# Adaptive Wavelet Estimation of a Biased Density for Strongly Mixing Sequences

Authors: Christophe Chesneau
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

The estimation of a biased density for exponentially strongly
mixing sequences is investigated. We construct a new adaptive wavelet
estimator based on a hard thresholding rule. We determine a sharp
upper bound of the associated mean integrated square error for a wide
class of functions.


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## Body

## 1. Introduction

In the standard density estimation problem, we observen random variables X1,…,Xn with common density function f. The goal is to estimate f from X1,…,Xn. However, in some applications, X1,…,Xn are not accessible; we only have n random variables Z1,…,Zn with the common density(1.1)g(x)=μ-1w(x)f(x),
where w denotes a known positive function and μ is the unknown normalization parameter: μ=∫w(y)f(y)dy. Our goal is to estimate the “biased density” f from Z1,…,Zn. Practical examples can be found in, for example, [1–3] and the survey by the author of [4].The standard i.i.d. case has been investigated in several papers. See, for example, [5–9]. To the best of our knowledge, the dependent case has only been examined in [10] for associated (positively or negatively) Z1,…,Zn. In this paper, we study another dependent (and realistic) structure which has not been addressed earlier: we suppose that Z1,…,Zn is a sample of a strictly stationary and exponentially strongly mixing process (Zi)i∈ℤ (to be defined in Section 2). Such a dependence condition arises for a wide class of GARCH-type time series models classically encountered in finance. See, for example, [11, 12] for an overview.We focus our attention on the wavelet methods because they provide a coherent set of procedures that are spatially adaptive and near optimal over a wide range of function spaces. See, for example, [13, 14] for a detailed coverage of wavelet theory in statistics. We develop two new wavelet estimators: a linear nonadaptive based on projections and a nonlinear adaptive using the hard thresholding rule introduced by [15]. We measure their performances by determining upper bounds of the mean integrated squared error (MISE) over Besov balls (to be defined in Section 3). We prove that our adaptive estimator attains a sharp rate of convergence, close to the one attained by the linear wavelet estimator (constructed in a nonadaptive fashion to minimize the MISE).The rest of the paper is organized as follows. Section2 is devoted to the assumptions on the model. In Section 3, we present wavelets and Besov balls. The considered wavelet estimators are defined in Section 4. Section 5 is devoted to the results. The proofs are postponed in Section 6.


## 2. Assumptions on the Model

We assume thatZ1,…,Zn coming from a strictly stationary process (Zi)i∈ℤ. For any m∈ℤ, we define the mth strongly mixing coefficient of (Zi)i∈ℤ by (2.1)am=sup⁡(A,B)∈F-∞,0Z×Fm,∞Z|P(A∩B)-P(A)P(B)|,
where, for any u∈ℤ, ℱ-∞,uZ is the σ-algebra generated by the random variables …,Zu-1,Zu and ℱu,∞Z is the σ-algebra generated by the random variables Zu,Zu+1,….We consider the exponentially strongly mixing case, that is, there exist three known constants,γ>0, c>0, and θ>0, such that, for any m∈ℤ,(2.2)am≤γexp⁡(-c|m|θ).
This assumption is satisfied by a large class of GARCH processes. See, for example, [11, 12, 16, 17].Note that, whenθ→∞, we are in the standard i.i.d. case.W.o.l.g., the support of the functionsf, and w are [0,1].There exist two constants,c>0 and C>0, such that(2.3)c≤inf⁡x∈[0,1]w(x),sup⁡x∈[0,1]w(x)≤C.
There exists a (known) constant C>0 such that(2.4)sup⁡x∈[0,1]f(x)≤C.
For any m∈{1,…,n}, let g(Z0,Zm) be the density of (Z0,Zm). There exists a constant C>0 such that(2.5)sup⁡m∈{1,…,n}sup⁡(x,y)∈[0,1]2|g(Z0,Zm)(x,y)-g(x)g(y)|≤C.The two first boundedness assumptions are standard in the estimation of biased densities. See, for example, [6–8].


## 3. Wavelets and Besov Balls

LetN be an integer ϕ and ψ be the initial wavelets of dbN (so supp(ϕ)=supp(ψ)=[1-N,N]). Set (3.1)ϕj,k(x)=2j/2ϕ(2jx-k),ψj,k(x)=2j/2ψ(2jx-k).
With an appropriate treatments at the boundaries, there exists an integer τ satisfying 2τ≥2N such that the collection ℬ={ϕτ,k(·), k∈{0,…,2τ-1}; ψj,k(·); j∈ℕ-{0,…,τ-1}, k∈{0,…,2j-1}}, is an orthonormal basis of 𝕃2([0,1]) (the space of square-integrable functions on [0,1]). See [18].For any integerℓ≥τ, any h∈𝕃2([0,1]) can be expanded on ℬ as (3.2)h(x)=∑k=02l-1αl,kϕl,k(x)+∑j=l∞∑k=02j-1βj,kψj,k(x),x∈[0,1],
where αj,k and βj,k are the wavelet coefficients of h defined by(3.3)αj,k=∫01h(x)ϕj,k(x)dx,βj,k=∫01h(x)ψj,k(x)dx.LetM>0, s>0, p≥1, and r≥1. A function h belongs to Bp,rs(M) if and only if there exists a constant M*>0 (depending on M) such that the associated wavelet coefficients (3.3) satisfy (3.4)2τ(1/2-1/p)(∑k=02τ-1|ατ,k|p)1/p+(∑j=τ∞(2j(s+1/2-1/p)(∑k=02j-1|βj,k|p)1/p)r)1/r≤M*.
In this expression, s is a smoothness parameter and p and r are norm parameters. For a particular choice of s, p, and r, Bp,rs(M) contains some classical sets of functions as the Hölder and Sobolev balls. See [19].


## 4. Estimators

Firstly, we consider the following estimator forμ:(4.1)μ̂=(1n∑i=1n1w(Zi))-1.
It is obtained by the method of moments (see Proposition 6.2 below).Then, for any integerj≥τ and any k∈{0,…,2j-1}, we estimate the unknown wavelet coefficient  (i)
αj,k=∫01f(x)ϕj,k(x)dx by
(4.2)α̂j,k=μ̂n∑i=1nϕj,k(Zi)w(Zi),(ii)
βj,k=∫01f(x)ψj,k(x)dx by
(4.3)β̂j,k=μ̂n∑i=1nψj,k(Zi)w(Zi).Note that they are those considered in the i.i.d. case (see, e.g., [8, 9]). Their statistical properties, with our dependent structure, are investigated in Propositions 6.2, 6.3, and 6.4 below.Assuming thatf∈Bp,rs(M) with p≥2, we define the linear estimator f̂L by(4.4)f̂L(x)=∑k=02j0-1α̂j0,kϕj0,k(x),x∈[0,1],
where α̂j,k is defined by (4.2) and j0 is the integer satisfying (4.5)12n1/(2s+1)<2j0≤n1/(2s+1).For a survey on wavelet linear estimators for various density models, we refer the reader to [20]. For the consideration of strongly mixing sequences, see, for example, [21, 22].We define the hard thresholding estimatorf̂H by(4.6)f̂H(x)=∑k=02τ-1α̂τ,kϕτ,k(x)+∑j=τj1∑k=02j-1β̂j,kI{|β̂j,k|≥κλn}ψj,k(x),x∈[0,1], where α̂τ,k is defined by (4.2) and β̂j,k by (4.3), for any random event 𝒜, 𝕀𝒜 is the indicator function on 𝒜, j1 is the integer satisfying (4.7)12n(ln⁡n)1+1/θ<2j1≤n(ln⁡n)1+1/θ,θ is the one in (2.2), κ is a large enough constant (the one in Proposition 6.4 below) and λn is the threshold(4.8)λn=(ln⁡n)1+1/θn.
The feature of the hard thresholding estimator is to only estimate the “large" unknown wavelet coefficients of f which contain his main characteristics.For the construction of hard thresholding wavelet estimators in the standard density model, see, for example, [15, 23].


## 5. Results

Theorem 5.1 (upper bound forf̂L).
Consider (1.1) under the assumptions of Section 2. Suppose that f∈Bp,rs(M) with s>0, p≥2, and r≥1. Let f̂L be (4.4). Then there exists a constant C>0 such that
(5.1)E(∫01(f̂L(x)-f(x))2dx)≤Cn-2s/(2s+1).The proof of Theorem5.1 uses a suitable decomposition of the MISE and a moment inequality on (4.2) (see Proposition 6.3 below).Note thatn-2s/(2s+1) is the optimal rate of convergence (in the minimax sense) for the standard density model in the independent case (see, e.g., [14, 23]).Theorem 5.2 (upper bound forf̂H).
Consider (1.1) under the assumptions of Section 2. Let f̂H be (4.6). Suppose that f∈Bp,rs(M) with r≥1, {p≥2 and s>0} or {p∈[1,2) and s>1/p}. Then there exists a constant C>0 such that
(5.2)E(∫01(f̂H(x)-f(x))2dx)≤C((ln⁡n)1+1/θn)2s/(2s+1).The proof of Theorem5.2 uses a suitable decomposition of the MISE, some moment inequalities on (4.2) and (4.3) (see Proposition 6.3 below), and a concentration inequality on (4.3) (see Proposition 6.4 below).Theorem5.2 shows that, besides being adaptive, f̂H attains a rate of convergence close to the one of f̂L. The only difference is the logarithmic term (lnn)(1+1/θ)(2s/(2s+1)).Note that, if we restrict our study to the independent case, that is,θ→∞, the rate of convergence attained by f̂H becomes the standard one: (logn/n)2s/(2s+1). See, for example, [14, 15, 23].


## 6. Proofs

In this section, we consider (1.1) under the assumptions of Section 2. Moreover, C denotes any constant that does not depend on j, k and n. Its value may change from one term to another and may depends on ϕ or ψ.


### 6.1. Auxiliary Results

Lemma 6.1.
For any integerj≥τ and any k∈{0,…,2j-1}, let α̂j,k be (4.2) and αj,k=∫01f(x)ϕj,k(x)dx. Then, under the assumptions of Section 2, there exists a constant C>0 such that
(6.1)|α̂j,k-αj,k|≤C(|μn∑i=1nϕj,k(Zi)w(Zi)-αj,k|+|1μ̂-1μ|).
This inequality holds for ψ instead of ϕ (and, a fortiori, β̂j,k defined by (4.3) instead of α̂j,k and βj,k=∫01f(x)ψj,k(x)dx instead of αj,k).Proof of Lemma6.1.
We have(6.2)α̂j,k-αj,k=μ̂μ(μn∑i=1nϕj,k(Zi)w(Zi)-αj,k)+αj,kμ̂(1μ-1μ̂).
Due to (2.3), we have |μ̂|≤C and |μ̂/μ|≤C. Therefore
(6.3)|α̂j,k-αj,k|≤C(|μn∑i=1nϕj,k(Zi)w(Zi)-αj,k|+|αj,k||1μ̂-1μ|).
Using (2.4) and the Cauchy-Schwarz inequality, we obtain
(6.4)|αj,k|≤∫01f(x)|ϕj,k(x)|dx≤C∫01|ϕj,k(x)|dx≤C(∫01(ϕj,k(x))2dx)1/2=C.
Hence
(6.5)|α̂j,k-αj,k|≤C(|μn∑i=1nϕj,k(Zi)w(Zi)-αj,k|+|1μ̂-1μ|).
Lemma 6.1 is proved.Proposition 6.2.
For any integerj≥τ such that 2j≤n and any k∈{0,…,2j-1}, let αj,k=∫01f(x)ϕj,k(x)dx and μ̂ be (4.1). Then,  (1)
one has(6.6)E(μn∑i=1nϕj,k(Zi)w(Zi))=αj,k,E(1μ̂)=1μ,(2)
there exists a constantC>0 such that
(6.7)V(μn∑i=1nϕj,k(Zi)w(Zi))≤C1n,(3)
there exists a constantC>0 such that
(6.8)V(1μ̂)≤C1n.These results hold for ψ instead of ϕ (and, a fortiori, βj,k=∫01f(x)ψj,k(x)dx instead of αj,k).Proof of Proposition6.2.
(1)  We have(6.9)E(μn∑i=1nϕj,k(Zi)w(Zi))=μE(ϕj,k(Z1)w(Z1))=μ∫01ϕj,k(x)w(x)g(x)dx=μ∫01ϕj,k(x)w(x)μ-1w(x)f(x)dx=∫01f(x)ϕj,k(x)dx=αj,k.
Since f is a density, we obtain
(6.10)E(1μ̂)=E(1n∑i=1n1w(Zi))=E(1w(Z1))=∫011w(x)g(x)dx=∫011w(x)μ-1w(x)f(x)dx=1μ∫01f(x)dx=1μ.
(2) We have(6.11)V(μn∑i=1nϕj,k(Zi)w(Zi))=μ2n2∑v=1n∑l=1nC(ϕj,k(Zv)w(Zv),ϕj,k(Zl)w(Zl))=μ2nV(ϕj,k(Z1)w(Z1))+2μ2n2∑v=2n∑l=1v-1C(ϕj,k(Zv)w(Zv),ϕj,k(Zl)w(Zl))≤μ2nV(ϕj,k(Z1)w(Z1))+2μ2n2|∑v=2n∑l=1v-1C(ϕj,k(Zv)w(Zv),ϕj,k(Zl)w(Zl))|.
Using (2.3) and (2.4), we have supx∈[0,1]g(x)≤C. Hence,
(6.12)V(ϕj,k(Z1)w(Z1))≤E((ϕj,k(Z1)w(Z1))2)≤CE((ϕj,k(Z1))2)=C∫01(ϕj,k(x))2g(x)dx≤C∫01(ϕj,k(x))2dx=C.
It follows from the stationarity of (Zi)i∈ℤ and 2j≤n that
(6.13)|∑v=2n∑l=1v-1C(ϕj,k(Zv)w(Zv),ϕj,k(Zl)w(Zl))|=|∑m=1n(n-m)C(ϕj,k(Z0)w(Z0),ϕj,k(Zm)w(Zm))|≤n∑m=1n|C(ϕj,k(Z0)w(Z0),ϕj,k(Zm)w(Zm))|=T1+T2,
where
(6.14)T1=n∑m=12j-1|C(ϕj,k(Z0)w(Z0),ϕj,k(Zm)w(Zm))|,T2=n∑m=2jn|C(ϕj,k(Z0)w(Z0),ϕj,k(Zm)w(Zm))|.
Let us now bound T1 and T2.Upper Bound forT1
Using (2.5), (2.3), and doing the change a variables y=2jx-k, we obtain
(6.15)|C(ϕj,k(Z0)w(Z0),ϕj,k(Zm)w(Zm))|=|∬01(g(Z0,Zm)(x,y)-g(x)g(y))ϕj,k(x)w(x)ϕj,k(y)w(y)dxdy|≤∬01|g(Z0,Zm)(x,y)-g(x)g(y)||ϕj,k(x)w(x)||ϕj,k(y)w(y)|dxdy≤C(∫01|ϕj,k(x)|dx)2=C(2-j/2∫01|ϕ(x)|dx)2=C2-j.
Therefore,
(6.16)T1≤Cn2-j2j=Cn.Upper Bound forT2
By the Davydov inequality for strongly mixing processes (see [24]), for any q∈(0,1), it holds that
(6.17)|C(ϕj,k(Z0)w(Z0),ϕj,k(Zm)w(Zm))|≤10amq(E(|ϕj,k(Z0)w(Z0)|2/(1-q)))1-q≤10amq(sup⁡x∈[0,1]|ϕj,k(x)w(x)|)2q(E((ϕj,k(Z0)w(Z0))2))1-q.
By (2.3), we have
(6.18)sup⁡x∈[0,1]|ϕj,k(x)w(x)|≤Csup⁡x∈[0,1]|ϕj,k(x)|≤C2j/2
and, by (6.12),
(6.19)E((ϕj,k(Z0)w(Z0))2)≤C.
Therefore,
(6.20)|C(ϕj,k(Z0)w(Z0),ϕj,k(Zm)w(Zm))|≤C2qjamq.
Since ∑m=2jnmqamq≤∑m=1∞mqamq=γq∑m=1∞mqexp(-cqmθ)<∞, we have
(6.21)T2≤Cn2qj∑m=2jnamq≤Cn∑m=2jnmqamq≤Cn.
It follows from (6.13), (6.16), and (6.21) that
(6.22)|∑v=2n∑l=1v-1C(ϕj,k(Zv)w(Zv),ϕj,k(Zl)w(Zl))|≤Cn.
Combining (6.11), (6.12), and (6.22), we obtain
(6.23)V(μn∑i=1nϕj,k(Zi)w(Zi))≤C1n.
(3) Proceeding in a similar fashion to 2-, we obtain(6.24)V(1μ̂)=V(1n∑i=1n1w(Zi))=1nV(1w(Z1))+21n2∑v=2n∑l=1v-1C(1w(Zv),1w(Zl))≤1nV(1w(Z1))+21n∑m=1n|C(1w(Z0),1w(Zm))|.
Using (2.3) (which implies supx∈[0,1](1/w(x))≤C) and applying the Davydov inequality, we obtain
(6.25)V(1μ̂)≤C1n(1+∑m=1namq)≤C1n.
The proof of Proposition 6.2 is complete.Proposition 6.3.
For any integerj≥τ such that 2j≤n and any k∈{0,…,2j-1}, let αj,k=∫01f(x)ϕj,k(x)dx and α̂j,k be (4.2). Then,  (1)
there exists a constantC>0 such that
(6.26)E((α̂j,k-αj,k)2)≤C1n;(2)
there exists a constantC>0 such that
(6.27)E((α̂j,k-αj,k)4)≤C2j1n.These inequalities hold for β̂j,k defined by (4.3) instead of α̂j,k, and βj,k=∫01f(x)ψj,k(x)dx instead of αj,k.Proof of Proposition6.3.
(1)  Applying Lemma6.1 and Proposition 6.2, we have
(6.28)E((α̂j,k-αj,k)2)≤C(E((μn∑i=1nϕj,k(Zi)w(Zi)-αj,k)2)+E((1μ̂-1μ)2))=C(V(μn∑i=1nϕj,k(Zi)w(Zi))+V(1μ̂))≤C1n.
(2)  We have(6.29)|α̂j,k-αj,k|≤|α̂j,k|+|αj,k|.
By (2.3), we have |μ̂|≤C and supx∈[0,1](1/w(x))≤C. So,
(6.30)|μ̂n∑i=1nϕj,k(Zi)w(Zi)|≤C1n∑i=1n|ϕj,k(Zi)w(Zi)|≤Csup⁡x∈[0,1]|ϕj,k(x)w(x)|≤Csup⁡x∈[0,1]|ϕj,k(x)|≤C2j/2.
By (6.4), we have |αj,k|≤C. Therefore
(6.31)|α̂j,k-αj,k|≤C(2j/2+1)≤C2j/2.
It follows from (6.31) and (6.28) that
(6.32)E((α̂j,k-αj,k)4)≤C2jE((α̂j,k-αj,k)2)≤C2j1n.
The proof of Proposition 6.3 is complete.Proposition 6.4.
For anyj∈{τ,…,j1} and any k∈{0,…,2j-1}, let βj,k=∫01f(x)ψj,k(x)dx, β̂j,k be (4.3) and λn be (4.8). Then there exist two constants, κ>0 and C>0, such that
(6.33)P(|β̂j,k-βj,k|≥κλn2)≤C1n4.Proof of Proposition6.4.
It follows from Lemma6.1 that
(6.34)P(|β̂j,k-βj,k|≥κλn2)≤P1+P2,
where
(6.35)P1=P(|μn∑i=1nψj,k(Zi)w(Zi)-βj,k|≥κCλn),P2=P(|1μ̂-1μ|≥κCλn).
In order to bound P1 and P2, let us present a Bernstein inequality for exponentially strongly mixing process. We refer to [25, 26].Lemma 6.5 (see [25, 26]).
Letγ>0, c>0, θ>1 and (Zi)i∈ℤ be a stationary process such that, for any m∈ℤ, the associated mth strongly mixing coefficient (2.2) satisfies am≤γexp(-c|m|θ). Let n∈ℕ*, h:ℝ→ℝ be a measurable function and, for any i∈ℤ, Ui=h(Zi). One assumes that 𝔼(U1)=0 and there exists a constant M>0 satisfying |U1|≤M<∞. Then, for any m∈{1,…,n} and any λ>4mM/n, one has
(6.36)P(|1n∑i=1nUi|≥λ)≤4exp⁡(-λ2nm(64E(U12)+8λM/3))+4γnmexp⁡(-cmθ).Upper Bound forP1
For anyi∈{1,…,n}, set
(6.37)Ui=μψj,k(Zi)w(Zi)-βj,k.
Then U1,…,Un are identically distributed, depend on the stationary strongly mixing process (Zi)i∈ℤ which satisfies (2.2), Proposition 6.2 gives
(6.38)E(U1)=0,E(U12)≤E((μψj,k(Z1)w(Z1))2)≤C
and, by (2.3) and (6.4),
(6.39)|U1|≤μsup⁡x∈[0,1]|ψj,k(x)w(x)|+|βj,k|≤C(sup⁡x∈[0,1]|ψj,k(x)|+1)≤C(2j/2+1)≤C2j/2.
It follows from Lemma 6.5 applied with U1,…,Un, λ=κCλn, λn=((lnn)1+1/θ/n)1/2, m=(ulnn)1/θ with u>0 (chosen later), M=C2j/2 and 2j≤2j1≤n/(lnn)1+1/θ, that
(6.40)P1=P(|1n∑i=1nUi|≥κCλn)≤4exp⁡(-Cκ2λn2nm(1+κλnM))+4γnmexp⁡(-cmθ)≤4exp⁡(-Cκ2(ln⁡n)1+1/θ(uln⁡n)1/θ(1+κ2j/2((ln⁡n)1+1/θ/n)1/2))+4γn(uln⁡n)1/θexp⁡(-culn⁡n)≤C(n-Cκ2/(u1/θ(1+κ))+n1-cu).
Therefore, for large enoughκ and u, we have(6.41)P1≤C1n4.Upper Bound forP2
For anyi∈{1,…,n}, set
(6.42)Ui=1w(Zi)-1μ.
Then U1,…,Un are identically distributed, depend on the stationary strongly mixing process (Zi)i∈ℤ which satisfies (2.2), Proposition 6.2 gives
(6.43)E(U1)=0,E(U12)≤E(1(w(Z1))2)≤C.
By (2.3), we have
(6.44)|U1|≤sup⁡x∈[0,1]1w(x)+|1μ|≤C.
It follows from Lemma 6.5 applied with U1,…,Un, λ=κCλn, λn=((lnn)1+1/θ/n)1/2, m=(ulnn)1/θ with u>0 (chosen later) and M=C that
(6.45)P2=P(|1n∑i=1nUi|≥κCλn)≤4exp⁡(-Cκ2λn2nm(1+κλnM))+4γnmexp⁡(-cmθ)≤4exp⁡(-Cκ2(ln⁡n)1+1/θ(uln⁡n)1/θ(1+κ((ln⁡n)1+1/θ/n)1/2))+4γn(uln⁡n)1/θexp⁡(-culn⁡n)≤C(n-Cκ2/u1/θ+n1-cu).
Therefore, for large enoughκ and u, we have(6.46)P2≤C1n4.
Putting (6.34), (6.41), and (6.46) together, this ends the proof of Proposition 6.4.


### 6.2. Proofs of the Main Results

Proof of Theorem5.1.
We expand the functionf on ℬ as
(6.47)f(x)=∑k=02j0-1αj0,kϕj0,k(x)+∑j=j0∞∑k=02j-1βj,kψj,k(x),x∈[0,1],
where αj0,k=∫01f(x)ϕj0,k(x)dx and βj,k=∫01f(x)ψj,k(x)dx.
We have, for anyx∈[0,1], (6.48)f̂L(x)-f(x)=∑k=02j0-1(α̂j0,k-αj0,k)ϕj0,k(x)-∑j=j0∞∑k=02j-1βj,kψj,k(x).
Since ℬ is an orthonormal basis of 𝕃2([0,1]), we have,
(6.49)E(∫01(f̂L(x)-f(x))2dx)=∑k=02j0-1E((α̂j0,k-αj0,k)2)+∑j=j0∞∑k=02j-1βj,k2.
Using Proposition 6.3, we obtain
(6.50)∑k=02j0-1E((α̂j0,k-αj0,k)2)≤C2j01n≤Cn-2s/(2s+1).
Since p≥2, we have Bp,rs(M)⊆B2,∞s(M). Hence
(6.51)∑j=j0∞∑k=02j-1βj,k2≤C2-2j0s≤Cn-2s/(2s+1).
Therefore,
(6.52)E(∫01(f̂L(x)-f(x))2dx)≤Cn-2s/(2s+1).
The proof of Theorem 5.1 is complete.Proof of Theorem5.2.
We expand the functionf on ℬ as
(6.53)f(x)=∑k=02τ-1ατ,kϕτ,k(x)+∑j=τ∞∑k=02j-1βj,kψj,k(x),x∈[0,1],
where ατ,k=∫01f(x)ϕτ,k(x)dx and βj,k=∫01f(x)ψj,k(x)dx.
We have, for anyx∈[0,1],(6.54)f̂H(x)-f(x)=∑k=02τ-1(α̂τ,k-ατ,k)ϕτ,k(x)+∑j=τj1∑k=02j-1(β̂j,kI{|β̂j,k|≥κλn}-βj,k)ψj,k(x)-∑j=j1+1∞∑k=02j-1βj,kψj,k(x).
Since ℬ is an orthonormal basis of 𝕃2([0,1]), we have
(6.55)E(∫01(f̂H(x)-f(x))2dx)=R+S+T,
where
(6.56)R=∑k=02τ-1E((α̂τ,k-ατ,k)2),S=∑j=τj1∑k=02j-1E((β̂j,kI{|β̂j,k|≥κλn}-βj,k)2),T=∑j=j1+1∞∑k=02j-1βj,k2.
Let us bound R, T, and S, in turn. Upper Bound forR
Using Proposition6.3 and 2s/(2s+1)<1, we obtain
(6.57)R≤C2τ1n≤C((ln⁡n)1+1/θn)2s/(2s+1).Upper Bound forT
Forr≥1 and p≥2, we have Bp,rs(M)⊆B2,∞s(M). Since 2s/(2s+1)<2s, we have
(6.58)T≤C∑j=j1+1∞2-2js≤C2-2j1s≤C((ln⁡n)1+1/θn)2s≤C((ln⁡n)1+1/θn)2s/(2s+1).
For r≥1 and p∈[1,2), we have Bp,rs(M)⊆B2,∞s+1/2-1/p(M). Since s>1/p, we have s+1/2-1/p>s/(2s+1). So
(6.59)T≤C∑j=j1+1∞2-2j(s+1/2-1/p)≤C2-2j1(s+1/2-1/p)≤C((ln⁡n)1+1/θn)2(s+1/2-1/p)≤C((ln⁡n)1+1/θn)2s/(2s+1).
Hence, for r≥1, {p≥2 and s>0} or {p∈[1,2) and s>1/p}, we have
(6.60)T≤C((ln⁡n)1+1/θn)2s/(2s+1).Upper Bound forS
Note that we can write the termS as
(6.61)S=S1+S2+S3+S4,
where
(6.62)S1=∑j=τj1∑k=02j-1E((β̂j,k-βj,k)2I{|β̂j,k|≥κλn}I{|βj,k|<κλn/2}),S2=∑j=τj1∑k=02j-1E((β̂j,k-βj,k)2I{|β̂j,k|≥κλn}I{|βj,k|≥κλn/2}),S3=∑j=τj1∑k=02j-1E(βj,k2I{|β̂j,k|<κλn}I{|βj,k|≥2κλn}),S4=∑j=τj1∑k=02j-1E(βj,k2I{|β̂j,k|<κλn}I{|βj,k|<2κλn}).
Let us investigate the bounds of S1, S2, S3, and S4 in turn.Upper Bounds forS1 and S3
We have(6.63){|β̂j,k|<κλn,|βj,k|≥2κλn}⊆{|β̂j,k-βj,k|>κλn2},{|β̂j,k|≥κλn,|βj,k|<κλn2}⊆{|β̂j,k-βj,k|>κλn2},{|β̂j,k|<κλn,|βj,k|≥2κλn}⊆{|βj,k|≤2|β̂j,k-βj,k|}.
So,
(6.64)max⁡(S1,S3)≤C∑j=τj1∑k=02j-1E((β̂j,k-βj,k)2I{|β̂j,k-βj,k|>κλn/2}).
It follows from the Cauchy-Schwarz inequality, Propositions 6.3 and 6.4, and 2j≤2j1≤n that
(6.65)E((β̂j,k-βj,k)2I{|β̂j,k-βj,k|>κλn/2})≤(E((β̂j,k-βj,k)4))1/2(P(|β̂j,k-βj,k|>κλn2))1/2≤C(2j1n)1/2(1n4)1/2≤C1n2.
Since 2s/(2s+1)<1, we have
(6.66)max⁡(S1,S3)≤C1n2∑j=τj12j≤C1n22j1≤C1n≤C((ln⁡n)1+1/θn)2s/(2s+1).Upper Bound forS2
Using again Proposition6.3, we obtain
(6.67)E((β̂j,k-βj,k)2)≤C1n≤C(ln⁡n)1+1/θn.
Hence,
(6.68)S2≤C(ln⁡n)1+1/θn∑j=τj1∑k=02j-1I{|βj,k|>κλn/2}.
Let j2 be the integer defined by
(6.69)12(n(ln⁡n)1+1/θ)1/(2s+1)<2j2≤(n(ln⁡n)1+1/θ)1/(2s+1).
We have
(6.70)S2≤S2,1+S2,2,
where
(6.71)S2,1=C(ln⁡n)1+1/θn∑j=τj2∑k=02j-1I{|βj,k|>κλn/2},S2,2=C(ln⁡n)1+1/θn∑j=j2+1j1∑k=02j-1I{|βj,k|>κλn/2}.
We have
(6.72)S2,1≤C(ln⁡n)1+1/θn∑j=τj22j≤C(ln⁡n)1+1/θn2j2≤C((ln⁡n)1+1/θn)2s/(2s+1).
For r≥1 and p≥2, since Bp,rs(M)⊆B2,∞s(M),
(6.73)S2,2≤C(ln⁡n)1+1/θnλn2∑j=j2+1j1∑k=02j-1βj,k2≤C∑j=j2+1∞∑k=02j-1βj,k2≤C2-2j2s≤C((ln⁡n)1+1/θn)2s/(2s+1).
For r≥1, p∈[1,2) and s>1/p, using 𝕀{|βj,k|>κλn/2}≤C|βj,k|p/λnp, Bp,rs(M)⊆B2,∞s+1/2-1/p(M) and (2s+1)(2-p)/2+(s+1/2-1/p)p=2s, we have
(6.74)S2,2≤C(ln⁡n)1+1/θnλnp∑j=j2+1j1∑k=02j-1|βj,k|p≤C((ln⁡n)1+1/θn)(2-p)/2∑j=j2+1∞2-j(s+1/2-1/p)p≤C((ln⁡n)1+1/θn)(2-p)/22-j2(s+1/2-1/p)p≤C((ln⁡n)1+1/θn)2s/(2s+1).
So, for r≥1, {p≥2 and s>0} or {p∈[1,2) and s>1/p}, we have
(6.75)S2≤C((ln⁡n)1+1/θn)2s/(2s+1).Upper Bound forS4
We have(6.76)S4≤∑j=τj1∑k=02j-1βj,k2I{|βj,k|<2κλn}.
Let j2 be the integer (6.69). Then
(6.77)S4≤S4,1+S4,2,
where
(6.78)S4,1=∑j=τj2∑k=02j-1βj,k2I{|βj,k|<2κλn},S4,2=∑j=j2+1j1∑k=02j-1βj,k2I{|βj,k|<2κλn}.
We have
(6.79)S4,1≤C∑j=τj22jλn2=C(ln⁡n)1+1/θn∑j=τj22j≤C(ln⁡n)1+1/θn2j2≤C((ln⁡n)1+1/θn)2s/(2s+1).
For r≥1 and p≥2, since Bp,rs(M)⊆B2,∞s(M), we have
(6.80)S4,2≤∑j=j2+1∞∑k=02j-1βj,k2≤C2-2j2s≤C((ln⁡n)1+1/θn)2s/(2s+1).
For r≥1, p∈[1,2) and s>1/p, using βj,k2𝕀{|βj,k|<2κλn}≤Cλn2-p|βj,k|p, Bp,rs(M)⊆B2,∞s+1/2-1/p(M) and (2s+1)(2-p)/2+(s+1/2-1/p)p=2s, we have
(6.81)S4,2≤Cλn2-p∑j=j2+1j1∑k=02j-1|βj,k|p=C((ln⁡n)1+1/θn)(2-p)/2∑j=j2+1j1∑k=02j-1|βj,k|p≤C((ln⁡n)1+1/θn)(2-p)/2∑j=j2+1∞2-j(s+1/2-1/p)p≤C((ln⁡n)1+1/θn)(2-p)/22-j2(s+1/2-1/p)p≤C((ln⁡n)1+1/θn)2s/(2s+1).
So, for r≥1, {p≥2 and s>0} or {p∈[1,2) and s>1/p}, we have
(6.82)S4≤C((ln⁡n)1+1/θn)2s/(2s+1).
It follows from (6.61), (6.66), (6.75), and (6.82) that
(6.83)S≤C((ln⁡n)1+1/θn)2s/(2s+1).
Combining (6.55), (6.57), (6.60), and (6.83), we have, for r≥1, {p≥2 and s>0} or {p∈[1,2) and s>1/p}, (6.84)E(∫01(f̂H(x)-f(x))2dx)≤C((ln⁡n)1+1/θn)2s/(2s+1).
The proof of Theorem 5.2 is complete.


## 6.1. Auxiliary Results

Lemma 6.1.
For any integerj≥τ and any k∈{0,…,2j-1}, let α̂j,k be (4.2) and αj,k=∫01f(x)ϕj,k(x)dx. Then, under the assumptions of Section 2, there exists a constant C>0 such that
(6.1)|α̂j,k-αj,k|≤C(|μn∑i=1nϕj,k(Zi)w(Zi)-αj,k|+|1μ̂-1μ|).
This inequality holds for ψ instead of ϕ (and, a fortiori, β̂j,k defined by (4.3) instead of α̂j,k and βj,k=∫01f(x)ψj,k(x)dx instead of αj,k).Proof of Lemma6.1.
We have(6.2)α̂j,k-αj,k=μ̂μ(μn∑i=1nϕj,k(Zi)w(Zi)-αj,k)+αj,kμ̂(1μ-1μ̂).
Due to (2.3), we have |μ̂|≤C and |μ̂/μ|≤C. Therefore
(6.3)|α̂j,k-αj,k|≤C(|μn∑i=1nϕj,k(Zi)w(Zi)-αj,k|+|αj,k||1μ̂-1μ|).
Using (2.4) and the Cauchy-Schwarz inequality, we obtain
(6.4)|αj,k|≤∫01f(x)|ϕj,k(x)|dx≤C∫01|ϕj,k(x)|dx≤C(∫01(ϕj,k(x))2dx)1/2=C.
Hence
(6.5)|α̂j,k-αj,k|≤C(|μn∑i=1nϕj,k(Zi)w(Zi)-αj,k|+|1μ̂-1μ|).
Lemma 6.1 is proved.Proposition 6.2.
For any integerj≥τ such that 2j≤n and any k∈{0,…,2j-1}, let αj,k=∫01f(x)ϕj,k(x)dx and μ̂ be (4.1). Then,  (1)
one has(6.6)E(μn∑i=1nϕj,k(Zi)w(Zi))=αj,k,E(1μ̂)=1μ,(2)
there exists a constantC>0 such that
(6.7)V(μn∑i=1nϕj,k(Zi)w(Zi))≤C1n,(3)
there exists a constantC>0 such that
(6.8)V(1μ̂)≤C1n.These results hold for ψ instead of ϕ (and, a fortiori, βj,k=∫01f(x)ψj,k(x)dx instead of αj,k).Proof of Proposition6.2.
(1)  We have(6.9)E(μn∑i=1nϕj,k(Zi)w(Zi))=μE(ϕj,k(Z1)w(Z1))=μ∫01ϕj,k(x)w(x)g(x)dx=μ∫01ϕj,k(x)w(x)μ-1w(x)f(x)dx=∫01f(x)ϕj,k(x)dx=αj,k.
Since f is a density, we obtain
(6.10)E(1μ̂)=E(1n∑i=1n1w(Zi))=E(1w(Z1))=∫011w(x)g(x)dx=∫011w(x)μ-1w(x)f(x)dx=1μ∫01f(x)dx=1μ.
(2) We have(6.11)V(μn∑i=1nϕj,k(Zi)w(Zi))=μ2n2∑v=1n∑l=1nC(ϕj,k(Zv)w(Zv),ϕj,k(Zl)w(Zl))=μ2nV(ϕj,k(Z1)w(Z1))+2μ2n2∑v=2n∑l=1v-1C(ϕj,k(Zv)w(Zv),ϕj,k(Zl)w(Zl))≤μ2nV(ϕj,k(Z1)w(Z1))+2μ2n2|∑v=2n∑l=1v-1C(ϕj,k(Zv)w(Zv),ϕj,k(Zl)w(Zl))|.
Using (2.3) and (2.4), we have supx∈[0,1]g(x)≤C. Hence,
(6.12)V(ϕj,k(Z1)w(Z1))≤E((ϕj,k(Z1)w(Z1))2)≤CE((ϕj,k(Z1))2)=C∫01(ϕj,k(x))2g(x)dx≤C∫01(ϕj,k(x))2dx=C.
It follows from the stationarity of (Zi)i∈ℤ and 2j≤n that
(6.13)|∑v=2n∑l=1v-1C(ϕj,k(Zv)w(Zv),ϕj,k(Zl)w(Zl))|=|∑m=1n(n-m)C(ϕj,k(Z0)w(Z0),ϕj,k(Zm)w(Zm))|≤n∑m=1n|C(ϕj,k(Z0)w(Z0),ϕj,k(Zm)w(Zm))|=T1+T2,
where
(6.14)T1=n∑m=12j-1|C(ϕj,k(Z0)w(Z0),ϕj,k(Zm)w(Zm))|,T2=n∑m=2jn|C(ϕj,k(Z0)w(Z0),ϕj,k(Zm)w(Zm))|.
Let us now bound T1 and T2.Upper Bound forT1
Using (2.5), (2.3), and doing the change a variables y=2jx-k, we obtain
(6.15)|C(ϕj,k(Z0)w(Z0),ϕj,k(Zm)w(Zm))|=|∬01(g(Z0,Zm)(x,y)-g(x)g(y))ϕj,k(x)w(x)ϕj,k(y)w(y)dxdy|≤∬01|g(Z0,Zm)(x,y)-g(x)g(y)||ϕj,k(x)w(x)||ϕj,k(y)w(y)|dxdy≤C(∫01|ϕj,k(x)|dx)2=C(2-j/2∫01|ϕ(x)|dx)2=C2-j.
Therefore,
(6.16)T1≤Cn2-j2j=Cn.Upper Bound forT2
By the Davydov inequality for strongly mixing processes (see [24]), for any q∈(0,1), it holds that
(6.17)|C(ϕj,k(Z0)w(Z0),ϕj,k(Zm)w(Zm))|≤10amq(E(|ϕj,k(Z0)w(Z0)|2/(1-q)))1-q≤10amq(sup⁡x∈[0,1]|ϕj,k(x)w(x)|)2q(E((ϕj,k(Z0)w(Z0))2))1-q.
By (2.3), we have
(6.18)sup⁡x∈[0,1]|ϕj,k(x)w(x)|≤Csup⁡x∈[0,1]|ϕj,k(x)|≤C2j/2
and, by (6.12),
(6.19)E((ϕj,k(Z0)w(Z0))2)≤C.
Therefore,
(6.20)|C(ϕj,k(Z0)w(Z0),ϕj,k(Zm)w(Zm))|≤C2qjamq.
Since ∑m=2jnmqamq≤∑m=1∞mqamq=γq∑m=1∞mqexp(-cqmθ)<∞, we have
(6.21)T2≤Cn2qj∑m=2jnamq≤Cn∑m=2jnmqamq≤Cn.
It follows from (6.13), (6.16), and (6.21) that
(6.22)|∑v=2n∑l=1v-1C(ϕj,k(Zv)w(Zv),ϕj,k(Zl)w(Zl))|≤Cn.
Combining (6.11), (6.12), and (6.22), we obtain
(6.23)V(μn∑i=1nϕj,k(Zi)w(Zi))≤C1n.
(3) Proceeding in a similar fashion to 2-, we obtain(6.24)V(1μ̂)=V(1n∑i=1n1w(Zi))=1nV(1w(Z1))+21n2∑v=2n∑l=1v-1C(1w(Zv),1w(Zl))≤1nV(1w(Z1))+21n∑m=1n|C(1w(Z0),1w(Zm))|.
Using (2.3) (which implies supx∈[0,1](1/w(x))≤C) and applying the Davydov inequality, we obtain
(6.25)V(1μ̂)≤C1n(1+∑m=1namq)≤C1n.
The proof of Proposition 6.2 is complete.Proposition 6.3.
For any integerj≥τ such that 2j≤n and any k∈{0,…,2j-1}, let αj,k=∫01f(x)ϕj,k(x)dx and α̂j,k be (4.2). Then,  (1)
there exists a constantC>0 such that
(6.26)E((α̂j,k-αj,k)2)≤C1n;(2)
there exists a constantC>0 such that
(6.27)E((α̂j,k-αj,k)4)≤C2j1n.These inequalities hold for β̂j,k defined by (4.3) instead of α̂j,k, and βj,k=∫01f(x)ψj,k(x)dx instead of αj,k.Proof of Proposition6.3.
(1)  Applying Lemma6.1 and Proposition 6.2, we have
(6.28)E((α̂j,k-αj,k)2)≤C(E((μn∑i=1nϕj,k(Zi)w(Zi)-αj,k)2)+E((1μ̂-1μ)2))=C(V(μn∑i=1nϕj,k(Zi)w(Zi))+V(1μ̂))≤C1n.
(2)  We have(6.29)|α̂j,k-αj,k|≤|α̂j,k|+|αj,k|.
By (2.3), we have |μ̂|≤C and supx∈[0,1](1/w(x))≤C. So,
(6.30)|μ̂n∑i=1nϕj,k(Zi)w(Zi)|≤C1n∑i=1n|ϕj,k(Zi)w(Zi)|≤Csup⁡x∈[0,1]|ϕj,k(x)w(x)|≤Csup⁡x∈[0,1]|ϕj,k(x)|≤C2j/2.
By (6.4), we have |αj,k|≤C. Therefore
(6.31)|α̂j,k-αj,k|≤C(2j/2+1)≤C2j/2.
It follows from (6.31) and (6.28) that
(6.32)E((α̂j,k-αj,k)4)≤C2jE((α̂j,k-αj,k)2)≤C2j1n.
The proof of Proposition 6.3 is complete.Proposition 6.4.
For anyj∈{τ,…,j1} and any k∈{0,…,2j-1}, let βj,k=∫01f(x)ψj,k(x)dx, β̂j,k be (4.3) and λn be (4.8). Then there exist two constants, κ>0 and C>0, such that
(6.33)P(|β̂j,k-βj,k|≥κλn2)≤C1n4.Proof of Proposition6.4.
It follows from Lemma6.1 that
(6.34)P(|β̂j,k-βj,k|≥κλn2)≤P1+P2,
where
(6.35)P1=P(|μn∑i=1nψj,k(Zi)w(Zi)-βj,k|≥κCλn),P2=P(|1μ̂-1μ|≥κCλn).
In order to bound P1 and P2, let us present a Bernstein inequality for exponentially strongly mixing process. We refer to [25, 26].Lemma 6.5 (see [25, 26]).
Letγ>0, c>0, θ>1 and (Zi)i∈ℤ be a stationary process such that, for any m∈ℤ, the associated mth strongly mixing coefficient (2.2) satisfies am≤γexp(-c|m|θ). Let n∈ℕ*, h:ℝ→ℝ be a measurable function and, for any i∈ℤ, Ui=h(Zi). One assumes that 𝔼(U1)=0 and there exists a constant M>0 satisfying |U1|≤M<∞. Then, for any m∈{1,…,n} and any λ>4mM/n, one has
(6.36)P(|1n∑i=1nUi|≥λ)≤4exp⁡(-λ2nm(64E(U12)+8λM/3))+4γnmexp⁡(-cmθ).Upper Bound forP1
For anyi∈{1,…,n}, set
(6.37)Ui=μψj,k(Zi)w(Zi)-βj,k.
Then U1,…,Un are identically distributed, depend on the stationary strongly mixing process (Zi)i∈ℤ which satisfies (2.2), Proposition 6.2 gives
(6.38)E(U1)=0,E(U12)≤E((μψj,k(Z1)w(Z1))2)≤C
and, by (2.3) and (6.4),
(6.39)|U1|≤μsup⁡x∈[0,1]|ψj,k(x)w(x)|+|βj,k|≤C(sup⁡x∈[0,1]|ψj,k(x)|+1)≤C(2j/2+1)≤C2j/2.
It follows from Lemma 6.5 applied with U1,…,Un, λ=κCλn, λn=((lnn)1+1/θ/n)1/2, m=(ulnn)1/θ with u>0 (chosen later), M=C2j/2 and 2j≤2j1≤n/(lnn)1+1/θ, that
(6.40)P1=P(|1n∑i=1nUi|≥κCλn)≤4exp⁡(-Cκ2λn2nm(1+κλnM))+4γnmexp⁡(-cmθ)≤4exp⁡(-Cκ2(ln⁡n)1+1/θ(uln⁡n)1/θ(1+κ2j/2((ln⁡n)1+1/θ/n)1/2))+4γn(uln⁡n)1/θexp⁡(-culn⁡n)≤C(n-Cκ2/(u1/θ(1+κ))+n1-cu).
Therefore, for large enoughκ and u, we have(6.41)P1≤C1n4.Upper Bound forP2
For anyi∈{1,…,n}, set
(6.42)Ui=1w(Zi)-1μ.
Then U1,…,Un are identically distributed, depend on the stationary strongly mixing process (Zi)i∈ℤ which satisfies (2.2), Proposition 6.2 gives
(6.43)E(U1)=0,E(U12)≤E(1(w(Z1))2)≤C.
By (2.3), we have
(6.44)|U1|≤sup⁡x∈[0,1]1w(x)+|1μ|≤C.
It follows from Lemma 6.5 applied with U1,…,Un, λ=κCλn, λn=((lnn)1+1/θ/n)1/2, m=(ulnn)1/θ with u>0 (chosen later) and M=C that
(6.45)P2=P(|1n∑i=1nUi|≥κCλn)≤4exp⁡(-Cκ2λn2nm(1+κλnM))+4γnmexp⁡(-cmθ)≤4exp⁡(-Cκ2(ln⁡n)1+1/θ(uln⁡n)1/θ(1+κ((ln⁡n)1+1/θ/n)1/2))+4γn(uln⁡n)1/θexp⁡(-culn⁡n)≤C(n-Cκ2/u1/θ+n1-cu).
Therefore, for large enoughκ and u, we have(6.46)P2≤C1n4.
Putting (6.34), (6.41), and (6.46) together, this ends the proof of Proposition 6.4.


## 6.2. Proofs of the Main Results

Proof of Theorem5.1.
We expand the functionf on ℬ as
(6.47)f(x)=∑k=02j0-1αj0,kϕj0,k(x)+∑j=j0∞∑k=02j-1βj,kψj,k(x),x∈[0,1],
where αj0,k=∫01f(x)ϕj0,k(x)dx and βj,k=∫01f(x)ψj,k(x)dx.
We have, for anyx∈[0,1], (6.48)f̂L(x)-f(x)=∑k=02j0-1(α̂j0,k-αj0,k)ϕj0,k(x)-∑j=j0∞∑k=02j-1βj,kψj,k(x).
Since ℬ is an orthonormal basis of 𝕃2([0,1]), we have,
(6.49)E(∫01(f̂L(x)-f(x))2dx)=∑k=02j0-1E((α̂j0,k-αj0,k)2)+∑j=j0∞∑k=02j-1βj,k2.
Using Proposition 6.3, we obtain
(6.50)∑k=02j0-1E((α̂j0,k-αj0,k)2)≤C2j01n≤Cn-2s/(2s+1).
Since p≥2, we have Bp,rs(M)⊆B2,∞s(M). Hence
(6.51)∑j=j0∞∑k=02j-1βj,k2≤C2-2j0s≤Cn-2s/(2s+1).
Therefore,
(6.52)E(∫01(f̂L(x)-f(x))2dx)≤Cn-2s/(2s+1).
The proof of Theorem 5.1 is complete.Proof of Theorem5.2.
We expand the functionf on ℬ as
(6.53)f(x)=∑k=02τ-1ατ,kϕτ,k(x)+∑j=τ∞∑k=02j-1βj,kψj,k(x),x∈[0,1],
where ατ,k=∫01f(x)ϕτ,k(x)dx and βj,k=∫01f(x)ψj,k(x)dx.
We have, for anyx∈[0,1],(6.54)f̂H(x)-f(x)=∑k=02τ-1(α̂τ,k-ατ,k)ϕτ,k(x)+∑j=τj1∑k=02j-1(β̂j,kI{|β̂j,k|≥κλn}-βj,k)ψj,k(x)-∑j=j1+1∞∑k=02j-1βj,kψj,k(x).
Since ℬ is an orthonormal basis of 𝕃2([0,1]), we have
(6.55)E(∫01(f̂H(x)-f(x))2dx)=R+S+T,
where
(6.56)R=∑k=02τ-1E((α̂τ,k-ατ,k)2),S=∑j=τj1∑k=02j-1E((β̂j,kI{|β̂j,k|≥κλn}-βj,k)2),T=∑j=j1+1∞∑k=02j-1βj,k2.
Let us bound R, T, and S, in turn. Upper Bound forR
Using Proposition6.3 and 2s/(2s+1)<1, we obtain
(6.57)R≤C2τ1n≤C((ln⁡n)1+1/θn)2s/(2s+1).Upper Bound forT
Forr≥1 and p≥2, we have Bp,rs(M)⊆B2,∞s(M). Since 2s/(2s+1)<2s, we have
(6.58)T≤C∑j=j1+1∞2-2js≤C2-2j1s≤C((ln⁡n)1+1/θn)2s≤C((ln⁡n)1+1/θn)2s/(2s+1).
For r≥1 and p∈[1,2), we have Bp,rs(M)⊆B2,∞s+1/2-1/p(M). Since s>1/p, we have s+1/2-1/p>s/(2s+1). So
(6.59)T≤C∑j=j1+1∞2-2j(s+1/2-1/p)≤C2-2j1(s+1/2-1/p)≤C((ln⁡n)1+1/θn)2(s+1/2-1/p)≤C((ln⁡n)1+1/θn)2s/(2s+1).
Hence, for r≥1, {p≥2 and s>0} or {p∈[1,2) and s>1/p}, we have
(6.60)T≤C((ln⁡n)1+1/θn)2s/(2s+1).Upper Bound forS
Note that we can write the termS as
(6.61)S=S1+S2+S3+S4,
where
(6.62)S1=∑j=τj1∑k=02j-1E((β̂j,k-βj,k)2I{|β̂j,k|≥κλn}I{|βj,k|<κλn/2}),S2=∑j=τj1∑k=02j-1E((β̂j,k-βj,k)2I{|β̂j,k|≥κλn}I{|βj,k|≥κλn/2}),S3=∑j=τj1∑k=02j-1E(βj,k2I{|β̂j,k|<κλn}I{|βj,k|≥2κλn}),S4=∑j=τj1∑k=02j-1E(βj,k2I{|β̂j,k|<κλn}I{|βj,k|<2κλn}).
Let us investigate the bounds of S1, S2, S3, and S4 in turn.Upper Bounds forS1 and S3
We have(6.63){|β̂j,k|<κλn,|βj,k|≥2κλn}⊆{|β̂j,k-βj,k|>κλn2},{|β̂j,k|≥κλn,|βj,k|<κλn2}⊆{|β̂j,k-βj,k|>κλn2},{|β̂j,k|<κλn,|βj,k|≥2κλn}⊆{|βj,k|≤2|β̂j,k-βj,k|}.
So,
(6.64)max⁡(S1,S3)≤C∑j=τj1∑k=02j-1E((β̂j,k-βj,k)2I{|β̂j,k-βj,k|>κλn/2}).
It follows from the Cauchy-Schwarz inequality, Propositions 6.3 and 6.4, and 2j≤2j1≤n that
(6.65)E((β̂j,k-βj,k)2I{|β̂j,k-βj,k|>κλn/2})≤(E((β̂j,k-βj,k)4))1/2(P(|β̂j,k-βj,k|>κλn2))1/2≤C(2j1n)1/2(1n4)1/2≤C1n2.
Since 2s/(2s+1)<1, we have
(6.66)max⁡(S1,S3)≤C1n2∑j=τj12j≤C1n22j1≤C1n≤C((ln⁡n)1+1/θn)2s/(2s+1).Upper Bound forS2
Using again Proposition6.3, we obtain
(6.67)E((β̂j,k-βj,k)2)≤C1n≤C(ln⁡n)1+1/θn.
Hence,
(6.68)S2≤C(ln⁡n)1+1/θn∑j=τj1∑k=02j-1I{|βj,k|>κλn/2}.
Let j2 be the integer defined by
(6.69)12(n(ln⁡n)1+1/θ)1/(2s+1)<2j2≤(n(ln⁡n)1+1/θ)1/(2s+1).
We have
(6.70)S2≤S2,1+S2,2,
where
(6.71)S2,1=C(ln⁡n)1+1/θn∑j=τj2∑k=02j-1I{|βj,k|>κλn/2},S2,2=C(ln⁡n)1+1/θn∑j=j2+1j1∑k=02j-1I{|βj,k|>κλn/2}.
We have
(6.72)S2,1≤C(ln⁡n)1+1/θn∑j=τj22j≤C(ln⁡n)1+1/θn2j2≤C((ln⁡n)1+1/θn)2s/(2s+1).
For r≥1 and p≥2, since Bp,rs(M)⊆B2,∞s(M),
(6.73)S2,2≤C(ln⁡n)1+1/θnλn2∑j=j2+1j1∑k=02j-1βj,k2≤C∑j=j2+1∞∑k=02j-1βj,k2≤C2-2j2s≤C((ln⁡n)1+1/θn)2s/(2s+1).
For r≥1, p∈[1,2) and s>1/p, using 𝕀{|βj,k|>κλn/2}≤C|βj,k|p/λnp, Bp,rs(M)⊆B2,∞s+1/2-1/p(M) and (2s+1)(2-p)/2+(s+1/2-1/p)p=2s, we have
(6.74)S2,2≤C(ln⁡n)1+1/θnλnp∑j=j2+1j1∑k=02j-1|βj,k|p≤C((ln⁡n)1+1/θn)(2-p)/2∑j=j2+1∞2-j(s+1/2-1/p)p≤C((ln⁡n)1+1/θn)(2-p)/22-j2(s+1/2-1/p)p≤C((ln⁡n)1+1/θn)2s/(2s+1).
So, for r≥1, {p≥2 and s>0} or {p∈[1,2) and s>1/p}, we have
(6.75)S2≤C((ln⁡n)1+1/θn)2s/(2s+1).Upper Bound forS4
We have(6.76)S4≤∑j=τj1∑k=02j-1βj,k2I{|βj,k|<2κλn}.
Let j2 be the integer (6.69). Then
(6.77)S4≤S4,1+S4,2,
where
(6.78)S4,1=∑j=τj2∑k=02j-1βj,k2I{|βj,k|<2κλn},S4,2=∑j=j2+1j1∑k=02j-1βj,k2I{|βj,k|<2κλn}.
We have
(6.79)S4,1≤C∑j=τj22jλn2=C(ln⁡n)1+1/θn∑j=τj22j≤C(ln⁡n)1+1/θn2j2≤C((ln⁡n)1+1/θn)2s/(2s+1).
For r≥1 and p≥2, since Bp,rs(M)⊆B2,∞s(M), we have
(6.80)S4,2≤∑j=j2+1∞∑k=02j-1βj,k2≤C2-2j2s≤C((ln⁡n)1+1/θn)2s/(2s+1).
For r≥1, p∈[1,2) and s>1/p, using βj,k2𝕀{|βj,k|<2κλn}≤Cλn2-p|βj,k|p, Bp,rs(M)⊆B2,∞s+1/2-1/p(M) and (2s+1)(2-p)/2+(s+1/2-1/p)p=2s, we have
(6.81)S4,2≤Cλn2-p∑j=j2+1j1∑k=02j-1|βj,k|p=C((ln⁡n)1+1/θn)(2-p)/2∑j=j2+1j1∑k=02j-1|βj,k|p≤C((ln⁡n)1+1/θn)(2-p)/2∑j=j2+1∞2-j(s+1/2-1/p)p≤C((ln⁡n)1+1/θn)(2-p)/22-j2(s+1/2-1/p)p≤C((ln⁡n)1+1/θn)2s/(2s+1).
So, for r≥1, {p≥2 and s>0} or {p∈[1,2) and s>1/p}, we have
(6.82)S4≤C((ln⁡n)1+1/θn)2s/(2s+1).
It follows from (6.61), (6.66), (6.75), and (6.82) that
(6.83)S≤C((ln⁡n)1+1/θn)2s/(2s+1).
Combining (6.55), (6.57), (6.60), and (6.83), we have, for r≥1, {p≥2 and s>0} or {p∈[1,2) and s>1/p}, (6.84)E(∫01(f̂H(x)-f(x))2dx)≤C((ln⁡n)1+1/θn)2s/(2s+1).
The proof of Theorem 5.2 is complete.



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