# A General Result on the Mean Integrated Squared Error of the Hard Thresholding Wavelet Estimator underα-Mixing Dependence

Authors: Christophe Chesneau
Journal: Journal of Probability and Statistics
 (2014)

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## Abstract

We consider the estimation of an unknown functionf for weakly dependent data (α-mixing) in a general setting. Our
contribution is theoretical: we prove that a hard thresholding wavelet 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.


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## Body

## 1. Introduction

A general nonparametric problem is adopted: we aim to estimate an unknown functionf via n random variables V1,…,Vn from a strictly stationary stochastic process (Vt)t∈ℤ. We suppose that (Vt)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., [1–3]). The problems where f is the density of V1 or a regression function have received a lot of attention. A partial list of related works includes Robinson [4], Roussas [5, 6], Truong and Stone [7], Tran [8], Masry [9, 10], Masry and Fan [11], Bosq [12], and Liebscher [13].For an efficient estimation off, 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 and Johnstone [14, 15] and Donoho et al. [16, 17]. 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, for example, Antoniadis [18] and Härdle et al. [19]. 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, for example, Leblanc [20], Tribouley and Viennet [21], Masry [22], Patil and Truong [23], Doosti et al. [24], Doosti and Niroumand [25], Doosti et al. [26], Cai and Liang [27], Niu and Liang [28], Benatia and Yahia [29], Chesneau [30–32], Chaubey and Shirazi [33], and Abbaszadeh and Emadi [34].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 functionsf (see, e.g., [14–17, 35]). Following the spirit of Kerkyacharian and Picard [36], we determine necessary assumptions on (Vt)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 ([30], Proposition 5.1) and ([31], Theorem 2), respectively. A last part is devoted to the regression model with random design. The obtained result completes the one of Patil and Truong [23].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 Section3. Applications of our general result are given in Section 4. The proofs are carried out in Section 5.


## 2. Wavelets and Besov Balls

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


### 2.1. Wavelet Basis

We consider the wavelet basis on[0,1] constructs from the Daubechies waveletsdb2N with N≥1 (see, e.g., [37]). 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 𝒞a for N>5a. For any j≥0, we set Λj={0,…,2j-1} and, for k∈Λj,
(1)ϕj,k(x)=2j/2ϕ(2jx-k),ψj,k(x)=2j/2ψ(2jx-k).With appropriated treatments at the boundaries, there exists an integerτ such that, for any integer ℓ≥τ, ℬ={ϕℓ,k,k∈Λℓ;ψj,k;j∈ℕ-{0,…,ℓ-1},k∈Λj} is an orthonormal basis of 𝕃2([0,1]), where
(2)𝕃2([0,1])={f:[0,1]→ℝ;∥f∥2=(∫01‍|f(x)|2dx)1/2<∞}.
For any integer ℓ≥τ and f∈𝕃2([0,1]), we have the following wavelet expansion:
(3)f(x)=∑k∈Λℓ‍cℓ,kϕℓ,k(x)+∑j=ℓ∞‍∑k∈Λj‍dj,kψj,k(x),lllllllllllllllllllllllllllllllllllllllllx∈[0,1],
where cj,k and dj,k denote the wavelet coefficients of f defined by
(4)cj,k=∫01‍f(x)ϕj,k(x)dx,dj,k=∫01‍f(x)ψj,k(x)dx.
Technical details can be found in, for example, Cohen et al. [38] and Mallat [39].In the main result of this paper, we will investigate the MISE rate of the proposed estimator by assuming that the unknown function of interestf belongs to a wide class of functions: the Besov class. Its definition in terms of wavelet coefficients is presented in the following.


### 2.2.  Besov Balls

We say thatf∈Bp,rs(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 (4) satisfy
(5)2τ(1/2-1/p)(∑k∈Λτ‍|cτ,k|p)1/p+(∑j=τ∞‍(2j(s+1/2-1/p)(∑k∈Λj‍|dj,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, Bp,rs(M) contains the classical Hölder and Sobolev balls (see, e.g., [40] and [19]).Remark 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.


## 2.1. Wavelet Basis

We consider the wavelet basis on[0,1] constructs from the Daubechies waveletsdb2N with N≥1 (see, e.g., [37]). 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 𝒞a for N>5a. For any j≥0, we set Λj={0,…,2j-1} and, for k∈Λj,
(1)ϕj,k(x)=2j/2ϕ(2jx-k),ψj,k(x)=2j/2ψ(2jx-k).With appropriated treatments at the boundaries, there exists an integerτ such that, for any integer ℓ≥τ, ℬ={ϕℓ,k,k∈Λℓ;ψj,k;j∈ℕ-{0,…,ℓ-1},k∈Λj} is an orthonormal basis of 𝕃2([0,1]), where
(2)𝕃2([0,1])={f:[0,1]→ℝ;∥f∥2=(∫01‍|f(x)|2dx)1/2<∞}.
For any integer ℓ≥τ and f∈𝕃2([0,1]), we have the following wavelet expansion:
(3)f(x)=∑k∈Λℓ‍cℓ,kϕℓ,k(x)+∑j=ℓ∞‍∑k∈Λj‍dj,kψj,k(x),lllllllllllllllllllllllllllllllllllllllllx∈[0,1],
where cj,k and dj,k denote the wavelet coefficients of f defined by
(4)cj,k=∫01‍f(x)ϕj,k(x)dx,dj,k=∫01‍f(x)ψj,k(x)dx.
Technical details can be found in, for example, Cohen et al. [38] and Mallat [39].In the main result of this paper, we will investigate the MISE rate of the proposed estimator by assuming that the unknown function of interestf belongs to a wide class of functions: the Besov class. Its definition in terms of wavelet coefficients is presented in the following.


## 2.2.  Besov Balls

We say thatf∈Bp,rs(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 (4) satisfy
(5)2τ(1/2-1/p)(∑k∈Λτ‍|cτ,k|p)1/p+(∑j=τ∞‍(2j(s+1/2-1/p)(∑k∈Λj‍|dj,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, Bp,rs(M) contains the classical Hölder and Sobolev balls (see, e.g., [40] and [19]).Remark 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.


## 3. Statistical Framework, Estimator and Result

### 3.1. Statistical Framework

As mentioned in Section1, a nonparametric estimation setting as general as possible is adopted: we aim to estimate an unknown function f∈𝕃2([0,1]) via n random variables (or vectors) V1,…,Vn from a strictly stationary stochastic process (Vt)t∈ℤ defined on a probability space (Ω,𝒜,ℙ). We suppose that (Vt)t∈ℤ has a α-mixing dependence structure with exponential decay rate; that is, there exist two constants γ>0 and θ>0 such that
(6)sup⁡m≥1(eθmαm)≤γ,
where αm=sup⁡(A,B)∈ℱ-∞,0V×ℱm,∞V|ℙ(A∩B)-ℙ(A)ℙ(B)|, ℱ-∞,0V is the σ-algebra generated by the random variables (or vectors) …,V-1,V0 and ℱm,∞V is the σ-algebra generated by the random variables (or vectors) Vm,Vm+1,….Theα-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, [1–3, 41]).The considered estimator forf is presented below.


### 3.2.  Estimator

We define the hard thresholding wavelet estimatorf^ by
(7)f^(x)=∑k∈Λj0‍c^j0,kϕj0,k(x)+∑j=j0j1‍∑k∈Λjd^j,k1{|d^j,k|≥κλj}ψj,k(x),
where
(8)c^j,k=1n∑i=1n‍q(ϕj,k,Vi),d^j,k=1n∑i=1n‍q(ψj,k,Vi),1 is the indicator function, κ>0 is a large enough constant, j0 is the integer satisfying
(9)2j0=[τln⁡n],
where [a] denotes the integer part of a and j1 is the integer satisfying
(10)2j1=[(n(ln⁡n)3)1/(2ρ+1)],(11)λj=2ρjln⁡nn.
Here it is supposed that there exists a function q:𝕃2([0,1])×V1(Ω)→ℂ such that(H1)
forγ∈{ϕ,ψ}, any integer j≥j0 and k∈Λj,
(12)𝔼(q(γj,k,V1))=∫01‍f(x)γj,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≥j0 and k∈Λj,         (i) sup⁡x∈V1(Ω)|q(γj,k,x)|≤C2ρj2j/2,          (ii) 𝔼(|q(γj,k,V1)|2)≤C22ρj,          (iii) for any m∈{1,…,n-1}≥1,
(13)|ℂov(q(γj,k,Vm+1),q(γj,k,V1))|≤C22ρj2-j,whereℂov denotes the covariance; that is, ℂov(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, hard thresholding wavelet estimators and important results can be found in, for example, Donoho and Johnstone [14, 15], Donoho et al. [16, 17], Delyon and Juditsky [35], Kerkyacharian and Picard [36], and Fan and Koo [42]. In the α-mixing context, f^ defined by (7) is a general and improved version of the estimator considered in Chesneau [30, 31]. 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 (8) are unbiased estimators for cj,k and dj,k given by (4), whereas (H2) is related to their good performance. See Proposition 10. These assumptions are not too restrictive. For instance, if we consider the standard density estimation problem where (Vt)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 (Vt)t∈ℤ, the covariance term in (H2)-(iii) is zero). The technical details are given in Donoho et al. [17].Lemma2 describes a simple situation in which assumption (H2)-(iii) is satisfied.Lemma 2.
We make the following assumptions.(F1)
Letu be the density of  V1 and let u(V1,Vm+1) be the density of (V1,Vm+1) for any m∈ℤ. We suppose that there exists a constant C>0 such that
(14)sup⁡m∈{1,…,n-1}ℤsup⁡(x,y)∈V1(Ω)×Vm+1(Ω)|u(V1,Vm+1)(x,y)lllllllllllllllllllllllllllllllllllllll-u(x)u(y)|≤C.(F2)
There exist two constants,C>0 and ρ≥0, satisfying, for γ∈{ϕ,ψ}, for any integer j≥j0 and k∈Λj,
(15)∫V1(Ω)‍|q(γj,k,x)|dx≤C2ρj2-j/2.Then, under (F1) and (F2), (H2)-(iii) is satisfied.


### 3.3.  Result

Theorem3 determines the rate of convergence attained by f^ under the MISE over Besov balls.Theorem 3.
We consider the general statistical setting described in Section3.1. Let f^ be (7) under (H1) and (H2). Suppose that f∈Bp,rs(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
(16)𝔼(∥f^-f∥22)≤C(ln⁡nn)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., [19, 43]). Moreover, note that Theorem 3 is flexible; the assumptions on (Vt)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.


## 3.1. Statistical Framework

As mentioned in Section1, a nonparametric estimation setting as general as possible is adopted: we aim to estimate an unknown function f∈𝕃2([0,1]) via n random variables (or vectors) V1,…,Vn from a strictly stationary stochastic process (Vt)t∈ℤ defined on a probability space (Ω,𝒜,ℙ). We suppose that (Vt)t∈ℤ has a α-mixing dependence structure with exponential decay rate; that is, there exist two constants γ>0 and θ>0 such that
(6)sup⁡m≥1(eθmαm)≤γ,
where αm=sup⁡(A,B)∈ℱ-∞,0V×ℱm,∞V|ℙ(A∩B)-ℙ(A)ℙ(B)|, ℱ-∞,0V is the σ-algebra generated by the random variables (or vectors) …,V-1,V0 and ℱm,∞V is the σ-algebra generated by the random variables (or vectors) Vm,Vm+1,….Theα-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, [1–3, 41]).The considered estimator forf is presented below.


## 3.2.  Estimator

We define the hard thresholding wavelet estimatorf^ by
(7)f^(x)=∑k∈Λj0‍c^j0,kϕj0,k(x)+∑j=j0j1‍∑k∈Λjd^j,k1{|d^j,k|≥κλj}ψj,k(x),
where
(8)c^j,k=1n∑i=1n‍q(ϕj,k,Vi),d^j,k=1n∑i=1n‍q(ψj,k,Vi),1 is the indicator function, κ>0 is a large enough constant, j0 is the integer satisfying
(9)2j0=[τln⁡n],
where [a] denotes the integer part of a and j1 is the integer satisfying
(10)2j1=[(n(ln⁡n)3)1/(2ρ+1)],(11)λj=2ρjln⁡nn.
Here it is supposed that there exists a function q:𝕃2([0,1])×V1(Ω)→ℂ such that(H1)
forγ∈{ϕ,ψ}, any integer j≥j0 and k∈Λj,
(12)𝔼(q(γj,k,V1))=∫01‍f(x)γj,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≥j0 and k∈Λj,         (i) sup⁡x∈V1(Ω)|q(γj,k,x)|≤C2ρj2j/2,          (ii) 𝔼(|q(γj,k,V1)|2)≤C22ρj,          (iii) for any m∈{1,…,n-1}≥1,
(13)|ℂov(q(γj,k,Vm+1),q(γj,k,V1))|≤C22ρj2-j,whereℂov denotes the covariance; that is, ℂov(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, hard thresholding wavelet estimators and important results can be found in, for example, Donoho and Johnstone [14, 15], Donoho et al. [16, 17], Delyon and Juditsky [35], Kerkyacharian and Picard [36], and Fan and Koo [42]. In the α-mixing context, f^ defined by (7) is a general and improved version of the estimator considered in Chesneau [30, 31]. 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 (8) are unbiased estimators for cj,k and dj,k given by (4), whereas (H2) is related to their good performance. See Proposition 10. These assumptions are not too restrictive. For instance, if we consider the standard density estimation problem where (Vt)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 (Vt)t∈ℤ, the covariance term in (H2)-(iii) is zero). The technical details are given in Donoho et al. [17].Lemma2 describes a simple situation in which assumption (H2)-(iii) is satisfied.Lemma 2.
We make the following assumptions.(F1)
Letu be the density of  V1 and let u(V1,Vm+1) be the density of (V1,Vm+1) for any m∈ℤ. We suppose that there exists a constant C>0 such that
(14)sup⁡m∈{1,…,n-1}ℤsup⁡(x,y)∈V1(Ω)×Vm+1(Ω)|u(V1,Vm+1)(x,y)lllllllllllllllllllllllllllllllllllllll-u(x)u(y)|≤C.(F2)
There exist two constants,C>0 and ρ≥0, satisfying, for γ∈{ϕ,ψ}, for any integer j≥j0 and k∈Λj,
(15)∫V1(Ω)‍|q(γj,k,x)|dx≤C2ρj2-j/2.Then, under (F1) and (F2), (H2)-(iii) is satisfied.


## 3.3.  Result

Theorem3 determines the rate of convergence attained by f^ under the MISE over Besov balls.Theorem 3.
We consider the general statistical setting described in Section3.1. Let f^ be (7) under (H1) and (H2). Suppose that f∈Bp,rs(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
(16)𝔼(∥f^-f∥22)≤C(ln⁡nn)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., [19, 43]). Moreover, note that Theorem 3 is flexible; the assumptions on (Vt)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.


## 4.  Applications

### 4.1. Density Deconvolution

Let(Vt)t∈ℤ be a strictly stationary stochastic process such that
(17)Vt=Xt+ϵt,t∈ℤ,
where (Xt)t∈ℤ is a strictly stationary stochastic process with unknown density f and (ϵt)t∈ℤ is a strictly stationary stochastic process with known density g. It is supposed that ϵt and Xt are independent for any t∈ℤ and (Vt)t∈ℤ is a α-mixing process with exponential decay rate (see Section 3.1 for a precise definition). Our aim is to estimate f via V1,…,Vn from (Vt)t∈ℤ. Some related works are Masry [44], Kulik [45], Comte et al. [46], and Van Zanten and Zareba [47].We formulate the following assumptions.(G1)
The support off is [0,1].(G2)
There exists a constantC>0 such that
(18)sup⁡x∈ℝf(x)≤C<∞.(G3)
Letu be the density of V1. We suppose that there exists a constant C>0 such that
(19)sup⁡x∈ℝu(x)≤C.(G4)
For anym∈ℤ, let u(V1,Vm+1) be the density of (V1,Vm+1). We suppose that there exists a constant C>0 such that
(20)sup⁡m∈ℤsup⁡(x,y)∈ℝ2u(V1,Vm+1)(x,y)≤C.(G5)
For any integrable functionγ, we define its Fourier transform by
(21)ℱ(γ)(x)=∫-∞∞‍γ(y)e-ixydy,x∈ℝ.We suppose that there exist three known constantsC>0, c>0, and δ>1 such that, for any x∈ℝ,(i)
the Fourier transform ofg satisfies
(22)|ℱ(g)(x)|≥c(1+x2)δ/2,(ii)
for anyℓ∈{0,1,2}, the ℓth derivative of the Fourier transform of g satisfies
(23)|(ℱ(g)(x))(ℓ)|≤C(1+|x|)δ+ℓ.We are now in the position to present the result.Theorem 4.
We consider the model (17). Suppose that (G1)–(G5) are satisfied. Let f^ be defined as in (7) with
(24)q(γ,x)=12π∫-∞∞‍ℱ(γ)¯(y)ℱ(g)(y)e-iyxdy,
where ℱ(γ)¯(y) denotes the complex conjugate of ℱ(γ)(y) and ρ=δ (appearing in(G5)).
Suppose thatf∈Bp,rs(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
(25)𝔼(∥f^-f∥22)≤C(ln⁡nn)2s/(2s+2δ+1).Theorem4 improves ([30], Proposition 5.1) in terms of rate of convergence; we gain a logarithmic term.Moreover, it is established that, in the i.i.d. setting, “((ln⁡n)/n)2s/(2s+2δ+1)” is(i)
exactly the rate of convergence attained by the hard thresholding wavelet estimator,(ii)
the near optimal rate of convergence in the minimax sense.The details can be found in Fan and Koo [42]. Thus, Theorem 4 can be viewed as an extension of this existing result to the weak dependent case.


### 4.2.  GARCH-Type Model

We consider the strictly stationary stochastic process(Vt)t∈ℤ where, for any t∈ℤ,
(26)Vt=σt2Zt,(σt2)t∈ℤ is a strictly stationary stochastic process with unknown density f, and (Zt)t∈ℤ is a strictly stationary stochastic process with known density g. It is supposed that σt2 and Zt are independent for any t∈ℤ and (Vt)t∈ℤ is a α-mixing process with exponential decay rate (see Section 3.1 for a precise definition). Our aim is to estimate f via V1,…,Vn from (Vt)t∈ℤ. Some related works are Comte et al. [46] and Chesneau [31].We formulate the following assumptions.(J1)
There exists a positive integerδ such that
(27)g(x)=1(δ-1)!(-ln⁡x)δ-1,x∈[0,1].
Let us remark that g is the density of ∏i=1δ‍Ui, where U1,…,Uδ are δ i.i.d. random variables having the common distribution 𝒰([0,1]).(J2)
The support off is [0,1] and f∈𝕃2([0,1]).(J3)
Letu be the density of V1. We suppose that there exists a constant C>0 such that
(28)sup⁡x∈ℝu(x)≤C.(J4)
For anym∈ℤ, let u(V1,Vm+1) be the density of (V1,Vm+1). We suppose that there exists a constant C>0 such that
(29)sup⁡m∈ℤsup⁡(x,y)∈ℝ2u(V1,Vm+1)(x,y)≤C. We are now in the position to present the result.Theorem 5.
We consider model (26). Suppose that (J1)–(J4) are satisfied. Let f^ be defined as in (7) with
(30)q(γ,x)=Tδ(γ)(x),
where, for any positive integer ℓ, T(γ)(x)=(xγ(x))′ and Tℓ(γ)(x)=T(Tℓ-1(γ))(x) and ρ=δ (appearing in (J1)).
Suppose thatf∈Bp,rs(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
(31)𝔼(∥f^-f∥22)≤C(ln⁡nn)2s/(2s+2δ+1).Theorem5 significantly improves ([31], Theorem 2) in terms of rate of convergence; we gain an exponent 1/2.


### 4.3.  Nonparametric Regression Model

We consider the strictly stationary stochastic process(Vt)t∈ℤ where, for any t∈ℤ, Vt=(Yt,Xt),
(32)Yt=f(Xt)+ξt,(Xt)t∈ℤ is a strictly stationary stochastic process with unknown density g, (ξt)t∈ℤ is a strictly stationary centered stochastic process, and f is the unknown regression function. It is supposed that Xt and ξt are independent for any t∈ℤ and (Vt)t∈ℤ is a α-mixing process with exponential decay rate (see Section 3.1 for a precise definition). Our aim is to estimate f via V1,…,Vn from (Vt)t∈ℤ. Applications of this problem can be found in Härdle [48]. Wavelet methods can be found in Patil and Truong [23], Doosti et al. [24], Doosti et al. [26], and Doosti and Niroumand [25].We formulate the following assumptions.(K1)
The support off and g is [0,1] and f and g∈𝕃2([0,1]).(K2)
ξ1(Ω) is bounded.(K3)
There exists a constantC>0 such that
(33)sup⁡x∈[0,1]|f(x)|≤C.(K4)
There exist two constantsc*>0 and C>0 such that
(34)c*≤inf⁡x∈[0,1]g(x),sup⁡x∈[0,1]g(x)≤C.(K5)
Letu be the density of V1. We suppose that there exists a constant C>0 such that
(35)sup⁡x∈ℝ×[0,1]u(x)≤C.(K6)
For anym∈ℤ, let u(V1,Vm+1) be the density of (V1,Vm+1). We suppose that there exists a constant C>0 such that
(36)sup⁡m∈ℤsup⁡(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.Theorem 6.
We consider the model (32). Suppose that (K1)–(K6) are satisfied. Let f^ be the truncated ratio estimator. Consider
(37)f^(x)=v^(x)g^(x)1{|g^(x)|≥c*/2},
where(i)
v^ is defined as in (7) with
(38)q(γ,(x,x*))=xγ(x*)
and ρ=0,(ii)
g^ is defined as in (7) with Xt instead of Vt,
(39)q(γ,x)=γ(x)
and ρ=0,(iii)
c* is the constant defined in (K4).
Suppose thatfg∈Bp,rs(M) and g∈Bp,rs(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
(40)𝔼(∥f^-f∥22)≤C(ln⁡nn)2s/(2s+1).The estimator (37) is derived by combining the procedure of Patil and Truong [23] with the truncated approach of Vasiliev [49].Theorem6 completes Patil and Truong [23] in terms of rates of convergence under the MISE over Besov balls.Remark 7.
The assumption(K2) can be relaxed with another strategy to the one developed in Theorem6. Some technical elements are given in Chesneau [32].
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.


## 4.1. Density Deconvolution

Let(Vt)t∈ℤ be a strictly stationary stochastic process such that
(17)Vt=Xt+ϵt,t∈ℤ,
where (Xt)t∈ℤ is a strictly stationary stochastic process with unknown density f and (ϵt)t∈ℤ is a strictly stationary stochastic process with known density g. It is supposed that ϵt and Xt are independent for any t∈ℤ and (Vt)t∈ℤ is a α-mixing process with exponential decay rate (see Section 3.1 for a precise definition). Our aim is to estimate f via V1,…,Vn from (Vt)t∈ℤ. Some related works are Masry [44], Kulik [45], Comte et al. [46], and Van Zanten and Zareba [47].We formulate the following assumptions.(G1)
The support off is [0,1].(G2)
There exists a constantC>0 such that
(18)sup⁡x∈ℝf(x)≤C<∞.(G3)
Letu be the density of V1. We suppose that there exists a constant C>0 such that
(19)sup⁡x∈ℝu(x)≤C.(G4)
For anym∈ℤ, let u(V1,Vm+1) be the density of (V1,Vm+1). We suppose that there exists a constant C>0 such that
(20)sup⁡m∈ℤsup⁡(x,y)∈ℝ2u(V1,Vm+1)(x,y)≤C.(G5)
For any integrable functionγ, we define its Fourier transform by
(21)ℱ(γ)(x)=∫-∞∞‍γ(y)e-ixydy,x∈ℝ.We suppose that there exist three known constantsC>0, c>0, and δ>1 such that, for any x∈ℝ,(i)
the Fourier transform ofg satisfies
(22)|ℱ(g)(x)|≥c(1+x2)δ/2,(ii)
for anyℓ∈{0,1,2}, the ℓth derivative of the Fourier transform of g satisfies
(23)|(ℱ(g)(x))(ℓ)|≤C(1+|x|)δ+ℓ.We are now in the position to present the result.Theorem 4.
We consider the model (17). Suppose that (G1)–(G5) are satisfied. Let f^ be defined as in (7) with
(24)q(γ,x)=12π∫-∞∞‍ℱ(γ)¯(y)ℱ(g)(y)e-iyxdy,
where ℱ(γ)¯(y) denotes the complex conjugate of ℱ(γ)(y) and ρ=δ (appearing in(G5)).
Suppose thatf∈Bp,rs(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
(25)𝔼(∥f^-f∥22)≤C(ln⁡nn)2s/(2s+2δ+1).Theorem4 improves ([30], Proposition 5.1) in terms of rate of convergence; we gain a logarithmic term.Moreover, it is established that, in the i.i.d. setting, “((ln⁡n)/n)2s/(2s+2δ+1)” is(i)
exactly the rate of convergence attained by the hard thresholding wavelet estimator,(ii)
the near optimal rate of convergence in the minimax sense.The details can be found in Fan and Koo [42]. Thus, Theorem 4 can be viewed as an extension of this existing result to the weak dependent case.


## 4.2.  GARCH-Type Model

We consider the strictly stationary stochastic process(Vt)t∈ℤ where, for any t∈ℤ,
(26)Vt=σt2Zt,(σt2)t∈ℤ is a strictly stationary stochastic process with unknown density f, and (Zt)t∈ℤ is a strictly stationary stochastic process with known density g. It is supposed that σt2 and Zt are independent for any t∈ℤ and (Vt)t∈ℤ is a α-mixing process with exponential decay rate (see Section 3.1 for a precise definition). Our aim is to estimate f via V1,…,Vn from (Vt)t∈ℤ. Some related works are Comte et al. [46] and Chesneau [31].We formulate the following assumptions.(J1)
There exists a positive integerδ such that
(27)g(x)=1(δ-1)!(-ln⁡x)δ-1,x∈[0,1].
Let us remark that g is the density of ∏i=1δ‍Ui, where U1,…,Uδ are δ i.i.d. random variables having the common distribution 𝒰([0,1]).(J2)
The support off is [0,1] and f∈𝕃2([0,1]).(J3)
Letu be the density of V1. We suppose that there exists a constant C>0 such that
(28)sup⁡x∈ℝu(x)≤C.(J4)
For anym∈ℤ, let u(V1,Vm+1) be the density of (V1,Vm+1). We suppose that there exists a constant C>0 such that
(29)sup⁡m∈ℤsup⁡(x,y)∈ℝ2u(V1,Vm+1)(x,y)≤C. We are now in the position to present the result.Theorem 5.
We consider model (26). Suppose that (J1)–(J4) are satisfied. Let f^ be defined as in (7) with
(30)q(γ,x)=Tδ(γ)(x),
where, for any positive integer ℓ, T(γ)(x)=(xγ(x))′ and Tℓ(γ)(x)=T(Tℓ-1(γ))(x) and ρ=δ (appearing in (J1)).
Suppose thatf∈Bp,rs(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
(31)𝔼(∥f^-f∥22)≤C(ln⁡nn)2s/(2s+2δ+1).Theorem5 significantly improves ([31], Theorem 2) in terms of rate of convergence; we gain an exponent 1/2.


## 4.3.  Nonparametric Regression Model

We consider the strictly stationary stochastic process(Vt)t∈ℤ where, for any t∈ℤ, Vt=(Yt,Xt),
(32)Yt=f(Xt)+ξt,(Xt)t∈ℤ is a strictly stationary stochastic process with unknown density g, (ξt)t∈ℤ is a strictly stationary centered stochastic process, and f is the unknown regression function. It is supposed that Xt and ξt are independent for any t∈ℤ and (Vt)t∈ℤ is a α-mixing process with exponential decay rate (see Section 3.1 for a precise definition). Our aim is to estimate f via V1,…,Vn from (Vt)t∈ℤ. Applications of this problem can be found in Härdle [48]. Wavelet methods can be found in Patil and Truong [23], Doosti et al. [24], Doosti et al. [26], and Doosti and Niroumand [25].We formulate the following assumptions.(K1)
The support off and g is [0,1] and f and g∈𝕃2([0,1]).(K2)
ξ1(Ω) is bounded.(K3)
There exists a constantC>0 such that
(33)sup⁡x∈[0,1]|f(x)|≤C.(K4)
There exist two constantsc*>0 and C>0 such that
(34)c*≤inf⁡x∈[0,1]g(x),sup⁡x∈[0,1]g(x)≤C.(K5)
Letu be the density of V1. We suppose that there exists a constant C>0 such that
(35)sup⁡x∈ℝ×[0,1]u(x)≤C.(K6)
For anym∈ℤ, let u(V1,Vm+1) be the density of (V1,Vm+1). We suppose that there exists a constant C>0 such that
(36)sup⁡m∈ℤsup⁡(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.Theorem 6.
We consider the model (32). Suppose that (K1)–(K6) are satisfied. Let f^ be the truncated ratio estimator. Consider
(37)f^(x)=v^(x)g^(x)1{|g^(x)|≥c*/2},
where(i)
v^ is defined as in (7) with
(38)q(γ,(x,x*))=xγ(x*)
and ρ=0,(ii)
g^ is defined as in (7) with Xt instead of Vt,
(39)q(γ,x)=γ(x)
and ρ=0,(iii)
c* is the constant defined in (K4).
Suppose thatfg∈Bp,rs(M) and g∈Bp,rs(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
(40)𝔼(∥f^-f∥22)≤C(ln⁡nn)2s/(2s+1).The estimator (37) is derived by combining the procedure of Patil and Truong [23] with the truncated approach of Vasiliev [49].Theorem6 completes Patil and Truong [23] in terms of rates of convergence under the MISE over Besov balls.Remark 7.
The assumption(K2) can be relaxed with another strategy to the one developed in Theorem6. Some technical elements are given in Chesneau [32].
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.


## 5. 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.


### 5.1. Key Lemmas

Let us present two lemmas which will be used in the proofs.Lemma8 shows a sharp covariance inequality under the α-mixing condition.Lemma 8 (see [50]).
Let(Wt)t∈ℤ be a strictly stationary α-mixing process with mixing coefficient αm, m≥0, and let h and k be two measurable functions. Let p>0 and q>0 satisfying 1/p+1/q<1 such that 𝔼(|h(W1)|p) and 𝔼(|k(W1)|q) exist. Then there exists a constant C>0 such that
(41)|ℂov(h(W1),k(Wm+1))|≤Cαm1-1/p-1/q(𝔼(|h(W1)|p))1/p(𝔼(|k(W1)|q))1/q.Lemma9 below presents a concentration inequality for α-mixing processes.Lemma 9 (see [13]).
Let(Wt)t∈ℤ be a strictly stationary process with the mth strongly mixing coefficient αm, m≥0, let n be a positive integer, let h:ℝ→ℂ be a measurable function, and, for any t∈ℤ, Ut=h(Wt). We assume that 𝔼(U1)=0 and there exists a constant M>0 satisfying |U1|≤M. Then, for any m∈{1,…,[n/2]} and λ>0, we have
(42)ℙ(1n|∑i=1n‍Ui|≥λ)≤4exp⁡(-λ2n16(Dm/m+λMm/3))+32Mλnαm,
where
(43)Dm=max⁡l∈{1,…,2m}𝕍(∑i=1l‍Ui).


### 5.2.  Intermediary Results

Proof of Lemma2.
Using a standard expression of the covariance, and (F1) as well as (F2), we obtain(44)|ℂov(q(γj,k,Vm+1),q(γj,k,V1))|=|∫V1(Ω)‍∫V1(Ω)‍q(γj,k,x)q(γj,k,y)¯×(u(V1,Vm+1)(x,y)-u(x)u(y))dxdy∫V1(Ω)‍|≤∫V1(Ω)‍∫V1(Ω)‍|q(γj,k,x)||q(γj,k,y)|×|u(V1,Vm+1)(x,y)-u(x)u(y)|dxdy≤C(∫V1(Ω)‍|q(γj,k,x)|dx)2≤C22ρj2-j.
This ends the proof of Lemma 2.Proposition10 proves probability and moments inequalities satisfied by the estimators (8).Proposition 10.
Letα^j,k and β^j,k be defined as in (8) under (H1) and (H2), let j0 be (9) and let j1 be (10).(a)
There exists a constantC>0 such that, for any j∈{j0,…,j1} and k∈Λj,
(45)𝔼(|c^j,k-cj,k|2)≤C22ρj1n,(46)𝔼(|d^j,k-dj,k|2)≤C22ρj1n.(b)
There exists a constantC>0 such that, for any j∈{j0,…,j1} and k∈Λj,
(47)𝔼(|d^j,k-dj,k|4)≤C24ρj.(c)
Letλj be defined as in (11). There exists a constant C>0 such that, for any κ large enough, j∈{j0,…,j1} and k∈Λj, we have
(48)ℙ(|d^j,k-dj,k|≥κλj2)≤C1n4.Proof of Proposition10.
(a) Using (H1) and the stationarity of(Vt)t∈ℤ, we obtain
(49)𝔼(|c^j,k-cj,k|2)=𝕍(c^j,k)=1n2∑i=1n‍𝕍(q(ϕj,k,Vi))+2n2×∑v=2n‍∑ℓ=1v-1‍Re(ℂov(q(ϕj,k,Vv),q(ϕj,k,Vℓ)))=1n2∑i=1n‍𝕍(q(ϕj,k,Vi))+2n2×∑m=1n-1‍(n-m)Re(ℂov(q(ϕj,k,Vm+1),q(ϕj,k,V1)))≤1n(∑m=1n-1‍𝔼(|q(ϕj,k,V1)|2)+2∑m=1n-1‍|ℂov(q(ϕj,k,Vm+1),q(ϕj,k,V1))|).
By (H2)-(ii) we get
(50)𝔼(|q(ϕj,k,V1)|2)≤C22ρj.
For the covariance term, note that
(51)∑m=1n-1‍|ℂov(q(ϕj,k,Vm+1),q(ϕj,k,V1))|=A+B,
where
(52)A=∑m=1[ln⁡n/θ]-1‍|ℂov(q(ϕj,k,Vm+1),q(ϕj,k,V1))|,B=∑m=[(ln⁡n)/θ]n-1‍|ℂov(q(ϕj,k,Vm+1),q(ϕj,k,V1))|.
It follows from (H2)-(iii) and 2-j≤2-j0<2(ln⁡n)-1 that
(53)A≤C22ρj2-j[ln⁡nθ]≤C22ρj.
The Davydov inequality described in Lemma 8 with p=q=4, (H2)-(i)-(ii), and 2j≤2j1≤n give
(54)B≤C𝔼(|q(ϕj,k,V1)|4)∑m=[(ln⁡n)/θ]n-1‍αm≤C2ρj2j/2𝔼(|q(ϕj,k,V1)|2)∑m=[(ln⁡n)/θ]∞‍e-θm/2=C22ρjne-(ln⁡n)/2≤C22ρj.
Thus
(55)∑m=1n-1‍|ℂov(q(ϕj,k,Vm+1),q(ϕj,k,V1))|≤C22ρj.
Putting (49), (50), and (55) 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|d^j,k|≤sup⁡x∈V1(Ω)|q(ψj,k,x)|≤C2ρj2j/2. It follows from the triangular inequality and |dj,k|≤∥f∥2≤C that
(56)|d^j,k-dj,k|≤|d^j,k|+|dj,k|≤C2ρj2j/2.
This inequality and the second result of (a) yield
(57)𝔼(|d^j,k-dj,k|4)≤C22ρj2j𝔼(|d^j,k-dj,k|2)≤C24ρj2j1n.
Using 2j≤2j1≤n, the proof of (b) is completed.
(c) We will use the Liebscher inequality described in Lemma9. Let us set
(58)Ui=q(ψj,k,Vi)-𝔼(q(ψj,k,V1)).
We have𝔼(U1)=0 and, by(H2)-(i) and 2j≤2j1≤n/(ln⁡n)3,
(59)|Ui|≤2sup⁡x∈V1(Ω)|q(ψj,k,x)|≤C2ρj2j/2≤C2ρjn(ln⁡n)3,
(so M=C2ρjn/(ln⁡n)3).
Proceeding as for the proofs of the bounds in(a), for any integerl≤Cln⁡n, since 2-j≤2-j0≤2(ln⁡n)-1, we show that
(60)𝕍(∑i=1lUi)=𝕍(∑i=1lq(ψj,k,Vi))≤C22ρj(l+l22-j)≤C22ρjl.
Therefore
(61)Dm=max⁡l∈{1,…,2m}𝕍(∑i=1l‍Ui)≤C22ρjm.
Owing to Lemma 9 applied with U1,…,Un, λ=κλj/2, m=[κln⁡n], M=C2ρjn/(ln⁡n)3, and the bound (61), we obtain
(62)ℙ(|d^j,k-dj,k|≥κλj2)≤C(exp⁡(-Cκ2λj2nDm/m+κλjmM)+Mλjne-θm)≤C(n/(ln⁡n)3(ln⁡n)/nexp⁡(-C((n(ln⁡n)3κ222ρjln⁡n)×(n(ln⁡n)322ρj+κ2ρj(ln⁡n)n×[κln⁡n]2ρjn(ln⁡n)3)-1))+n/(ln⁡n)3(ln⁡n)/nne-θ[κln⁡n])≤C(n-Cκ2/(1+κ3/2)+n2-θκ).
Taking κ large enough, the last term is bounded by C/n4. This completes the proof of(c).
This completes the proof of Proposition10.Proof of Theorem3.
Theorem3 can be proved by combining arguments of ([36], Theorem 5.1) and ([51], Theorem 4.2). It is close to ([30], Proof of Theorem 2) by taking θ→∞. The interested reader can find the details below.
We consider the following wavelet decomposition forf:
(63)f(x)=∑k∈Λj0‍cj0,kϕj0,k(x)+∑j=j0∞‍∑k∈Λj‍dj,kψj,k(x),
where cj0,k=∫01‍f(x)ϕj0,k(x)dx and dj,k=∫01‍f(x)ψj,k(x)dx.
Using the orthonormality of the wavelet basisℬ, the MISE of f^ can be expressed as
(64)𝔼(∥f^-f∥22)=P+Q+R,
where
(65)P=∑k∈Λj0‍𝔼(|c^j0,k-cj0,k|2),Q=∑j=j0j1‍∑k∈Λj‍𝔼(|d^j,k1{|d^j,k|≥κλj}-dj,k|2),R=∑j=j1+1∞‍∑k∈Λj‍dj,k2.
Let us now investigate sharp upper bounds forP, R and Q successively.
Upper Bound for  P. The point (a) of Proposition 10 and 2s/(2s+2ρ+1)<1 yield
(66)P≤C2j0n≤Cln⁡nn≤C(ln⁡nn)2s/(2s+2ρ+1).Upper Bound forR(i)
Forr≥1 and p≥2, we have f∈Bp,rs(M)⊆B2,∞s(M). Using 2s/(2s+2ρ+1)<2s/(2ρ+1), we obtain
(67)R≤C∑j=j1+1∞‍2-2js≤C((ln⁡n)3n)2s/(2ρ+1)≤C(ln⁡nn)2s/(2s+2ρ+1).(ii)
Forr≥1 and p∈[1,2), we have f∈Bp,rs(M)⊆B2,∞s+1/2-1/p(M). The condition s>(2ρ+1)/p implies that (s+1/2-1/p)/(2ρ+1)>s/(2s+2ρ+1). Thus
(68)R≤C∑j=j1+1∞‍2-2j(s+1/2-1/p)≤C((ln⁡n)3n)2(s+1/2-1/p)/(2ρ+1)≤C(ln⁡nn)2s/(2s+2ρ+1). Hence, for r≥1, {p≥2 and s>0} or {p∈[1,2), and s>(2ρ+1)/p}, we have
(69)R≤C(ln⁡nn)2s/(2s+2ρ+1).
Upper Bound for  Q. Adopting the notation D^j,k=d^j,k-dj,k, Q can be written as
(70)Q=∑i=14‍Qi,
where
(71)Q1=∑j=j0j1‍∑k∈Λj‍𝔼(|D^j,k|21{|d^j,k|≥κλj,|dj,k|<κλj/2}),Q2=∑j=j0j1‍∑k∈Λj‍𝔼(|D^j,k|21{|d^j,k|≥κλj,|dj,k|≥κλj/2}),Q3=∑j=j0j1‍∑k∈Λj‍𝔼(dj,k21{|d^j,k|<κλj,|dj,k|≥2κλj}),Q4=∑j=j0j1‍∑k∈Λj‍𝔼(dj,k21{|d^j,k|<κλj,|dj,k|<2κλj}).
Upper Bound for  Q1+Q3. Owing to the inequalities 1{|d^j,k|<κλj,|dj,k|≥2κλj}≤1{|D^j,k|>κλj/2}, 1{|d^j,k|≥κλj,|dj,k|<κλj/2}≤1{|D^j,k|>κλj/2} and 1{|d^j,k|<κλj,|dj,k|≥2κλj}≤1{|dj,k|≤2|D^j,k|}, the Cauchy-Schwarz inequality, and the points (b) and (c) of Proposition 10, we have
(72)Q1+Q3≤C∑j=j0j1‍∑k∈Λj‍𝔼(|D^j,k|21{|D^j,k|>κλj/2})≤C∑j=j0j1‍∑k∈Λj‍(𝔼(|D^j,k|4))1/2(ℙ(|D^j,k|>κλj2))1/2≤C1n2∑j=j0j1‍2j(1+2ρ)≤C1n≤C(ln⁡nn)2s/(2s+2ρ+1).
Upper Bound for  Q2. It follows from the point (a) of Proposition 10 that
(73)Q2≤∑j=j0j1‍∑k∈Λj‍𝔼(|D^j,k|2)1{|dj,k|≥κλj/2}≤C1n∑j=j0j1‍22ρj∑k∈Λj‍1{|dj,k|>κλj/2}.
Let us now introduce the integer j* defined by
(74)2j*=[(nln⁡n)1/(2s+2ρ+1)].
Note that j*∈{j0,…,j1} for n large enough.
ThenQ2 can be bounded as
(75)Q2≤Q2,1+Q2,2,
where
(76)Q2,1=C1n∑j=j0j*‍22ρj∑k∈Λj‍1{|dj,k|>κλj/2},Q2,2=C1n∑j=j*+1j1‍22ρj∑k∈Λj‍1{|dj,k|>κλj/2}.
On the one hand we have
(77)Q2,1≤Cln⁡nn∑j=j0j*‍2j(1+2ρ)≤C(ln⁡nn)2s/(2s+2ρ+1).
On the other hand, we have the following.(i)
Forr≥1 and p≥2, the Markov inequality and f∈Bp,rs(M)⊆B2,∞s(M) yield
(78)Q2,2≤Cln⁡nn∑j=j*+1j1‍22ρj1λj2∑k∈Λj‍dj,k2≤C∑j=j*+1∞‍∑k∈Λj‍dj,k2≤C∑j=j*+1∞‍2-2js≤C(ln⁡nn)2s/(2s+2ρ+1).(ii)
Forr≥1, p∈[1,2) and s>(2ρ+1)/p, the Markov inequality, f∈Bp,rs(M), and (2s+2ρ+1)(2-p)/2+(s+1/2-1/p+ρ-2ρ/p)p=2s imply that
(79)Q2,2≤Cln⁡nn∑j=j*+1j1‍22ρj1λjp∑k∈Λj‍|dj,k|p≤C(ln⁡nn)(2-p)/2∑j=j*+1∞‍2jρ(2-p)2-j(s+1/2-1/p)p≤C(ln⁡nn)(2-p)/22-j*(s+1/2-1/p+ρ-2ρ/p)p≤C(ln⁡nn)2s/(2s+2ρ+1).Therefore, for r≥1, {p≥2 and s>0} or {p∈[1,2), and s>(2ρ+1)/p}, we have
(80)Q2≤C(ln⁡nn)2s/(2s+2ρ+1).
Upper Bound for  Q4. We have
(81)Q4≤∑j=j0j1‍∑k∈Λj‍dj,k21{|dj,k|<2κλj}.
Let j* be the integer (74). Then Q4 can be bound as
(82)Q4≤Q4,1+Q4,2,
where
(83)Q4,1=∑j=j0j*‍∑k∈Λj‍dj,k21{|dj,k|<2κλj},Q4,2=∑j=j*+1j1‍∑k∈Λj‍dj,k21{|dj,k|<2κλj}.
On the one hand, we have
(84)Q4,1≤C∑j=j0j*‍2jλj2=Cln⁡nn∑j=j0j*‍2j(1+2ρ)≤C(ln⁡nn)2s/(2s+2ρ+1).
On the other hand, we have the following(i)
Forr≥1 and p≥2, since f∈Bp,rs(M)⊆B2,∞s(M), we have
(85)Q4,2≤∑j=j*+1∞‍∑k∈Λj‍dj,k2≤C∑j=j*+1∞‍2-2js≤C(ln⁡nn)2s/(2s+2ρ+1).(ii)
Forr≥1, p∈[1,2) and s>(2ρ+1)/p, owing to the Markov inequality, f∈Bp,rs(M) and (2s+2ρ+1)(2-p)/2+(s+1/2-1/p+ρ-2ρ/p)p=2s, we get
(86)Q4,2≤C∑j=j*+1j1‍λj2-p∑k∈Λj‍|dj,k|p=C(ln⁡nn)(2-p)/2∑j=j*+1j1‍2jρ(2-p)∑k∈Λj‍|dj,k|p≤C(ln⁡nn)(2-p)/2∑j=j*+1∞‍2jρ(2-p)2-j(s+1/2-1/p)p≤C(ln⁡nn)(2-p)/22-j*(s+1/2-1/p+ρ-2ρ/p)p≤C(ln⁡nn)2s/(2s+2ρ+1).So, for r≥1, {p≥2 and s>0} or {p∈[1,2), and s>(2ρ+1)/p}, we have
(87)Q4≤C(ln⁡nn)2s/(2s+2ρ+1).
Putting (70), (72), (80), and (87) together, for r≥1, {p≥2 and s>0} or {p∈[1,2) and s>(2ρ+1)/p}, we obtain
(88)Q≤C(ln⁡nn)2s/(2s+2ρ+1).
Combining (64), (66), (69), and (88), we complete the proof of Theorem 3.Proof of Theorem4.
The proof of Theorem4 is a direct application of Theorem 3: under (G1)–(G5), the function q defined by (24) satisfies (H1) see ([42], equation (2)) and (H2): (i) see ([42], Lemma 6), (ii) see, ([42], equation (11)) and (iii) see ([30], Proof of Proposition 6.1), with ρ=δ.Proof of Theorem5.
The proof of Theorem5 is a consequence of Theorem 3: under (J1)–(J4), the function q defined by (30) satisfies (H1) and (H2): (i)-(ii) see ([31], Proposition 1) and (iii) see ([52], equation (26)), with ρ=δ.Proof of Theorem6.
Setv(x)=f(x)g(x). Following the methodology of [49], we have
(89)f^(x)-f(x)=S(x)-T(x),
where
(90)S(x)=1g^(x)(f(x)(g(x)-g^(x))v^(x)-v(x)+f(x)(g(x)-g^(x)))1{|g^(x)|≥c*/2}T(x)=f(x)1{|g^(x)|<c*/2}.
Using (K3) and the indicator function, we have
(91)|S(x)|≤C(|v^(x)-v(x)|+|g^(x)-g(x)|).
It follows from {|g^(x)|<c*/2}∩{|g(x)|>c*}⊆{|g^(x)-g(x)|>c*/2}, (K3), (K4), and the Markov inequality that
(92)|T(x)|≤C1{|g^(x)-g(x)|>c*/2}≤C|g^(x)-g(x)|.
The triangular inequality yields
(93)|f^(x)-f(x)|≤C(|v^(x)-v(x)|+|g^(x)-g(x)|).
The elementary inequality (a+b)2≤2(a2+b2) implies that
(94)𝔼(∥f^-f∥22)≤C(𝔼(∥v^-v∥22)+𝔼(∥g^-g∥22)).
We now bound this two MISEs via Theorem 3.
Upper Bound for the MISE of  v^. Under (K1)–(K6), the function q defined by (38) satisfies the following.(H1) Withv instead of f: since ξ1 and X1 are independent with 𝔼(ξ1)=0,
(95)𝔼(q(γj,k,V1))=𝔼(Y1γj,k(X1))=𝔼(f(X1)γj,k(X1))=∫01‍f(x)γj,k(x)g(x)dx=∫01‍v(x)γj,k(x)dx,(H2): (i)-(ii)-(iii) withρ=0:
(i) sinceY1(Ω) is bounded thanks to (K2) and (K3), say |Y1|≤M with M>0, we have
(96)sup⁡(x,x*)∈V1(Ω)|q(γj,k,(x,x*))|=sup⁡(x,x*)∈[-M,M]×[0,1]|xγj,k(x*)|≤Msup⁡x*∈[0,1]|γj,k(x*)|≤C2j/2
(ii) using the boundedness ofY1(Ω), then (K4), we have
(97)𝔼(|q(γj,k,V1)|2)=𝔼(Y12(γj,k(X1))2)≤C𝔼((γj,k(X1))2)=C∫01‍(γj,k(x))2g(x)dx≤C∫01‍(γj,k(x))2dx≤C
(iii) using the boundedness ofY1(Ω) and making the change of variables y=2jx-k, we obtain
(98)∫V1(Ω)|q(γj,k,x)|dx=(∫-MM‍|x|dx)(∫01‍|γj,k(x*)|dx*)=M2∫01‍|γj,k(x)|dx≤C2-j/2.
We conclude by applying Lemma 2 with ρ=0; (K5) and (K6) imply (F1), and the previous inequality implies (F2).
Therefore, assuming thatv∈Bp,rs(M) with r≥1, {p≥2 and s∈(0,N)} or {p∈[1,2) and s∈(1/p,N)}, Theorem 3 proves the existence of a constant C>0 satisfying
(99)𝔼(∥v^-v∥22)≤C(ln⁡nn)2s/(2s+1).
Upper Bound for the MISE of  g^. Under (K1)–(K6), proceeding as the previous point, we show that the function q defined by (39) satisfies (H1) with g instead of f and Xt instead of Vt, and (H2): (i)-(ii)-(iii) with ρ=0.
Therefore, assuming thatg∈Bp,rs(M) with r≥1, {p≥2 and s∈(0,N)} or {p∈[1,2), and s∈(1/p,N)}, Theorem 3 proves the existance of a constant C>0 satisfying
(100)𝔼(∥g^-g∥22)≤C(ln⁡nn)2s/(2s+1).
Combining (94), (99), and (100), we end the proof of Theorem 6.


## 5.1. Key Lemmas

Let us present two lemmas which will be used in the proofs.Lemma8 shows a sharp covariance inequality under the α-mixing condition.Lemma 8 (see [50]).
Let(Wt)t∈ℤ be a strictly stationary α-mixing process with mixing coefficient αm, m≥0, and let h and k be two measurable functions. Let p>0 and q>0 satisfying 1/p+1/q<1 such that 𝔼(|h(W1)|p) and 𝔼(|k(W1)|q) exist. Then there exists a constant C>0 such that
(41)|ℂov(h(W1),k(Wm+1))|≤Cαm1-1/p-1/q(𝔼(|h(W1)|p))1/p(𝔼(|k(W1)|q))1/q.Lemma9 below presents a concentration inequality for α-mixing processes.Lemma 9 (see [13]).
Let(Wt)t∈ℤ be a strictly stationary process with the mth strongly mixing coefficient αm, m≥0, let n be a positive integer, let h:ℝ→ℂ be a measurable function, and, for any t∈ℤ, Ut=h(Wt). We assume that 𝔼(U1)=0 and there exists a constant M>0 satisfying |U1|≤M. Then, for any m∈{1,…,[n/2]} and λ>0, we have
(42)ℙ(1n|∑i=1n‍Ui|≥λ)≤4exp⁡(-λ2n16(Dm/m+λMm/3))+32Mλnαm,
where
(43)Dm=max⁡l∈{1,…,2m}𝕍(∑i=1l‍Ui).


## 5.2.  Intermediary Results

Proof of Lemma2.
Using a standard expression of the covariance, and (F1) as well as (F2), we obtain(44)|ℂov(q(γj,k,Vm+1),q(γj,k,V1))|=|∫V1(Ω)‍∫V1(Ω)‍q(γj,k,x)q(γj,k,y)¯×(u(V1,Vm+1)(x,y)-u(x)u(y))dxdy∫V1(Ω)‍|≤∫V1(Ω)‍∫V1(Ω)‍|q(γj,k,x)||q(γj,k,y)|×|u(V1,Vm+1)(x,y)-u(x)u(y)|dxdy≤C(∫V1(Ω)‍|q(γj,k,x)|dx)2≤C22ρj2-j.
This ends the proof of Lemma 2.Proposition10 proves probability and moments inequalities satisfied by the estimators (8).Proposition 10.
Letα^j,k and β^j,k be defined as in (8) under (H1) and (H2), let j0 be (9) and let j1 be (10).(a)
There exists a constantC>0 such that, for any j∈{j0,…,j1} and k∈Λj,
(45)𝔼(|c^j,k-cj,k|2)≤C22ρj1n,(46)𝔼(|d^j,k-dj,k|2)≤C22ρj1n.(b)
There exists a constantC>0 such that, for any j∈{j0,…,j1} and k∈Λj,
(47)𝔼(|d^j,k-dj,k|4)≤C24ρj.(c)
Letλj be defined as in (11). There exists a constant C>0 such that, for any κ large enough, j∈{j0,…,j1} and k∈Λj, we have
(48)ℙ(|d^j,k-dj,k|≥κλj2)≤C1n4.Proof of Proposition10.
(a) Using (H1) and the stationarity of(Vt)t∈ℤ, we obtain
(49)𝔼(|c^j,k-cj,k|2)=𝕍(c^j,k)=1n2∑i=1n‍𝕍(q(ϕj,k,Vi))+2n2×∑v=2n‍∑ℓ=1v-1‍Re(ℂov(q(ϕj,k,Vv),q(ϕj,k,Vℓ)))=1n2∑i=1n‍𝕍(q(ϕj,k,Vi))+2n2×∑m=1n-1‍(n-m)Re(ℂov(q(ϕj,k,Vm+1),q(ϕj,k,V1)))≤1n(∑m=1n-1‍𝔼(|q(ϕj,k,V1)|2)+2∑m=1n-1‍|ℂov(q(ϕj,k,Vm+1),q(ϕj,k,V1))|).
By (H2)-(ii) we get
(50)𝔼(|q(ϕj,k,V1)|2)≤C22ρj.
For the covariance term, note that
(51)∑m=1n-1‍|ℂov(q(ϕj,k,Vm+1),q(ϕj,k,V1))|=A+B,
where
(52)A=∑m=1[ln⁡n/θ]-1‍|ℂov(q(ϕj,k,Vm+1),q(ϕj,k,V1))|,B=∑m=[(ln⁡n)/θ]n-1‍|ℂov(q(ϕj,k,Vm+1),q(ϕj,k,V1))|.
It follows from (H2)-(iii) and 2-j≤2-j0<2(ln⁡n)-1 that
(53)A≤C22ρj2-j[ln⁡nθ]≤C22ρj.
The Davydov inequality described in Lemma 8 with p=q=4, (H2)-(i)-(ii), and 2j≤2j1≤n give
(54)B≤C𝔼(|q(ϕj,k,V1)|4)∑m=[(ln⁡n)/θ]n-1‍αm≤C2ρj2j/2𝔼(|q(ϕj,k,V1)|2)∑m=[(ln⁡n)/θ]∞‍e-θm/2=C22ρjne-(ln⁡n)/2≤C22ρj.
Thus
(55)∑m=1n-1‍|ℂov(q(ϕj,k,Vm+1),q(ϕj,k,V1))|≤C22ρj.
Putting (49), (50), and (55) 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|d^j,k|≤sup⁡x∈V1(Ω)|q(ψj,k,x)|≤C2ρj2j/2. It follows from the triangular inequality and |dj,k|≤∥f∥2≤C that
(56)|d^j,k-dj,k|≤|d^j,k|+|dj,k|≤C2ρj2j/2.
This inequality and the second result of (a) yield
(57)𝔼(|d^j,k-dj,k|4)≤C22ρj2j𝔼(|d^j,k-dj,k|2)≤C24ρj2j1n.
Using 2j≤2j1≤n, the proof of (b) is completed.
(c) We will use the Liebscher inequality described in Lemma9. Let us set
(58)Ui=q(ψj,k,Vi)-𝔼(q(ψj,k,V1)).
We have𝔼(U1)=0 and, by(H2)-(i) and 2j≤2j1≤n/(ln⁡n)3,
(59)|Ui|≤2sup⁡x∈V1(Ω)|q(ψj,k,x)|≤C2ρj2j/2≤C2ρjn(ln⁡n)3,
(so M=C2ρjn/(ln⁡n)3).
Proceeding as for the proofs of the bounds in(a), for any integerl≤Cln⁡n, since 2-j≤2-j0≤2(ln⁡n)-1, we show that
(60)𝕍(∑i=1lUi)=𝕍(∑i=1lq(ψj,k,Vi))≤C22ρj(l+l22-j)≤C22ρjl.
Therefore
(61)Dm=max⁡l∈{1,…,2m}𝕍(∑i=1l‍Ui)≤C22ρjm.
Owing to Lemma 9 applied with U1,…,Un, λ=κλj/2, m=[κln⁡n], M=C2ρjn/(ln⁡n)3, and the bound (61), we obtain
(62)ℙ(|d^j,k-dj,k|≥κλj2)≤C(exp⁡(-Cκ2λj2nDm/m+κλjmM)+Mλjne-θm)≤C(n/(ln⁡n)3(ln⁡n)/nexp⁡(-C((n(ln⁡n)3κ222ρjln⁡n)×(n(ln⁡n)322ρj+κ2ρj(ln⁡n)n×[κln⁡n]2ρjn(ln⁡n)3)-1))+n/(ln⁡n)3(ln⁡n)/nne-θ[κln⁡n])≤C(n-Cκ2/(1+κ3/2)+n2-θκ).
Taking κ large enough, the last term is bounded by C/n4. This completes the proof of(c).
This completes the proof of Proposition10.Proof of Theorem3.
Theorem3 can be proved by combining arguments of ([36], Theorem 5.1) and ([51], Theorem 4.2). It is close to ([30], Proof of Theorem 2) by taking θ→∞. The interested reader can find the details below.
We consider the following wavelet decomposition forf:
(63)f(x)=∑k∈Λj0‍cj0,kϕj0,k(x)+∑j=j0∞‍∑k∈Λj‍dj,kψj,k(x),
where cj0,k=∫01‍f(x)ϕj0,k(x)dx and dj,k=∫01‍f(x)ψj,k(x)dx.
Using the orthonormality of the wavelet basisℬ, the MISE of f^ can be expressed as
(64)𝔼(∥f^-f∥22)=P+Q+R,
where
(65)P=∑k∈Λj0‍𝔼(|c^j0,k-cj0,k|2),Q=∑j=j0j1‍∑k∈Λj‍𝔼(|d^j,k1{|d^j,k|≥κλj}-dj,k|2),R=∑j=j1+1∞‍∑k∈Λj‍dj,k2.
Let us now investigate sharp upper bounds forP, R and Q successively.
Upper Bound for  P. The point (a) of Proposition 10 and 2s/(2s+2ρ+1)<1 yield
(66)P≤C2j0n≤Cln⁡nn≤C(ln⁡nn)2s/(2s+2ρ+1).Upper Bound forR(i)
Forr≥1 and p≥2, we have f∈Bp,rs(M)⊆B2,∞s(M). Using 2s/(2s+2ρ+1)<2s/(2ρ+1), we obtain
(67)R≤C∑j=j1+1∞‍2-2js≤C((ln⁡n)3n)2s/(2ρ+1)≤C(ln⁡nn)2s/(2s+2ρ+1).(ii)
Forr≥1 and p∈[1,2), we have f∈Bp,rs(M)⊆B2,∞s+1/2-1/p(M). The condition s>(2ρ+1)/p implies that (s+1/2-1/p)/(2ρ+1)>s/(2s+2ρ+1). Thus
(68)R≤C∑j=j1+1∞‍2-2j(s+1/2-1/p)≤C((ln⁡n)3n)2(s+1/2-1/p)/(2ρ+1)≤C(ln⁡nn)2s/(2s+2ρ+1). Hence, for r≥1, {p≥2 and s>0} or {p∈[1,2), and s>(2ρ+1)/p}, we have
(69)R≤C(ln⁡nn)2s/(2s+2ρ+1).
Upper Bound for  Q. Adopting the notation D^j,k=d^j,k-dj,k, Q can be written as
(70)Q=∑i=14‍Qi,
where
(71)Q1=∑j=j0j1‍∑k∈Λj‍𝔼(|D^j,k|21{|d^j,k|≥κλj,|dj,k|<κλj/2}),Q2=∑j=j0j1‍∑k∈Λj‍𝔼(|D^j,k|21{|d^j,k|≥κλj,|dj,k|≥κλj/2}),Q3=∑j=j0j1‍∑k∈Λj‍𝔼(dj,k21{|d^j,k|<κλj,|dj,k|≥2κλj}),Q4=∑j=j0j1‍∑k∈Λj‍𝔼(dj,k21{|d^j,k|<κλj,|dj,k|<2κλj}).
Upper Bound for  Q1+Q3. Owing to the inequalities 1{|d^j,k|<κλj,|dj,k|≥2κλj}≤1{|D^j,k|>κλj/2}, 1{|d^j,k|≥κλj,|dj,k|<κλj/2}≤1{|D^j,k|>κλj/2} and 1{|d^j,k|<κλj,|dj,k|≥2κλj}≤1{|dj,k|≤2|D^j,k|}, the Cauchy-Schwarz inequality, and the points (b) and (c) of Proposition 10, we have
(72)Q1+Q3≤C∑j=j0j1‍∑k∈Λj‍𝔼(|D^j,k|21{|D^j,k|>κλj/2})≤C∑j=j0j1‍∑k∈Λj‍(𝔼(|D^j,k|4))1/2(ℙ(|D^j,k|>κλj2))1/2≤C1n2∑j=j0j1‍2j(1+2ρ)≤C1n≤C(ln⁡nn)2s/(2s+2ρ+1).
Upper Bound for  Q2. It follows from the point (a) of Proposition 10 that
(73)Q2≤∑j=j0j1‍∑k∈Λj‍𝔼(|D^j,k|2)1{|dj,k|≥κλj/2}≤C1n∑j=j0j1‍22ρj∑k∈Λj‍1{|dj,k|>κλj/2}.
Let us now introduce the integer j* defined by
(74)2j*=[(nln⁡n)1/(2s+2ρ+1)].
Note that j*∈{j0,…,j1} for n large enough.
ThenQ2 can be bounded as
(75)Q2≤Q2,1+Q2,2,
where
(76)Q2,1=C1n∑j=j0j*‍22ρj∑k∈Λj‍1{|dj,k|>κλj/2},Q2,2=C1n∑j=j*+1j1‍22ρj∑k∈Λj‍1{|dj,k|>κλj/2}.
On the one hand we have
(77)Q2,1≤Cln⁡nn∑j=j0j*‍2j(1+2ρ)≤C(ln⁡nn)2s/(2s+2ρ+1).
On the other hand, we have the following.(i)
Forr≥1 and p≥2, the Markov inequality and f∈Bp,rs(M)⊆B2,∞s(M) yield
(78)Q2,2≤Cln⁡nn∑j=j*+1j1‍22ρj1λj2∑k∈Λj‍dj,k2≤C∑j=j*+1∞‍∑k∈Λj‍dj,k2≤C∑j=j*+1∞‍2-2js≤C(ln⁡nn)2s/(2s+2ρ+1).(ii)
Forr≥1, p∈[1,2) and s>(2ρ+1)/p, the Markov inequality, f∈Bp,rs(M), and (2s+2ρ+1)(2-p)/2+(s+1/2-1/p+ρ-2ρ/p)p=2s imply that
(79)Q2,2≤Cln⁡nn∑j=j*+1j1‍22ρj1λjp∑k∈Λj‍|dj,k|p≤C(ln⁡nn)(2-p)/2∑j=j*+1∞‍2jρ(2-p)2-j(s+1/2-1/p)p≤C(ln⁡nn)(2-p)/22-j*(s+1/2-1/p+ρ-2ρ/p)p≤C(ln⁡nn)2s/(2s+2ρ+1).Therefore, for r≥1, {p≥2 and s>0} or {p∈[1,2), and s>(2ρ+1)/p}, we have
(80)Q2≤C(ln⁡nn)2s/(2s+2ρ+1).
Upper Bound for  Q4. We have
(81)Q4≤∑j=j0j1‍∑k∈Λj‍dj,k21{|dj,k|<2κλj}.
Let j* be the integer (74). Then Q4 can be bound as
(82)Q4≤Q4,1+Q4,2,
where
(83)Q4,1=∑j=j0j*‍∑k∈Λj‍dj,k21{|dj,k|<2κλj},Q4,2=∑j=j*+1j1‍∑k∈Λj‍dj,k21{|dj,k|<2κλj}.
On the one hand, we have
(84)Q4,1≤C∑j=j0j*‍2jλj2=Cln⁡nn∑j=j0j*‍2j(1+2ρ)≤C(ln⁡nn)2s/(2s+2ρ+1).
On the other hand, we have the following(i)
Forr≥1 and p≥2, since f∈Bp,rs(M)⊆B2,∞s(M), we have
(85)Q4,2≤∑j=j*+1∞‍∑k∈Λj‍dj,k2≤C∑j=j*+1∞‍2-2js≤C(ln⁡nn)2s/(2s+2ρ+1).(ii)
Forr≥1, p∈[1,2) and s>(2ρ+1)/p, owing to the Markov inequality, f∈Bp,rs(M) and (2s+2ρ+1)(2-p)/2+(s+1/2-1/p+ρ-2ρ/p)p=2s, we get
(86)Q4,2≤C∑j=j*+1j1‍λj2-p∑k∈Λj‍|dj,k|p=C(ln⁡nn)(2-p)/2∑j=j*+1j1‍2jρ(2-p)∑k∈Λj‍|dj,k|p≤C(ln⁡nn)(2-p)/2∑j=j*+1∞‍2jρ(2-p)2-j(s+1/2-1/p)p≤C(ln⁡nn)(2-p)/22-j*(s+1/2-1/p+ρ-2ρ/p)p≤C(ln⁡nn)2s/(2s+2ρ+1).So, for r≥1, {p≥2 and s>0} or {p∈[1,2), and s>(2ρ+1)/p}, we have
(87)Q4≤C(ln⁡nn)2s/(2s+2ρ+1).
Putting (70), (72), (80), and (87) together, for r≥1, {p≥2 and s>0} or {p∈[1,2) and s>(2ρ+1)/p}, we obtain
(88)Q≤C(ln⁡nn)2s/(2s+2ρ+1).
Combining (64), (66), (69), and (88), we complete the proof of Theorem 3.Proof of Theorem4.
The proof of Theorem4 is a direct application of Theorem 3: under (G1)–(G5), the function q defined by (24) satisfies (H1) see ([42], equation (2)) and (H2): (i) see ([42], Lemma 6), (ii) see, ([42], equation (11)) and (iii) see ([30], Proof of Proposition 6.1), with ρ=δ.Proof of Theorem5.
The proof of Theorem5 is a consequence of Theorem 3: under (J1)–(J4), the function q defined by (30) satisfies (H1) and (H2): (i)-(ii) see ([31], Proposition 1) and (iii) see ([52], equation (26)), with ρ=δ.Proof of Theorem6.
Setv(x)=f(x)g(x). Following the methodology of [49], we have
(89)f^(x)-f(x)=S(x)-T(x),
where
(90)S(x)=1g^(x)(f(x)(g(x)-g^(x))v^(x)-v(x)+f(x)(g(x)-g^(x)))1{|g^(x)|≥c*/2}T(x)=f(x)1{|g^(x)|<c*/2}.
Using (K3) and the indicator function, we have
(91)|S(x)|≤C(|v^(x)-v(x)|+|g^(x)-g(x)|).
It follows from {|g^(x)|<c*/2}∩{|g(x)|>c*}⊆{|g^(x)-g(x)|>c*/2}, (K3), (K4), and the Markov inequality that
(92)|T(x)|≤C1{|g^(x)-g(x)|>c*/2}≤C|g^(x)-g(x)|.
The triangular inequality yields
(93)|f^(x)-f(x)|≤C(|v^(x)-v(x)|+|g^(x)-g(x)|).
The elementary inequality (a+b)2≤2(a2+b2) implies that
(94)𝔼(∥f^-f∥22)≤C(𝔼(∥v^-v∥22)+𝔼(∥g^-g∥22)).
We now bound this two MISEs via Theorem 3.
Upper Bound for the MISE of  v^. Under (K1)–(K6), the function q defined by (38) satisfies the following.(H1) Withv instead of f: since ξ1 and X1 are independent with 𝔼(ξ1)=0,
(95)𝔼(q(γj,k,V1))=𝔼(Y1γj,k(X1))=𝔼(f(X1)γj,k(X1))=∫01‍f(x)γj,k(x)g(x)dx=∫01‍v(x)γj,k(x)dx,(H2): (i)-(ii)-(iii) withρ=0:
(i) sinceY1(Ω) is bounded thanks to (K2) and (K3), say |Y1|≤M with M>0, we have
(96)sup⁡(x,x*)∈V1(Ω)|q(γj,k,(x,x*))|=sup⁡(x,x*)∈[-M,M]×[0,1]|xγj,k(x*)|≤Msup⁡x*∈[0,1]|γj,k(x*)|≤C2j/2
(ii) using the boundedness ofY1(Ω), then (K4), we have
(97)𝔼(|q(γj,k,V1)|2)=𝔼(Y12(γj,k(X1))2)≤C𝔼((γj,k(X1))2)=C∫01‍(γj,k(x))2g(x)dx≤C∫01‍(γj,k(x))2dx≤C
(iii) using the boundedness ofY1(Ω) and making the change of variables y=2jx-k, we obtain
(98)∫V1(Ω)|q(γj,k,x)|dx=(∫-MM‍|x|dx)(∫01‍|γj,k(x*)|dx*)=M2∫01‍|γj,k(x)|dx≤C2-j/2.
We conclude by applying Lemma 2 with ρ=0; (K5) and (K6) imply (F1), and the previous inequality implies (F2).
Therefore, assuming thatv∈Bp,rs(M) with r≥1, {p≥2 and s∈(0,N)} or {p∈[1,2) and s∈(1/p,N)}, Theorem 3 proves the existence of a constant C>0 satisfying
(99)𝔼(∥v^-v∥22)≤C(ln⁡nn)2s/(2s+1).
Upper Bound for the MISE of  g^. Under (K1)–(K6), proceeding as the previous point, we show that the function q defined by (39) satisfies (H1) with g instead of f and Xt instead of Vt, and (H2): (i)-(ii)-(iii) with ρ=0.
Therefore, assuming thatg∈Bp,rs(M) with r≥1, {p≥2 and s∈(0,N)} or {p∈[1,2), and s∈(1/p,N)}, Theorem 3 proves the existance of a constant C>0 satisfying
(100)𝔼(∥g^-g∥22)≤C(ln⁡nn)2s/(2s+1).
Combining (94), (99), and (100), we end the proof of Theorem 6.



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*Source: 403764-2014-01-19.xml*