator under the MISE over Besov balls. This result is flexible; it can be applied for a wide class of statistical models. Moreover, the obtained rate of convergence is sharp; it can correspond to the near optimal one in the minimax sense for the standard i.i.d. case. Some recent results on sophisticated statistical problems are improved. Thanks to its flexibility, the perspectives of applications of our theoretical result in other contexts are numerous.

Proofs

In this section, C denotes any constant that does not depend on j, k and n. Its value may change from one term to another and may depend on φ or ψ.

imsart-generic ver. 2009/12/15 file: mixing2.tex date: January 4, 2014

Key lemmas

Let us present two lemmas which will be used in the proofs.

Lemma 5.1 below shows a sharp covariance inequality under the α-mixing condition.

Lemma 5.1 [START_REF] Davydov | The invariance principle for stationary processes[END_REF]). Let (W t ) t∈Z be a strictly stationary α-mixing process with mixing coefficient α m , m ≥ 0, and h and k be two measurable functions. Let p > 0 and q > 0 satisfying 1/p + 1/q < 1, such that E(|h(W 1 )| p ) and E(|k(W 1 )| q ) exist. Then there exists a constant C > 0 such that

|C ov (h(W 1 ), k(W m+1 ))| ≤ Cα 1-1/p-1/q m (E(|h(W 1 )| p )) 1/p (E(|k(W 1 )| q )) 1/q .
Lemma 5.2 below presents a concentration inequality for α-mixing processes.

Lemma 5.2 [START_REF] Liebscher | Estimation of the density and the regression function under mixing conditions[END_REF]). Let (W t ) t∈Z be a strictly stationary process with the m-th strongly mixing coefficient α m , m ≥ 0, n be a positive integer, h : R → C be a measurable function and, for any t ∈ Z, U t = h(W t ). We assume that E(U 1 ) = 0 and there exists a constant M > 0 satisfying |U 1 | ≤ M . Then, for any m ∈ {1, . . . , [n/2]} and λ > 0, we have

P 1 n n i=1 U i ≥ λ ≤ 4 exp - λ 2 n 16(D m /m + λM m/3) + 32 M λ nα m ,
where

D m = max l∈{1,...,2m} V l i=1 U i .

Intermediary results

Proof of Lemma 3.1. Using a standard expression of the covariance, and (F1) and (F2), we obtain

|C ov (q(γ j,k , V m+1 ), q(γ j,k , V 1 )) | = V1(Ω) V1(Ω) q(γ j,k , x)q(γ j,k , y)(u (V1,Vm+1) (x, y) -u(x)u(y))dxdy ≤ V1(Ω) V1(Ω) |q(γ j,k , x)||q(γ j,k , y)||u (V1,Vm+1) (x, y) -u(x)u(y)|dxdy ≤ C V1(Ω) |q(γ j,k , x)|dx 2 ≤ C2 2ρj 2 -j .
This ends the proof of Lemma 3.1.

Proposition 5.1 below proves probability and moments inequalities satisfied by the estimators (3.2).

imsart-generic ver. 2009/12/15 file: mixing2.tex date: January 4, 2014 Proposition 5.1. Let αj,k and βj,k be defined as in (3.2) under (H1) and (H2), j 0 be (3.3) and j 1 be (3.4).

(a) There exists a constant C > 0 such that, for any j ∈ {j 0 , . . . , j 1 } and k ∈ Λ j ,

E |ĉ j,k -c j,k | 2 ≤ C2 2ρj 1 n and E dj,k -d j,k 2 ≤ C2 2ρj 1 n .
(b) There exists a constant C > 0 such that, for any j ∈ {j 0 , . . . , j 1 } and

k ∈ Λ j , E dj,k -d j,k 4 ≤ C2 4ρj .
(c) Let λ j be defined as in (3.5). There exists a constant C > 0 such that, for any κ large enough, j ∈ {j 0 , . . . , j 1 } and k ∈ Λ j , we have

P | dj,k -d j,k | ≥ κλ j /2 ≤ C 1 n 4 .
Proof of Proposition 5.1.

(a) Using (H1) and the stationarity of (V t ) t∈Z , we obtain

E |ĉ j,k -c j,k | 2 = V (ĉ j,k ) = 1 n 2 n i=1 V(q(φ j,k , V i )) + 2 n 2 n v=2 v-1 ℓ=1 Re (C ov (q(φ j,k , V v ), q(φ j,k , V ℓ ))) = 1 n 2 n i=1 V(q(φ j,k , V i )) + 2 n 2 n-1 m=1 (n -m)Re (C ov (q(φ j,k , V m+1 ), q(φ j,k , V 1 ))) ≤ 1 n E |q(φ j,k , V 1 )| 2 + 2 n-1 m=1 |C ov (q(φ j,k , V m+1 ), q(φ j,k , V 1 ))| .
(5.1)

By (H2)-(ii) we get E |q(φ j,k , V 1 )| 2 ≤ C2 2ρj . (5.2)
For the covariance term, note that

n-1 m=1 |C ov (q(φ j,k , V m+1 ), q(φ j,k , V 1 ))| = A + B,
where and

A = [(ln n)/θ]-1 m=1 |C ov (q(φ j,k , V m+1 ), q(φ j,k , V 1 ))| imsart-
B = n-1 m=[(ln n)/θ] |C ov (q(φ j,k , V m+1 ), q(φ j,k , V 1 ))|. It follows from (H2)-(iii) and 2 -j ≤ 2 -j0 < 2(ln n) -1 that A ≤ C2 2ρj 2 -j [(ln n)/θ] ≤ C2 2ρj .
The Davydov inequality described in Lemma 5.1 with p = q = 4, (H2)-(i)-(ii) and 2 j ≤ 2 j1 ≤ n give

B ≤ C E |q(φ j,k , V 1 )| 4 n-1 m=[(ln n)/θ] √ α m ≤ C2 ρj 2 j/2 E |q(φ j,k , V 1 )| 2 ∞ m=[(ln n)/θ] e -θm/2 = C2 2ρj √ ne -(ln n)/2 ≤ C2 2ρj . Thus n-1 m=1 |C ov (q(φ j,k , V m+1 ), q(φ j,k , V 1 ))| ≤ C2 2ρj .
(5.3) Putting (5.1), (5.2) and ( 5.3) together, the first point in (a) is proved.

The proof of the second point is identical with ψ instead of φ.

(b) Thanks to (H2)-(i), we have | dj,k | ≤ sup x∈V1(Ω) |q(ψ j,k , x)| ≤ C2 ρj 2 j/2 .
It follows from the triangular inequality and

|d j,k | ≤ ||f || 2 ≤ C that | dj,k -d j,k | ≤ | dj,k | + |d j,k | ≤ C2 ρj 2 j/2 .
This inequality and the second result of (a) yield

E | dj,k -d j,k | 4 ≤ C2 2ρj 2 j E | dj,k -d j,k | 2 ≤ C2 4ρj 2 j 1 n .
Using 2 j ≤ 2 j1 ≤ n, the proof of (b) is completed. (c) We will use the Liebscher inequality described in Lemma 5.2. Let us set

U i = q(ψ j,k , V i ) -E(q(ψ j,k , V 1 )).
We have E(U 1 ) = 0 and, by (H2)-(i) and 2 j ≤ 2 j1 ≤ n/(ln n) 3 , Proceeding as for the proofs of the bounds in (a), for any integer l ≤ C ln n, since 2 -j ≤ 2 -j0 ≤ 2(ln n) -1 , we show that

|U i | ≤ 2 sup x∈V1(Ω) |q(ψ j,k , x)| ≤ C2 ρj 2 j/2 ≤ C2 ρj n (ln n) 3 , (so M = C2 ρj n/(ln n) 3 ).
V l i=1 U i = V l i=1 q(ψ j,k , V i ) ≤ C2 2ρj (l + l 2 2 -j ) ≤ C2 2ρj l. Therefore D m = max l∈{1,...,2m} V l i=1 U i ≤ C2 2ρj m. (5.4) Owing to Lemma 5.2 applied with U 1 , . . . , U n , λ = κλ j /2, m = [ √ κ ln n],
M = C2 ρj n/(ln n) 3 and the bound (5.4), we obtain

P | dj,k -d j,k | ≥ κλ j /2 ≤ C exp -C κ 2 λ 2 j n D m /m + κλ j mM + M λ j ne -θm ≤ C exp -C κ 2 2 2ρj ln n 2 2ρj + κ2 ρj (ln n)/n[ √ κ ln n]2 ρj n/(ln n) 3 + n/(ln n) 3 (ln n)/n ne -θ[ √ κ ln n] ≤ C n -Cκ 2 /(1+κ 3/2 ) + n 2-θ √ κ .
Taking κ large enough, the last term is bounded by C/n 4 . This completes the proof of (c).

This completes the proof of Proposition 5.1.

Proof of Theorem 3.1

Theorem 3.1 can be proved by combining arguments of (Kerkyacharian and Picard, 2000, Theorem 5.1) and (Chesneau, 2008, Theorem 4.2). It is close to [START_REF] Chesneau | On the adaptive wavelet deconvolution of a density for strong mixing sequences[END_REF], Proof of Theorem 2) by taking θ → ∞. The interested reader can find the details below. We consider the following wavelet decomposition for f :

f (x) = k∈Λj 0 c j0,k φ j0,k (x) + ∞ j=j0 k∈Λj d j,k ψ j,k (x), where c j0,k = 1 0 f (x)φ j0,k (x)dx and d j,k = 1 0 f (x)ψ j,k (x)dx.
imsart-generic ver. 2009/12/15 file: mixing2.tex date: January 4, 2014

Using the orthonormality of the wavelet basis B, the MISE of f can be expressed as

E f -f 2 2 = P + Q + R, (5.5)
where

P = k∈Λj 0 E |ĉ j0,k -c j0,k | 2 , Q = j1 j=j0 k∈Λj E dj,k 1 {| dj,k |≥κλj } -d j,k 2 and R = ∞ j=j1+1 k∈Λj d 2 j,k .
Let us now investigate sharp upper bounds for P , R and Q sucessively. Upper bound for P :

The point (a) of Proposition 5.1 and 2s/(2s + 2ρ + 1) < 1 yield

P ≤ C 2 j0 n ≤ C ln n n ≤ C ln n n 2s/(2s+2ρ+1)
.

(5.6)

Upper bound for R:

• For r ≥ 1 and p ≥ 2, we have f ∈ B s p,r (M ) ⊆ B s 2,∞ (M ). Using 2s/(2s + 2ρ + 1) < 2s/(2ρ + 1), we obtain R ≤ C ∞ j=j1+1 2 -2js ≤ C (ln n) 3 n 2s/(2ρ+1) ≤ C ln n n 2s/(2s+2ρ+1)
.

• For r ≥ 1 and p ∈ [1, 2), we have f ∈ B s p,r (M ) ⊆ B s+1/2-1/p 2,∞ (
M ). The condition s > (2ρ + 1)/p implies that (s + 1/2 -1/p)/(2ρ + 1) > s/(2s + 2ρ + 1). Thus

R ≤ C ∞ j=j1+1 2 -2j(s+1/2-1/p) ≤ C (ln n) 3 n 2(s+1/2-1/p)/(2ρ+1) ≤ C ln n n 2s/(2s+2ρ+1)
.

Hence, for r ≥ 1, {p ≥ 2 and s > 0} or {p ∈ [1, 2) and s > (2ρ + 1)/p}, we have

R ≤ C ln n n 2s/(2s+2ρ+1)
.

(5.7)

Upper bound for Q:

imsart-generic ver. 2009/12/15 file: mixing2.tex date: January 4, 2014

Adopting the notation Dj,k = dj,k -d j,k , Q can be written as

Q = 4 i=1 Q i , (5.8)
where

Q 1 = j1 j=j0 k∈Λj E | Dj,k | 2 1 {| dj,k |≥κλj , |d j,k |<κλj /2} , Q 2 = j1 j=j0 k∈Λj E | Dj,k | 2 1 {| dj,k |≥κλj , |d j,k |≥κλj /2} , Q 3 = j1 j=j0 k∈Λj E d 2 j,k 1 {| dj,k |<κλj , |d j,k |≥2κλj } and Q 4 = j1 j=j0 k∈Λj E d 2 j,k 1 {| dj,k |<κλj , |d j,k |<2κλj } . Upper bound for Q 1 + Q 3 : Owing to the inequalities 1 {| dj,k |<κλj , |d j,k |≥2κλj } ≤ 1 {| Dj,k |>κλj /2} , 1 {| dj,k |≥κλj , |d j,k |<κλj /2} ≤ 1 {| Dj,k |>κλj /2} and 1 {| dj,k |<κλj , |d j,k |≥2κλj } ≤ 1 {|dj,k|≤2| Dj,k |}
, the Cauchy-Schwarz inequality, the points (b) and (c) of Proposition 5.1, we have

Q 1 + Q 3 ≤ C j1 j=j0 k∈Λj E | Dj,k | 2 1 {| Dj,k |>κλj /2} ≤ C j1 j=j0 k∈Λj E | Dj,k | 4 1/2 P | Dj,k | > κλ j /2 1/2 ≤ C 1 n 2 j1 j=j0 2 j(1+2ρ) ≤ C 1 n ≤ C ln n n 2s/(2s+2ρ+1)
.

(5.9)

Upper bound for Q 2 : It follows from the point (a) of Proposition 5.1 that

Q 2 ≤ j1 j=j0 k∈Λj E | Dj,k | 2 1 {|d j,k |≥κλj /2} ≤ C 1 n j1 j=j0 2 2ρj k∈Λj 1 {|d j,k |>κλj /2} .
Let us now introduce the integer j * defined by

2 j * = n ln n 1/(2s+2ρ+1)
.

(5.10) Note that j * ∈ {j 0 , . . . , j 1 } for n large enough.

Then Q 2 can be bounded as

Q 2 ≤ Q 2,1 + Q 2,2 ,
where

Q 2,1 = C 1 n j * j=j0 2 2ρj k∈Λj 1 {|d j,k |>κλj /2}
and

Q 2,2 = C 1 n j1 j=j * +1 2 2ρj k∈Λj 1 {|d j,k |>κλj /2} .
On the one hand we have

Q 2,1 ≤ C ln n n j * j=j0 2 j(1+2ρ) ≤ C ln n n 2s/(2s+2ρ+1)
.

On the other hand,

• for r ≥ 1 and p ≥ 2, the Markov inequality and

f ∈ B s p,r (M ) ⊆ B s 2,∞ (M ) yield Q 2,2 ≤ C ln n n j1 j=j * +1 2 2ρj 1 λ 2 j k∈Λj d 2 j,k ≤ C ∞ j=j * +1 k∈Λj d 2 j,k ≤ C ∞ j=j * +1 2 -2js ≤ C ln n n 2s/(2s+2ρ+1)
.

• for r ≥ 1, p ∈ [1, 2) and s > (2ρ+1)/p, the Markov inequality, f ∈ B s p,r (M ) and (2s + 2ρ + 1)(2 -p)/2 + (s + 1/2 -1/p + ρ -2ρ/p)p = 2s imply that

Q 2,2 ≤ C ln n n j1 j=j * +1 2 2ρj 1 λ p j k∈Λj |d j,k | p ≤ C ln n n (2-p)/2 ∞ j=j * +1 2 jρ(2-p) 2 -j(s+1/2-1/p)p ≤ C ln n n (2-p)/2 2 -j * (s+1/2-1/p+ρ-2ρ/p)p ≤ C ln n n 2s/(2s+2ρ+1)
.

Therefore, for r ≥ 1, {p ≥ 2 and s > 0} or {p ∈ [1, 2) and s > (2ρ + 1)/p}, we have

Q 2 ≤ C ln n n 2s/(2s+2ρ+1)
.

(5.11) Upper bound for Q 4 :

We have

Q 4 ≤ j1 j=j0 k∈Λj d 2 j,k 1 {|d j,k |<2κλj } .
Let j * be the integer (5.10). Then Q 4 can be bound as

Q 4 ≤ Q 4,1 + Q 4,2 ,
where

Q 4,1 = j * j=j0 k∈Λj d 2 j,k 1 {|d j,k |<2κλj } , Q 4,2 = j1 j=j * +1 k∈Λj d 2 j,k 1 {|d j,k |<2κλj } .
On the one hand, we have

Q 4,1 ≤ C j * j=j0 2 j λ 2 j = C ln n n j * j=j0 2 j(1+2ρ) ≤ C ln n n 2s/(2s+2ρ+1)
.

On the other hand,

• for r ≥ 1 and p ≥ 2, since f ∈ B s p,r (M ) ⊆ B s 2,∞ (M ), we have Q 4,2 ≤ ∞ j=j * +1 k∈Λj d 2 j,k ≤ C ∞ j=j * +1 2 -2js ≤ C ln n n 2s/(2s+2ρ+1)
.

• for r ≥ 1, p ∈ [1, 2) and s > (2ρ + 1)/p, owing to the Markov inequality, f ∈ B s p,r (M ) and (2s + 2ρ + 1)(2 -p)/2 + (s + 1/2 -1/p + ρ -2ρ/p)p = 2s, we get

Q 4,2 ≤ C j1 j=j * +1 λ 2-p j k∈Λj |d j,k | p = C ln n n (2-p)/2 j1 j=j * +1 2 jρ(2-p) k∈Λj |d j,k | p ≤ C ln n n (2-p)/2 ∞ j=j * +1 2 jρ(2-p) 2 -j(s+1/2-1/p)p ≤ C ln n n (2-p)/2 2 -j * (s+1/2-1/p+ρ-2ρ/p)p ≤ C ln n n 2s/(2s+2ρ+1)
.

So, for r ≥ 1, {p ≥ 2 and s > 0} or {p ∈ [1, 2) and s > (2ρ + 1)/p}, we have

Q 4 ≤ C ln n n 2s/(2s+2ρ+1)
.

(5.12) Putting (5.8), (5.9), (5.11) and (5.12) together, for r ≥ 1, {p ≥ 2 and s > 0} or {p ∈ [1, 2) and s > (2ρ + 1)/p}, we obtain

Q ≤ C ln n n 2s/(2s+2ρ+1)
.

(5.13) Combining (5.5), (5.6), (5.7) and (5.13), we complete the proof of Theorem 3.1.

Proof of Theorem 4.1

The proof of Theorem 4.1 is a direct application of Theorem 3.1: under (G1)-(G5), the function q defined by (4.2) satisfies (H1), see (Fan and Koo, 2002, equation (2)) and (H2): (i) see (Fan and Koo, 2002, Lemma 6), (ii) see (Fan and Koo, 2002, equation (11)) and (iii) see [START_REF] Chesneau | On the adaptive wavelet deconvolution of a density for strong mixing sequences[END_REF], Proof of Proposition 6.1), with ρ = δ.

Proof of Theorem 4.2

The proof of Theorem 4.2 is a consequence of Theorem 3.1: under (J1)-(J4), the function q defined by (4.4) satisfies (H1) and (H2): (i)-(ii) see (Chesneau, 2013a, Proposition 1) and (iii) see (Chesneau and Doosti, 2012, equation ( 26)), with ρ = δ.

Proof of Theorem 4.3

Set v(x) = f (x)g(x). Following the methodology of Vasiliev (2012), we have

f (x) -f (x) = S(x) -T (x), where S(x) = 1 ĝ(x) (v(x) -v(x) + f (x)(g(x) -ĝ(x))) 1 {|ĝ(x)|≥c * /2} and T (x) = f (x)1 {|ĝ(x)|<c * /2} .
Using (K3) and the indicator function, we have The triangular inequality yields

|S(x)| ≤ C (|v(x) -v(x)| + |ĝ(x) -g(x)|) . It follows from {|ĝ(x)| < c * /2} ∩ {|g(x)| ≥ c * } ⊆ {|ĝ(x) -g(x)| > c * /2}, (
| f (x) -f (x)| ≤ |S(x)| + |T (x)| ≤ C (|v(x) -v(x)| + |ĝ(x) -g(x)|) .
The elementary inequality (a + b) 2 ≤ 2(a 2 + b 2 ) implies that

E f -f 2 2 ≤ C E v -v 2 2 + E ĝ -g 2 2 .
(5.14)

We now bound this two MISEs via Theorem 3.1.

Upper bound for the MISE of v: Under (K1)-(K6), the function q defined by (4.7) satisfies * (H1) with v instead of f : since ξ 1 and X 1 are independent with E(ξ 1 ) = 0, We conclude by applying Lemma 3.1 with ρ = 0; (K5) and (K6) imply (F1), and the previous inequality implies (F2). .

E (q(γ j,k , V 1 )) = E (Y 1 γ j,k (X 1 )) = E (f (X 1 )γ j,k (X 1 )) = 1 0 f (x)γ j,k (
(5.15)

Upper bound for the MISE of ĝ: Under (K1)-(K6), proceeding as the previous point, we show that the function q defined by (4.8) satisfies (H1) with g instead of f and X t instead of V t , and (H2): .

(5.16)

Combining (5.14), (5.15) and (5.16), we end the proof of Theorem 4.3.

average estimators. The Annals of Statistics, 21, 1040-1057. Tribouley, K. and Viennet, G. (1998). L p adaptive density estimation in a βmixing framework. Ann. Inst. H. Poincar Probab. Statist., 34, 179-208. Truong, Y.K. and Stone, C.J. (1992). Nonparametric function estimation involving time series. The Annals of Statistics, 20, 77-97. Tsybakov, A.B. (2004). Introduction à l'estimation non-paramétrique, Springer, New York. van Zanten, H. and Zareba, P. (2008). A note on wavelet density deconvolution for weakly dependent data. Stat. Inference Stoch. Process., 11, 207-219. Vasiliev, V.A. (2012). One investigation method of a ratios type estimators.

16th IFAC Symposium on System Identification, Brussels, Belgium, July 11-13, pages 1-6, 2012 (in progress).

  imsart-generic ver. 2009/12/15 file: mixing2.tex date: January 4, 2014 (G3) Let u be the density of V 1 . We suppose that there exists a constant C > 0 such that sup x∈R u(x) ≤ C.

  δ i=1 U i , where U 1 , . . . , U δ are δ i.i.d. random variables having the common distribution U ([0, 1]). (J2) The support of f is [0, 1] and f ∈ L 2 ([0, 1]). (J3) Let u be the density of V 1 . We suppose that there exists a constant C

)

  There exists a constant C > 0 such that sup x∈[0,1] |f (x)| ≤ C. (K4) There exist two constants c * > 0 and C > 0 such that

  imsart-generic ver. 2009/12/15 file: mixing2.tex date: January 4, 2014

  x)g(x)dx = 1 0 v(x)γ j,k (x)dx, * (H2): (i)-(ii)-(iii) with ρ = 0:• (i): since Y 1 (Ω) is bounded thanks to (K2) and (K3), say |Y 1 | ≤ M with M > 0, we have sup (x,x * )∈V1(Ω) |q(γ j,k , (x, x * ))| = sup (x,x * )∈[-M,M ]×[0,1] |xγ j,k (x * )| ≤ M sup x * ∈[0,1] |γ j,k (x * )| ≤ C2 j/2 .• (ii): using the boundedness of Y 1 (Ω), then (K4), we haveE |q(γ j,k , V 1 )| 2 = E Y 2 1 (γ j,k (X 1 ))) 2 ≤ CE (γ j,k (X 1 ))) ,k (x)) 2 dx ≤ C.• (iii): using the boundedness of Y 1 (Ω) and making the change of variables y = 2 j x -k, we obtain k (x)|dx ≤ C2 -j/2 .

  imsart-generic ver. 2009/12/15 file: mixing2.tex date: January 4, 2014 Therefore, assuming that v ∈ B s p,r (M ) with r ≥ 1, {p ≥ 2 and s ∈ (0, N )} or {p ∈ [1, 2) and s ∈ (1/p, N )}, Theorem 3.1 proves the existance of a constant C

  (i)-(ii)-(iii) with ρ = 0. Therefore, assuming that g ∈ B s p,r (M ) with r ≥ 1, {p ≥ 2 and s ∈ (0, N )} or {p ∈ [1, 2) and s ∈ (1/p, N )}, Theorem 3.1 proves the existance of a constant C > 0 satisfying E ĝ -g 2 2 ≤ C ln n n 2s/(2s+1)

imsart-generic ver. 2009/12/15 file: mixing2.tex date: January 4, 2014

Acknowledgments

The author is thankful to the reviewers for their comments which have helped in improving the presentation.