# Nonparametric Pointwise Estimation for a Regression Model with Multiplicative Noise

Authors: Jia Chen; Junke Kou
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we consider a general nonparametric regression estimation model with the feature of having multiplicative noise. We propose a linear estimator and nonlinear estimator by wavelet method. The convergence rates of those regression estimators under pointwise error over Besov spaces are proved. It turns out that the obtained convergence rates are consistent with the optimal convergence rate of pointwise nonparametric functional estimation.


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## Body

## 1. Introduction

The regression estimation plays important roles in practical applications. The classical regression estimation model considers a strictly stationary random processUi,Vi,i∈ℤ, which defined on 0,1d×ℝ and has a common density function g. Then, a regression function is defined by
(1)rx≔EρV∣U=x=∫ℝρygx,ydyhx,x∈0,1d,where ρy be a known function and hx stands for the density function of random variable U. This classical model is aimed at estimating the unknown regression function rx by the data U1,V1,U2,V2,⋯,Un,Vn.However, the above dataU1,V1,U2,V2,⋯,Un,Vn is not easily observed in practical applications. In other words, the observed data contains some noise. In view of this case, this paper considers a regression estimation model with multiplicative noise. Let Xi,Yi,i∈ℤ be a strictly stationary random process with a common density function
(2)fx,y=ωx,ygx,yμ,x,y∈0,1d×ℝ,whereωx,y be a multiplicative noise function, gx,y stands for the density function of the unobserved random variable U,V, and μ≔EωU,V<∞. Then, we want to estimate the unknown regression function rx based on the observed noisy data X1,Y1,X2,Y2,⋯,Xn,Yn.This paper is aimed at considering the pointwise error of wavelet regression estimators under some mild assumptions. Chaubey et al. [1, 2] construct linear and nonlinear wavelet estimators based on dependent data and evaluate its rate of convergence under the mean integrated square error, respectively. The convergence rates of wavelet estimators in independent case are discussed by Chesneau and Shirazi [3]. An optimal convergence rate of regression estimator in independent case is proved by Kou and Liu [4]. Guo and Kou [5] consider wavelet estimations for negatively associated case. The mean integrated square error of regression derivative estimators based on strong mixing data are discussed by Kou et al. [6]. But those above results all only focus on global error. There is a lack of result on pointwise error for the regression model with multiplicative noise ((1) and (2)). Hence, this paper will study the convergence rates of wavelet estimators under pointwise risk.


### 1.1. Strong Mixing and Assumptions

In many practical applications, the observed data is usual not independent. This paper will consider strong mixing case, which is an important dependent relationship in time series. Now we will introduce the definition of strong mixing in the following.Definition 1.
LetXi,i∈ℤ be a strict stationary sequence of random vectors and
(3)limk⟶∞αk=limk⟶∞supℙA∩B−ℙAℙB,A∈ϝ−∞0,B∈ϝk∞=0,where ϝ−∞0 denotes the σ field generated by Xi,i<0 and ϝk∞ does by Xi,i≥k. Then, the sequence Xi,i∈ℤ is strong mixing.Note that when the random sample is independent,ℙA∩B=ℙAℙB and αk≡0. Hence, the independent and identically distributed i.i.d. data must be strong mixing. Gorodetskii [7] shows that linear processes are strong mixing under certain conditions. Genon-Catahot et al. [8] prove that some continuous time diffusion models and stochastic volatility models are strong mixing, which include some most popular models in the pricing theories of financial assets, such as the Black Scholes pricing theory of options. Moreover, strong mixing also naturally appears in a large class of stationary time series such as ARMA and GARCH models.Next, we formulate some assumptions for the above regression model. Note that the assumptions C1 and C2 are the standard conditions for nonparametric regression estimation model with multiplicative noise [1, 3]. C3 is a technical condition, which plays an important role in the proof of some lemmas. For the assumptions C4 and C5, the functions αk≡0 and hkx,y,x′,y′≡0 when the random sample is independent. Hence, C4 and C5 may be considered as other explanation of strong mixing.C1. The density function hx satisfies infx∈0,1dhx≥c1 with a positive constant c1.C2. For any x,y∈0,1d×ℝ, there exist two positive constant c2<c3<∞ such that
(4)c2≤ωx,y≤c3.C3. The function ρy satisfies ρy∈Lℝ∩L∞ℝ.C4. The strong mixing coefficient of Xi,Yi,i=1,2,⋯,n satisfies αk≤γe−c4k with γ>0 and c4>0.C5. Let fX1,Y1,Xk+1,Yk+1 be the density function of X1,Y1,Xk+1,Yk+1k≥1 and fX1,Y1 be the density function of X1,Y1. Then, for x,y,x′,y′∈0,1d×ℝ×0,1d×ℝ,
(5)supk≥1supx,y,x′,y′∈0,1d×ℝ×0,1d×ℝhkx,y,x′,y′≤c4,with c4>0 and hk≔fX1,Y1,Xk+1,Yk+1−fX1,Y1fXk+1,Yk+1.


### 1.2. Wavelets and Besov Space

In this paper, we construct regression estimators by wavelet method. Now let us briefly present the wavelet basis ofL2ℝd by tensor product method. For a scaling function φx, there is M=2d−1 wavelet function ψll=1,2,⋯,M such that for each fx∈L2ℝd,
(6)fx=∑k∈ℤdαj0,kφj0,kx+∑j=j0∞∑l=1M∑k∈ℤdβj,klψj,klx,holds in L2ℝd sense, where αj0,k=f,φj0,k, βj,kl=f,ψj,kl, and
(7)φj0,kx=2j0d/2φ2j0dx−k,ψj,klx=2jd/2ψl2jdx−k.If a scaling functionφx satisfies Condition θ, i.e., ∑k∈ℤdφx−k∈L∞ℝd, then the functions φx∈Lℝd∩L∞ℝd and ∑k∈ℤdφx−kφy−k¯ converge absolutely almost everywhere. In this paper, we use the Daubechies wavelet, which has compact supports and satisfies Condition θ. Then, it can be shown that for fx∈Lpℝd1≤p<∞,
(8)Pjfx=∑k∈ℤdαj,kφj,kx,holds almost everywhere on ℝd.Besov spaces are important in theory and applications, which contain Hölder andL2 lebesgue spaces as special examples [9–11]. The next lemma provides equivalent definitions of Besov space [12].Lemma 2.
Letφx be a scaling function, ψlxl=1,2,⋯,M be the corresponding wavelet function, and fx∈Lpℝd1≤p<∞. For 1≤p,q≤+∞ and s>0, then the following assertions are equivalent:
(9)f∈Bp,qsℝd,2jsPjf−fp∈lq,2js−d/p+d/2βj.p∈lq.
The Besov norm off can be defined by
(10)fBp,qs=αj0p+2js−d/p+d/2βj.pj≥j0q,with αj0pp=∑k∈ℤdαj0,kp and βj.pp=∑l=1M∑k∈ℤdβj,klp.
In this paper, we assume that the regression functionrx belongs to Besov ball with H>0, i.e.,
(11)rx∈Bp,qsH≔f∈Bp,qsℝd,fBp,qs≤H.


## 1.1. Strong Mixing and Assumptions

In many practical applications, the observed data is usual not independent. This paper will consider strong mixing case, which is an important dependent relationship in time series. Now we will introduce the definition of strong mixing in the following.Definition 1.
LetXi,i∈ℤ be a strict stationary sequence of random vectors and
(3)limk⟶∞αk=limk⟶∞supℙA∩B−ℙAℙB,A∈ϝ−∞0,B∈ϝk∞=0,where ϝ−∞0 denotes the σ field generated by Xi,i<0 and ϝk∞ does by Xi,i≥k. Then, the sequence Xi,i∈ℤ is strong mixing.Note that when the random sample is independent,ℙA∩B=ℙAℙB and αk≡0. Hence, the independent and identically distributed i.i.d. data must be strong mixing. Gorodetskii [7] shows that linear processes are strong mixing under certain conditions. Genon-Catahot et al. [8] prove that some continuous time diffusion models and stochastic volatility models are strong mixing, which include some most popular models in the pricing theories of financial assets, such as the Black Scholes pricing theory of options. Moreover, strong mixing also naturally appears in a large class of stationary time series such as ARMA and GARCH models.Next, we formulate some assumptions for the above regression model. Note that the assumptions C1 and C2 are the standard conditions for nonparametric regression estimation model with multiplicative noise [1, 3]. C3 is a technical condition, which plays an important role in the proof of some lemmas. For the assumptions C4 and C5, the functions αk≡0 and hkx,y,x′,y′≡0 when the random sample is independent. Hence, C4 and C5 may be considered as other explanation of strong mixing.C1. The density function hx satisfies infx∈0,1dhx≥c1 with a positive constant c1.C2. For any x,y∈0,1d×ℝ, there exist two positive constant c2<c3<∞ such that
(4)c2≤ωx,y≤c3.C3. The function ρy satisfies ρy∈Lℝ∩L∞ℝ.C4. The strong mixing coefficient of Xi,Yi,i=1,2,⋯,n satisfies αk≤γe−c4k with γ>0 and c4>0.C5. Let fX1,Y1,Xk+1,Yk+1 be the density function of X1,Y1,Xk+1,Yk+1k≥1 and fX1,Y1 be the density function of X1,Y1. Then, for x,y,x′,y′∈0,1d×ℝ×0,1d×ℝ,
(5)supk≥1supx,y,x′,y′∈0,1d×ℝ×0,1d×ℝhkx,y,x′,y′≤c4,with c4>0 and hk≔fX1,Y1,Xk+1,Yk+1−fX1,Y1fXk+1,Yk+1.


## 1.2. Wavelets and Besov Space

In this paper, we construct regression estimators by wavelet method. Now let us briefly present the wavelet basis ofL2ℝd by tensor product method. For a scaling function φx, there is M=2d−1 wavelet function ψll=1,2,⋯,M such that for each fx∈L2ℝd,
(6)fx=∑k∈ℤdαj0,kφj0,kx+∑j=j0∞∑l=1M∑k∈ℤdβj,klψj,klx,holds in L2ℝd sense, where αj0,k=f,φj0,k, βj,kl=f,ψj,kl, and
(7)φj0,kx=2j0d/2φ2j0dx−k,ψj,klx=2jd/2ψl2jdx−k.If a scaling functionφx satisfies Condition θ, i.e., ∑k∈ℤdφx−k∈L∞ℝd, then the functions φx∈Lℝd∩L∞ℝd and ∑k∈ℤdφx−kφy−k¯ converge absolutely almost everywhere. In this paper, we use the Daubechies wavelet, which has compact supports and satisfies Condition θ. Then, it can be shown that for fx∈Lpℝd1≤p<∞,
(8)Pjfx=∑k∈ℤdαj,kφj,kx,holds almost everywhere on ℝd.Besov spaces are important in theory and applications, which contain Hölder andL2 lebesgue spaces as special examples [9–11]. The next lemma provides equivalent definitions of Besov space [12].Lemma 2.
Letφx be a scaling function, ψlxl=1,2,⋯,M be the corresponding wavelet function, and fx∈Lpℝd1≤p<∞. For 1≤p,q≤+∞ and s>0, then the following assertions are equivalent:
(9)f∈Bp,qsℝd,2jsPjf−fp∈lq,2js−d/p+d/2βj.p∈lq.
The Besov norm off can be defined by
(10)fBp,qs=αj0p+2js−d/p+d/2βj.pj≥j0q,with αj0pp=∑k∈ℤdαj0,kp and βj.pp=∑l=1M∑k∈ℤdβj,klp.
In this paper, we assume that the regression functionrx belongs to Besov ball with H>0, i.e.,
(11)rx∈Bp,qsH≔f∈Bp,qsℝd,fBp,qs≤H.


## 2. Wavelet Estimators and Theorem

Let us define linear wavelet estimator as follows:(12)r^nlinx≔∑k∈Ωα^j0,kφj0,kx,where
(13)μ^n=1n∑i=1n1ωXi,Yi−1,α^j0,k=μ^nn∑i=1nρYiφj0,kXiωXi,YihXi.Using hard thresholding method, our nonlinear wavelet estimator is defined by(14)r^nnonx≔∑k∈Ωα^j0,kφj0,kx+∑j=j0j1∑l=1M∑k∈Ωjβ^j,klIβ^j,kl≥κtnψj,klx,with tn=lnn/n, the indicator function Ix and
(15)β^j,kl=μ^nn∑i=1nρYiψj,klXiωXi,YihXi.The positive parametricsj0, j1, and κ will be given in later discussion. In addition, Ω≔k∈ℤ,suppr∩suppφj0,k≠∅ and Ωj≔k∈ℤ,suppr∩suppψj,kl≠∅. Because of the compactly supported of regression functions rx,φj0,kx and ψj,klx, the cardinality of Ω and Ωj satisfies that Ω~2j0d and Ωj~2jd, respectively. Here and after, AB denotes A≤cB for some positive constant c, and A~B stands for both AB and BA.Now we state our main theorem in the following, which investigates the performance of linear and nonlinear wavelet estimators in terms of rates of convergence under pointwise risk over Besov space.Theorem 3.
Consider the regression problems (1) and (2) under the assumptions C1-C5 and rx∈Bp,qsHp,q∈1,+∞,s>d/p. Then, the linear wavelet estimator r^nlinx with 2j0~n1/2s−d/p+d satisfies
(16)Er∧nlinx−rx2n−2s−d/p/2s−d/p+d.The nonlinear one with 2j0~n1/2m+dm>s and 2j1~n/lnn31/d does
(17)Er∧nnonx−rx2lnn3n−2s−d/p/2s−d/p+d.Remark 4.
Note that the linear and nonlinear wavelet estimators all can attain the optimal convergence rate for pointwise risk [13].Remark 5.
Compared with the linear wavelet estimatorr^nlinx, the convergence rate of the nonlinear estimator r^nnonx keeps the same as the linear one up to a lnn factor. Moreover, the nonlinear wavelet estimator is adaptive, which means that it only depends on the observed data.


## 3. Auxiliary Results

This section will provide some results for the proof of Theorem3.Lemma 6.
For the problem defined in (1) and (2), we have
(18)E1μ^n=1μ,EμρYiφj0,kXiωXi,YihXi=αj0,k,EμρYiψj,klXiωXi,YihXi=βj,kl.Lemma 7.
LetXi,Yi,i=1,2,⋯,n be a stationary strong mixing process. If C1-C5 hold and 2jd≤n,
(19)Var∑i=1n1ωXi,Yi≲n,(20)Var∑i=1nρYiφj0,kXiωXi,YihXi≲n,(21)Var∑i=1nρYiψj,klXiωXi,YihXi≲n.Proof.
According to the properties of variance, we know(22)Var∑i=1n1ωXi,Yi=∑i=1nVar1ωXi,Yi+2∑1≤i<v≤nCov1ωXi,Yi,1ωXv,Yv≲nVar1ωXi,Yi+∑v=2n∑i=1v−1Cov1ωXi,Yi,1ωXv,Yv.
Now we estimate the first term. It follows from the property of variance and C2 that(23)Var1ωXi,Yi≤E1ωXi,Yi2=∫0,1d×ℝ1ωx,y2fx,ydxdy≲1.
Using the property of strictly stationary,(24)∑v=2n∑i=1v−1Cov1ωXi,Yi,1ωXv,Yv≲∑i=1n−1n−iCov1ωX1,Y1,1ωXi+1,Yi+1≲n∑i=1n−1Cov1ωX1,Y1,1ωXi+1,Yi+1.
Then, we need the following Davydov inequality [14]. Let Xii∈Z be strong mixing with mixing coefficient αk and f and g be two measurable functions. If EfX1p and EgX1q exist for p,q>0 and 1/p+1/q<1,
(25)CovfX1,gXk+1≲αk1−1/p−1/qE∣fX1p1/pE∣gX1q1/q.
Using the Davydov inequality withp=q=4,
(26)Cov1ωX1,Y1,1ωXi+1,Yi+1≲αi1/2E∣1ωX1,Y141/2.
This with C2 showsCov1/ωX1,Y1,1/ωXi+1,Yi+1≲αi1/2. Furthermore, it follows from C4 that
(27)∑i=1n−1Cov1ωX1,Y1,1ωXi+1,Yi+1≲∑i=1n−1αi1/2≤∑i=1n−1γ1/2e−c4/2i≲∑i=1∞e−c4/2i≲1.
Combining this result with and (22)–(24) and ((27)), one gets that
(28)Var∑i=1n1ωXi,Yi≲n.
For the estimation of (20), we know that Var∑i=1nρYiφj0,kXi/ωXi,YihXi≲nVarρYiφj0,kXi/ωXi,YihXi+∑v=2n∑i=1v−1CovρYiφj0,kXi/ωXi,YihXi,ρYvφj0,kXv/ωXv,YvhXv.Then, it is easy to see from the property of variance and the assumptions C1-C3 that
(29)nVarρYiφj0,kXiωXi,YihXi≤nEρYiφj0,kXiωXi,YihXi2=n∫0,1d×ℝρyφj0,kxωx,yhx2fx,ydxdy=n∫0,1d×ℝρy2φj0,kx2gx,yμωx,yhx2dxdy≲n∫0,1d×ℝφj0,kx2gx,yhxdxdy=n∫0,1dφj0,kx2dx≲n.
Now it remains to prove(30)∑v=2n∑i=1v−1CovρYiφj0,kXiωXi,YihXi,ρYvφj0,kXvωXv,YvhXv≲n.
Similar to the arguments of (24), one gets that ∑v=2n∑i=1v−1CovρYiφj0,kXi/ωXi,YihXi,ρYvφj0,kXv/ωXv,YvhXvn∑i=1n−1CovρY1φj0,kX1/ωX1,Y1hX1,ρYi+1φj0,kXi+1/ωXi+1,Yi+1hXi+1. Because of 2jd≤n, we can obtain that
(31)∑v=2n∑i=1v−1CovρYiφj0,kXiωXi,YihXi,ρYvφj0,kXvωXv,YvhXv≲nT1+T2,(32)T1≔∑i=12j0d−1CovρY1φj0,kX1ωX1,Y1hX1,ρYi+1φj0,kXi+1ωXi+1,Yi+1hXi+1,(33)T2≔∑i=2j0dn−1CovρY1φj0,kX1ωX1,Y1hX1,ρYi+1φj0,kXi+1ωXi+1,Yi+1hXi+1.
First, estimateT1. By the property of covariance, CovρY1φj0,kX1/ωX1,Y1hX1,ρYi+1φj0,kXi+1/ωXi+1,Yi+1hXi+1=EρY1φj0,kX1/ωX1,Y1hX1ρYi+1φj0,kXi+1/ωXi+1,Yi+1hXi+1−EρY1φj0,kX1/ωX1,Y1hX1EρYi+1φj0,kXi+1/ωXi+1,Yi+1hXi+1. Furthermore, using the assumptions C1-C3 and C5, one can get that
(34)CovρY1φj0,kX1ωX1,Y1hX1,ρYi+1φj0,kXi+1ωXi+1,Yi+1hXi+1≤≤∫0,1d×ℝ×0,1d×ℝρyφj0,kxωx,yhxρy′φj0,kx′ωx′,y′hx′hix,y,x′,y′dxdydx′dy′≲∫ℝ∣ρy∣dy2∫0,1d∣φj0,kx∣dx2≲∫0,1d∣φj0,kx∣dx2≲2−j0d.
Then, one have(35)T1=∑i=12j0d−1∣CovρY1φj0,kX1ωX1,Y1hX1,ρYi+1φj0,kXi+1ωXi+1,Yi+1hXi+1∣≲∑i=12j0d−12−j0d≲1.
Now we estimateT2. Using the above Davydov inequality with p=q=4,
(36)CovρY1φj0,kX1ωX1,Y1hX1,ρYi+1φj0,kXi+1ωXi+1,Yi+1hXi+1≲αi1/2EρY1φj0,kX1ωX1,Y1hX141/2≲≲αi1/2supρY1φj0,kX1ωX1,Y1hX1EρY1φj0,kX1ωX1,Y1hX121/2.
It follows from C1-C3 that(37)supρY1φj0,kX1ωX1,Y1hX1≲2j0d.
On the other hand,(38)EρY1φj0,kX1ωXi,YihX12=∫0,1d×ℝρyφj0,kxωx,yhx2ωx,ygx,yμdxdy≲∫0,1dφj0,kx2dx≲1.
Hence,(39)T2≲∑i=2j0dn−1αi122j0d≤∑i=2j0dn−1γ12e−c4/2i2j0d≲∑i=1∞e−c4/2ii12≲1.
Now the above results (31), (35), and (39) imply the desired inequality
(40)∑v=2n∑i=1v−1CovρYiφj0,kXiωXi,YihXi,ρYvφj0,kXvωXv,YvhXv≲n.
The prove of the last inequality (21) is similar to (20).Lemma 8.
Let assumptions C1-C5 hold and2jd≤n. Then,
(41)Eα∧j0,k−αj0,k2≲n−1,Eβ∧j,kl−βj,kl2≲n−1.Proof.
By the definition ofα^j0,k,
(42)α^j0,k−αj0,k=μ^nn∑i=1nρYiωXi,YihXiφj0,kXi−αj0,k≤μ^nμμn∑i=1nρYiωXi,YihXiφj0,kXi−αj0,k+αj0,kμ^n1μ−1μ^n.
Note that C3 and the definition of regression functionrx imply the boundedness of rx. Furthermore, αj0,k=∫0,1drxφj0,kxdx≲1 by Hölder inequality and the orthonormality of φj0,kx. On the other hand, μ^n/μ≲1 and μ≲1 because of C2. Then, one can obtain that
(43)Eα∧j0,k−αj0,k2≲E1n∑i=1nμρYiωXi,YihXiφj0,kXi−αj0,k2+E1μ−1μ∧n2.
It is easy to see from Lemmas6 and 7 that
(44)Eα∧j0,k−αj0,k2≲Var1n∑i=1nμρYiωXi,YihXiφj0,kXi+Var1μ^n≲1n2Var∑i=1nρYiωXi,YihXiφj0,kXi+1n2Var∑i=1n1ωXi,Yi≲n−1.
For the second inequality, according to the definition ofβ^j,kl,
(45)β^j,kl−βj,kl≤μ^nμμn∑i=1nρYiωXi,YihXiψj,klXi−βj,kl+βj,klμ^n1μ−1μ^n.
Then, it follows from the similar arguments of (43) that
(46)Eβ∧j,kl−βj,kl2≲E1n∑i=1nμρYiωXi,YihXiψj,klXi−βj,kl2+E1μ−1μ∧n2.
Using Lemmas6 and 7, one can get
(47)Eβ∧j,kl−βj,kl2≲Var1n∑i=1nμρYiωXi,YihXiψj,klXi+Var1μ^n≲n−1.Lemma 9.
Letβ^j,kl be defined by (15). If C1-C5 hold, tn=lnn/n and 2jd≤n/lnn3; then for each w>0, there exists a constant κ>1 such that
(48)ℙβ^j,kl−βj,kl≥κtn≲2−wj.Proof.
According to the arguments of (43) and the definition of β^j,kl,
(49)β^j,kl−βj,kl≤μ^nμμn∑i=1nρYiωXi,YihXiψj,klXi−βj,kl+βj,klμ^n1μ−1μ^n≲1n∑i=1nμρYiωXi,YihXiψj,klXi−βj,kl+1n∑i=1n1ωXi,Yi−1μ.
Then, we only need to prove the following two inequalities, respectively.(50)ℙ1n∑i=1nμρYiωXi,YihXiψj,klXi−βj,kl≥κ2tn≲2−wj,(51)ℙ1n∑i=1n1ωXi,Yi−1μ≥κ2tn≲2−wj.(i)
Proof of (50). Define ηi≔μρYi/ωXi,YihXiψj,klXi−βj,kl. Then, Eηi=0 thanks to Lemma 6. Moreover, η1,η2,⋯,ηn are strong mixing with the mixing coefficients αk≤γe−c4k. Using the assumptions C1-C3, one gets that ∣ηi∣≤∣μρYi/ωXi,YihXiψj,klXi∣+∣βj,kl∣≲2jd/2. On the other hand, it follows from Lemma 7 that(52)max1≤j≤2mVar∑i=1jηi=max1≤j≤2mVar∑i=1jμρYiωXi,YihXiψj,klXi≲m.
Because strong mixing is much more complicated than independent, we need to use a new technique. In this paper, we use the following Bernstein type inequality [15]. Let Xii∈Z be a strong mixing process with mixing coefficient αk, EXi=0, ∣Xi∣≤M<∞, and Dm≔max1≤j≤2mVar∑i=1jXi. Then, for ε>0 and n,m∈N with 0<m≤n/2,
(53)ℙ∑i=1nXi≥εexp−ε216nm−1Dm+13εMm−1+Mεnαm.
According to the above Bernstein type inequality withm=ulnn and M=2jd/2,
(54)ℙ1n∑i=1nηi≥κ2tn=ℙ∑i=1nηi≥κ2ntn≲exp−κ/2ntn216nm−1m+13κ2ntn2jd/2m−1+2jd/2κ/2ntnnαm≲exp−κntn264n+κu6ntn2jd/2lnn−1+2jd/2tne−culnn.
Then, it follows fromtn=lnn/n and 2jd≤n/lnn3≤n that
(55)exp−κntn264n+κu6ntn2jd/2lnn−1≲exp−κ2lnn641+κu6−1.
Obviously, there exists sufficiently largeκ>1 such that
(56)exp−κ2lnn641+κu6−1≲2−wj.
On the other hand, note that2jd/2/tne−culnn≲n1−cu≲2jd1−cu. Choosing u such that d1−cu<−w, then
(57)2jd/2tne−culnn≲2−wj.
Together with the above results, one can obtain the desired result (50)
(58)ℙ1n∑i=1nηi≥κ2tn≲2−wj.(ii)
Proof of (51). Define ξi≔1/ωXi,Yi−1/μ. Then, Eξi=0 thanks to Lemma 6. Moreover, it follows from the assumption C4 that ξ1,ξ2,⋯,ξn are strong mixing with the mixing coefficients αk≤γe−c4k. Note that ξi≤1/ωXi,Yi+1/μ≲1 with C2. According to Lemma 7, max1≤j≤2mVar∑i=1jξi=max1≤j≤2mVar∑i=1j1/ωXi,Yi≲m. Then, similar to the proof of (50), one can get the desired result (51).


## 4. Proof of Theorem

This last section is devoted to proving Theorem3.Proof of (16).
It is easy to see that(59)Er∧nlinx−rx2≤Er∧nlinx−Pj0rx+Pj0rx−rx2≲Er∧nlinx−Pj0rx2+EPj0rx−rx2.(i)
Upper bound ofEr∧nlinx−Pj0rx2. Note that(60)Er∧nlinx−Pj0rx2≤E∑k∈Ωα∧j0,k−αj0,kφj0,kx2.
Then, it follows from Cauchy-Schwarz inequality and Conditionθ that
(61)Er∧nlinx−Pj0rx2≤E∑k∈Ωα∧j0,k−αj0,k2φj0,kx∑k∈Ωφj0,kx≲2j0d/2∑k∈Ωφj0,kxEα∧j0,k−αj0,k2.
Using Lemma8 and 2j0~n1/2s−d/p+d,
(62)Er∧nlinx−Pj0rx2≲2j0d/2n−1∑k∈Ωφj0,kx≲2j0dn−1≲n−2s−d/p/2s−d/p+d.(ii)
Upper bound ofEPj0rx−rx2. It is easy to see from Hölder inequality that(63)EPj0rx−rx2=∑j=j0∞∑l=1M∑k∈Ωjβj,klψj,klx2≤∑j=j0∞∑l=1M∑k∈Ωjβj,klpψj,klx1/p∑l=1M∑k∈Ωjψj,klx1/p′2.
Furthermore, Conditionθ and the boundedness of ψj,klx imply that
(64)EPj0rx−rx2≲∑j=j0∞2jd/2∑l=1M∑k∈Ωjβj,klp1/p2=∑j=j0∞2jd/2βj.p2.
Now usingrx∈Bp,qsH, Lemma 2, and 2j0~n1/2s−d/p+d, one can get that
(65)EPj0rx−rx2≲∑j=j0∞2−js−d/p2≲2−2j0s−d/p~n−2s−d/p/2s−d/p+d.
Combining with (59), (62), and (65), the desired result is obtained by
(66)Er∧nlinx−rx2≲n−2s−d/p/2s−d/p+d.Proof of (17).
By the definition ofr^nnonx,
(67)r^nnonx−rxr^nlinx−Pj0rx+Pj1rx−rx++∑j=j0j1∑l=1M∑k∈Ωjβ^j,klIβ^j,kl≥κtn−βj,klψj,klx.
Hence,(68)Er∧nnonx−rx2≲Er∧nlinx−Pj0rx2+EPj1rx−rx2++E∑j=j0j1∑l=1M∑k∈Ωjβ∧j,klIβ∧j,kl≥κtn−βj,klψj,klx2.
According to the arguments of (62), 2j0~n1/2m+d and s>d/p,
(69)Er∧nlinx−Pj0rx22j0dn−1≲n−2m/2m+d≲n−2s−d/p/2s−d/p+d.
Similar to the arguments of (65) and 2j1~n/lnn31/d,
(70)EPj1rx−rx2≲∑j=j1∞2jd/2βj.p2≲2−2j1s−d/p≲nlnn3−2s−d/p/dlnn3≲n−2s−d/p/2s−d/p+d.
Then, we only need to estimate the upper bound of(71)Q≔E∑j=j0j1∑l=1M∑k∈Ωjβ∧j,klIβ∧j,kl≥κtn−βj,klψj,klx2.
Note that(72)Q≤j1−j0+1E∑j=j0j1∑l=1M∑k∈Ωjβ∧j,klIβ∧j,kl≥κtn−βj,klψj,klx2≲j1−j0+1Q1+Q2+Q3+Q4,
where(73)Q1≔E∑j=j0j1∑l=1M∑k∈Ωjβ∧j,kl−βj,klIβ∧j,kl≥κtnIβj,kl≥κ/2tnψj,klx2,Q2≔E∑j=j0j1∑l=1M∑k∈Ωjβ∧j,kl−βj,klIβ∧j,kl≥κtnIβj,kl<κ/2tnψj,klx2,Q3≔E∑j=j0j1∑l=1M∑k∈Ωjβj,klIβ∧j,kl≤κtnIβj,kl≥2κtnψj,klx2,Q4≔E∑j=j0j1∑l=1M∑k∈Ωjβj,klIβ∧j,kl≤κtnIβj,kl≤2κtnψj,klx2.(i)
Upper Bound ofQ2andQ3. It follows from Cauchy-Schwarz inequality that(74)maxQ2,Q3≲E∑j=j0j1∑l=1M∑k∈Ωjβ∧j,kl−βj,klIβ∧j,kl−βj,kl≥κ/2tnψj,klx2≲E∑j=j0j1∑l=1M∑k∈Ωjβ∧j,kl−βj,kl2Iβ^j,kl−βj,kl≥κ/2tnψj,klx∑l=1M∑k∈Ωjψj,klx≲∑j=j0j12jd/2∑l=1M∑k∈Ωjψj,klxEβ∧j,kl−βj,kl2Iβ^j,kl−βj,kl≥κ/2tn.
Using Lemmas8 and 9, one gets
(75)Eβ∧j,kl−βj,kl2Iβ^j,kl−βj,kl≥κ/2tn≤Eβ∧j,kl−βj,kl41/2ℙβ∧j,kl−βj,kl≥κ2tn1/2≤2jd/2Eβ∧j,kl−βj,kl21/2ℙβ∧j,kl−βj,kl≥κ2tn1/2≲2jd/2−w/2n−1/2.
Then, choosingw such that w>2d+2m, one has that
(76)maxQ2,Q3≲∑j=j0j12jd/2−w/2n−1/2∑l=1M∑k∈Ωjψj,klx∑j=j0j12j3d/2−w/2n−1/2≲2j03d/2−w/2n−1/2n−w/2+m−d/2m+dn−2m/2m+dn−2s−d/p/2s−d/p+d.(ii)
Upper Bound ofQ1. To estimate Q1, one defines(77)2j∗~n1/2s−d/p+d.
Note thatQ1 can be decomposed into the following two terms:
(78)Q1=E∑j=j0j∗−1+∑j=j∗j1∑l=1M∑k∈Ωjβ∧j,kl−βj,klIβ∧j,kl≥κtnIβj,kl≥k/2tnψj,klx2≔Q11+Q12.
Using Cauchy-Schwarz inequality and Lemma8,
(79)Q11≲E∑j=j0j∗−1∑l=1M∑k∈Ωjβ∧j,kl−βj,klψj,klx2≲E∑j=j0j∗−1∑l=1M∑k∈Ωjβ∧j,kl−βj,kl2ψj,klx∑l=1M∑k∈Ωjψj,klx≲∑j=j0j∗−12jd/2∑l=1M∑k∈Ωjψj,klxEβ∧j,kl−βj,kl2≲∑j=j0j∗−12jd/2n−1∑l=1M∑k∈Ωjψj,klx≲∑j=j0j∗−12jdn−1n−2s−d/p/2s−d/p+d.
On the other hand, it is easy to see that(80)Q12≲E∑j=j∗j1∑l=1M∑k∈Ωjβ∧j,kl−βj,klβj,kltnψj,klx2≲nlnnE∑j=j∗j1βj.∞2∑l=1M∑k∈Ωjβ∧j,kl−βj,kl2ψj,klx∑l=1M∑k∈Ωjψj,klx≲n∑j=j∗j12jd2βj.∞2∑l=1M∑k∈Ωjψj,klxEβ∧j,kl−βj,kl2≲∑j=j∗j12jdβj.p2.
Then, one knows(81)Q12≲∑j=j∗j1≲2−2js−d/p2−2j∗s−d/p≲n−2s−d/p/2s−d/p+d,with Lemma 2. Now together with (78), (79), and (81),
(82)Q1≲n−2s−d/p/2s−d/p+d.(iii)
Upper Bound ofQ4. Note that(83)Q4=E∑j=j0j∗−1+∑j=j∗j1∑l=1M∑k∈Ωjβj,klIβ∧j,kl≤κtnIβj,kl≤2κtnψj,klx2≔Q41+Q42.
By Conditionθ and tn=lnn/n,
(84)Q41≲∑j=j0j∗−1∑l=1M∑k∈Ωjtnψj,klx2≲lnnn∑j=j0j∗−12jdlnnn2j∗dlnnn−2s−d/p/2s−d/p+d.
Moreover, according to Hölder inequality and Lemma2, we can get
(85)Q42≲∑j=j∗j1∑l=1M∑k∈Ωjβj,klψj,klx2≲∑j=j∗j1∑l=1M∑k∈Ωjβj,klpψj,klx1/p∑l=1M∑k∈Ωjψj,klx1/p′2≲∑j=j∗j12jdβj.p2∑j=j∗j12−2js−d/pn−2s−d/p/2s−d/p+d.
Combining this with (83) and (84),
(86)Q4≲lnnn−2s−d/p/2s−d/p+d.
Now the results (72), (76), (82), and (86) imply that
(87)Q≲lnn3n−2s−d/p/2s−d/p+d.
This with (68)–(70) shows
(88)Er∧nnonx−rx2≲lnn3n−2s−d/p/2s−d/p+d.



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