consistency and uniformly asymptotic normality of wavelet estimator in regression model with...

Statistics and Probability Letters 78 (2008) 2947–2956

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Statistics and Probability Letters

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Consistency and uniformly asymptotic normality of wavelet estimator inregression model with associated samplesI

Yongming Li a,∗, Shanchao Yang b, Yong Zhou ca Department of Mathematics, Shangrao Normal College, Shangrao, Jiangxi 334001, Chinab Department of Mathematics, Guangxi Normal University, Guilin, Guangxi 541004, Chinac Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China

a r t i c l e i n f o

Article history:Received 18 August 2007Received in revised form 5 March 2008Accepted 6 May 2008Available online 13 May 2008

MSC:62G0562G0862E20

a b s t r a c t

Consider the wavelet estimator of a nonparametric fixed design regression function whenerrors are strictly stationary and associated random variables. We establish pointwiseweak consistency and uniformly asymptotic normality of wavelet estimator of regressionfunction. We give rates of uniformly asymptotic normality for associated samples. Therates are near n−1/4 and n−1/6 when their third moments are finite for NA and PA cases,respectively.

Crown Copyright© 2008 Published by Elsevier B.V. All rights reserved.

1. Introduction

In regression, nonparametric regression is often more appropriate than parametric regression when this functionalrelationship is of a complex or subtle nature. We will consider the following nonparametric regression model

Yni = g(xni)+ εni, 1 ≤ i ≤ n, (1.1)

where regression function g is an unknown bounded real valued function on A, which is a compact set of R, the nonrandomdesign points xn1, . . . , xnn ∈ A, and εn1, . . . , εnn are regression errors with zeromean and finite variance.We assume that,for each n, εn1, . . . , εnn have the same joint distribution as ξ1, . . . , ξn, where ξt , t = 0,±1, . . . , is a strictly stationarytime series defined on a probability space (Ω,=, P) and taking values on R. Without loss of generality, we can suppose thatA = [0, 1], and xn1 ≤ xn2 ≤ · · · ≤ xnn.It is well known that regression function estimation is an important method in data analysis and has a wide range of

applications in filtering and prediction in communications and control systems, pattern recognition and classification, andeconometrics. So model (1.1) has been widely studied by many authors in the literature. Some recent works about theasymptotic normality of weighted function estimate of regression function can be found in the work of Roussas and Tran(1992), Yang (2003), and Yang and Li (2006).In the last decade, wavelet techniques, due to their ability to adapt to local features of curves, received much

attention from mathematicians, engineers and statisticians. Many authors have applied wavelet procedures to estimatenonparametric and semiparametric models. See for example recent works by Antoniadis et al. (1994), Donoho et al. (1996),

I This research was supported by the Natural Science Foundation of China (10661003), the Natural Science Foundation of Jiangxi 2008G and the ScienceFoundation of Jiangxi Educational ([2007]347).∗ Corresponding author.E-mail addresses: [email protected] (Y. Li), [email protected] (S. Yang), [email protected] (Y. Zhou).

0167-7152/$ – see front matter Crown Copyright© 2008 Published by Elsevier B.V. All rights reserved.doi:10.1016/j.spl.2008.05.004

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2948 Y. Li et al. / Statistics and Probability Letters 78 (2008) 2947–2956

Qian and Chai (1999), Chai and Liu (2001), Xue (2002), Lin and Zhang (2004), Sun and Chai (2004) and Zhou and You (2004),and so on.Related to wavelet estimation for model (1.1), Antoniadis et al. (1994) constructed the wavelet estimator of g,

gn(t) =n∑i=1

Yni

∫AiEm(t, s)ds, (1.2)

where Ai = [si−1, si) is a partition of interval A = [0, 1]with xni ∈ Ai. Wavelet kernel Em(t, s) = 2mE0(2mt, 2ms), E0(t, s) =∑j∈Z φ(t − j)φ(s − j),m = m(n) > 0 is an integer depending only on n, φ is a scaling function. Antoniadis et al. (1994)

considered the consistency and asymptotic normality of (1.2) under i.i.d. processes. Xue (2002) and Sun and Chai (2004)studied the consistency, strong consistency and convergence rate of (1.2) for mixing processes.The concepts of positively associated (PA) and negatively associated (NA) are due to Esary et al. (1967), Joag-Dev and

Proschan (1983), respectively. PA andNA randomvariables have foundwide applicability in reliability, statisticalmechanics,probability/stochastic processes and statistics which were detailed by Roussas (1999). Asymptotic normality of the smoothestimate for distribution function and probability density function under association has been considered by Roussas (1995,2000). Cai and Roussas (1999) stated Berry–Esseen bounds for smooth estimator of a distribution function under association.Yang (2003, 2005) established the uniformly asymptotic normality of the regression weighted estimator under association.Here our purpose is to establish pointwise weak consistency and uniformly asymptotic normality of (1.2) for PA and NA

processes, and to derive its rate of uniformly asymptotic normality.The paper is organized as follows. In Section 2, we give assumptions and main results. Sections 3–5 contain the proofs.

Appendix is the appendix that contains some known results used in the proofs.

2. Notations, assumptions and main results

Let Sn(t) = σ−1n (t)gn(t) − Egn(t), Fn(u) = P(Sn(t) < u), σ 2n (t) = Var(gn(t)), s2n =∑kj=1 Var(Ynj), Φ(u) be

the standard normal distribution function. For convenient writing, we omit everywhere the argument t , and set Zni =σ−1n εni

∫AiEm(t, s)ds, i = 1, . . . , n, so that Sn =

∑ni=1 Zni. We employ Bernstein’s big-block and small-block procedure.

We partition the set 1, . . . , n into 2kn + 1 subsets with large blocks of size p = pn and small blocks of size q = qn and setk = kn = [n/(pn + qn)]. Then Sn may be split as

Sn = S ′n + S′′

n + S′′′

n , (2.1)

where S ′n =∑kj=1 ynj, S

′′n =

∑km=1 y

′

nj, S′′′n = y

′

nk+1, and ynj =∑kj+p−1i=kj

Zni, y′nj =∑lj+q−1i=lj

Zni, y′nk+1 =∑ni=k(p+q)+1 Zni, kj =

(j− 1)(p+ q)+ 1, lj = (j− 1)(p+ q)+ p+ 1, j = 1, . . . , k.

Assumption 2.1. (i) For each n, the joint distribution of εni, i = 1, . . . , n is the same as that of ξ1, . . . , ξn, where ξi, i =1, . . . , n is a NA or PA time series with zero mean and finite second moment, supj≥1 E(ξ 2j ) < ∞; (ii) supj≥1 E|ξj|

2+δ < ∞for some δ > 0.

Assumption 2.2. (i) φ(·) is l-regular (l is a positive integer), satisfies the Lipschitz condition with order 1 and has a compactsupport. Furthermore, |φ(ξ)− 1| = O(ξ) as ξ →∞, where φ(·) is the Fourier transform of φ(·); (ii) max1≤i≤n |si − si−1| =O(n−1); (iii) n−12m = O(n−θ ) for some 1/2 < θ < 1 as n→∞.

Assumption 2.3. There exist positive integers p and q such that for sufficiently large n

p+ q ≤ n, qp−1 ≤ C <∞, (2.2)

and as n→∞

γ1n = 2mqp−1 → 0, γ2n = 2mpn−1 → 0, (2.3)

γ3n = qp−1 → 0, γ4n = pn−1 → 0. (2.4)

Here, we are now ready to establish the pointwise weak consistency and the rates of uniformly asymptotic normality ofthe estimator (1.2) described as follows.

Theorem 2.1. Let Assumption 2.2(i), (ii) be satisfied and 2m = O(n1/3). If ξi; i ≥ 1 are NA random variables with zero mean,E|ξi|p <∞ (∀p > 3

2 ), then we have

g(t)P→ g(t).

Y. Li et al. / Statistics and Probability Letters 78 (2008) 2947–2956 2949

Theorem 2.2. Let ξj : j ≥ 1 be a NA sequence, and u(q) = supj≥1∑|i−j|≥q |Cov(ξi, ξj)|. If Assumptions 2.1 and 2.2, and

relations (2.2) and (2.3) are satisfied, then

supu|Fn(u)− Φ(u)| ≤ C(γ

1/21n + γ

1/22n )(log n log log n)

1/2+ γ

δ/22n + u

1/3(q)+ n−1

+ ((nqp−1)−δ/2 + p−δ/2)(log n log log n)(δ−2)/2.

Corollary 2.1. Suppose the assumptions of Theorem 2.2 are fulfilled. If u(1) <∞, then

supu|Fn(u)− Φ(u)| = (1).

Corollary 2.2. Suppose the assumptions of Theorem 2.2 are fulfilled. If supj≥1 E(ξ 3j ) < ∞, u(n) = O(n−(3θ−3ρ)/(4ρ−2)), where

12 < ρ ≤ θ ≤ 2ρ , then

supu|Fn(u)− Φ(u)| = O

(n−(θ−ρ)/2(log n log log n)1/2

). (2.5)

Remark 2.1. From Corollary 2.2, the rate of convergence is near n−1/4, if θ ≈ 1, and ρ ≈ 1/2.

Theorem 2.3. Let ξj : j ≥ 1 be a PA sequence, and v(n) =∑∞

j=n Cov(ξ1, ξj+1). If Assumptions 2.1 and 2.2(i), (ii), and relations(2.2) and (2.4) are satisfied, v(n) = O(n−(r−2)(r+δ)/(2δ)) for some r > 2, then

supu|Fn(u)− Φ(u)| ≤ C

γ1/33n + γ

1/34n + γ

(r−2)/24n + u1/3(q)

.

Corollary 2.3. Suppose all the assumptions of Theorem 2.3 are fulfilled, and (i) r = 3; (ii) u(n) = O(n−(3+δ)/(2δ)). Then

supu|Fn(u)− Φ(u)| = O

(n−(3+δ)/(18+12δ)

).

Remark 2.2. From Corollary 2.3, the rate of convergence is near n−1/6, if δ→ 0.

Remark 2.3. Here are some comments on the assumptions. (i) Assumption 2.3 is easily satisfied, if p, q and 2m are chosenas follows: let p ∼ nδ1 , q ∼ nδ2 , and 2m ∼ nδ3 , where an ∼ bn means that, as n → ∞, an/bn tends to a constant, andδ1, δ2, δ3 satisfy

(a) 0 < δ2 < δ1 < 1; (b) 0 < δ1 + δ2 < 1; (c) 0 < δ3 < minδ1 − δ2, 1− δ1.

For example, take δ1 = 0.65, δ2 = 0.25 and δ3 = 0.1, then Assumption 2.3 is satisfied. In addition, we know (2.4) is weakerthan those used in Yang (2005) for p, q.(ii) It is well known that Assumption 2.2 gives mild regularity conditions for wavelet smoothing in the recent literature,

see Antoniadis et al. (1994), Chai and Liu (2001), Xue (2002) and Sun and Chai (2004).(iii) Assumption 2.1 is the same basic assumption as that used in Yang (2003).From (i)–(iii) above, we can see that the conditions in this paper are suitable and reasonable.

Remark 2.4. Up until now, we have not found a result on uniformly asymptotic normality of wavelet estimation for model(1.1). But, in this paper, not only do we establish asymptotic normality, but also we gain the rate of uniformly asymptoticnormality under very weak assumptions. Furthermore, from Remarks 2.1 and 2.2, the rate here is as fast as in Yang (2003,2005) for NA and PA cases, respectively. These results show that thewaveletmethod is valid formodel (1.1) under associatederrors.

3. Proof of the weak consistency

We write that

|g(t)− g(t)| ≤ |g(t)− Eg(t)| + |Eg(t)− g(t)|. (3.1)

From Theorem 3.2 in Antoniadis et al. (1994), we can get |Eg(t)− g(t)| → 0. So it remains to prove

|g(t)− Eg(t)|P→ 0. (3.2)


Let ani =∫AiEm(t, s)ds, then g(t) − Eg(t) =

∑ni=1 aniεni. Let a

+

ni = max0, ani and a−

ni = max0,−ani be the positiveand negative functions of ani, respectively. Obviously,

∑ni=1 aniεni =

∑ni=1 a

+

niεni −∑ni=1 a

−

niεni. Without loss of generality,we can assume that ani ≥ 0, i = 1, . . . , n and denote$ni = aniεni, i = 1, . . . , n. So $ni, i = 1, . . . , n still are negativeassociated random variables. Hence, for any µ > 0, by applying the Markov inequality, we have

P

(∣∣∣∣∣ n∑i=1

$ni

∣∣∣∣∣ ≥ µ)≤ µ−pE

∣∣∣∣∣ n∑i=1

$ni

∣∣∣∣∣p

. (3.3)

If 32 < p ≤ 2, by Lemma A.1, we have

E

∣∣∣∣∣ n∑i=1

$ni

∣∣∣∣∣p

≤ C ·n∑i=1

∣∣∣∣∣∫AiEm(t, s)

n∑j=1

I(sj−1 < t < sj)ds

∣∣∣∣∣p

· E|εni|p

≤ Cn∑i=1

∣∣∣∣∣∫Ai

n∑j=1

(Em(t, s)− Em(xnj, s))I(sj−1 < t < sj)ds

∣∣∣∣∣p

· E|εni|p

+ Cn∑i=1

∣∣∣∣∣∫Ai

n∑j=1

Em(xnj, s)I(sj−1 < t < sj)ds

∣∣∣∣∣p

· E|εni|p =: I1 + I2. (3.4)

Noting that E0(t, s) satisfies the Lipschitz condition of order 1 on t , we have

I1 ≤ C1n∑i=1

∣∣∣∣∣∫Ai

n∑j=1

22m

nI(sj−1 < t ≤ sk)ds

∣∣∣∣∣p

E|εni|p ≤ C2n−43 p

n∑i=1

E|εni|p, (3.5)

and by Lemma A.5,

I2 = Cn∑i=1

∣∣∣∣∣ n∑j=1

I(sj−1 < t < sj)∫AiEm(xnj, s)ds

∣∣∣∣∣p

· E|εni|p

≤ C3n∑i=1

∣∣∣∣∫AiEm(xnj, s)ds

∣∣∣∣p · E|εni|p ≤ C4 · n− 23 p · n∑i=1

E|εni|p. (3.6)

Thus, (3.3)–(3.6) imply the conclusion of (3.2).If p > 2, by Lemma A.1, we have

E

∣∣∣∣∣ n∑i=1

$ni

∣∣∣∣∣p

≤ C

n∑i=1

E∣∣∣∣εni ∫

AiEm(t, s)ds

∣∣∣∣p +(

n∑i=1

E(εni

∫AiEm(t, s)ds

)2) p2 =: I3 + I4. (3.7)

By (3.4)–(3.6), we get

I3 = C5 ·(n−

43 p + n−

23 p)·

n∑i=1

E|εni|p, I4 = C6 ·

(n−

83 + ·n−

43

)·

n∑i=1

E|εni|2 p2

. (3.8)

Thus, (3.7) and (3.8) imply (3.2). Therefore, Theorem 2.1 is verified from (3.1) and (3.2).

4. Proof of the uniformly asymptotic normality for the NA case

In order to prove the main results for the NA case, we first present several lemmas.

Lemma 4.1. Let Assumptions 2.1 and 2.2 be satisfied, then

Var(g(t)) = O(2mn−1).

Proof. By applying Lemmas A.1 and A.5 in the Appendix, we obtain

Var(g(t)) ≤ C2n∑i=1

E(εni

∫AiEm(t, s)ds

)2≤ C2

n∑i=1

(∫AiEm(t, s)ds

)2≤ O(2mn−1).

Thus Lemma 4.1 is verified.


Lemma 4.2. Let Assumption 2.1(i), Assumption 2.2 and relation (2.2) be satisfied, then

E(S ′′n )2≤ Cγ1n, E(S ′′′n )

2≤ Cγ2n. (4.1)

Further, if Assumption 2.1(ii) holds, then

E(S ′′n )2+δ≤ Cγ 1+δ/21n , E(S ′′′n )

2+δ≤ Cγ 1+δ/22n . (4.2)

Proof. By the first inequality in Lemma A.1, we have

E(S ′′n )2≤ C

k∑j=1

lj+q−1∑i=lj

σ−2n

(∫AiEm(t, s)ds

)2≤ C

k∑j=1

lj+q−1∑i=lj

∣∣∣∣∫AiEm(t, s)ds

∣∣∣∣≤ Ckq2mn−1 = C2mq(p+ q)−1 = Cγ1n,

E(S ′′′n )2≤ C

n∑i=k(p+q)+1

σ−2n

(∫AiEm(t, s)ds

)2≤ C

n∑i=k(p+q)+1


∣∣∣∣≤ C[n− k(p+ q)]2mn−1 ≤ C[n(p+ q)−1 − k](p+ q)2mn−1 ≤ C2mpn−1 = Cγ2n.

Thus (4.1) holds. Analogously, we obtain (4.2) by the second inequality in Lemma A.1.

Lemma 4.3. Under Assumptions 2.1 and 2.2,

|s2n − 1| ≤ C(γ1/21n + γ

1/22n + u(q)). (4.3)

Proof. Let Γn =∑1≤i<j≤k Cov(yni, ynj). Clearly s

2n = E(S

′n)2− 2Γn, and since E(Sn)2 = 1, by Lemma 4.2, we have

|E(S ′n)2− 1| = |E(S ′′n + S

′′′

n )2− 2E[Sn(S ′′n + S

′′′

n )]| ≤ C(γ1/21n + γ

1/22n ). (4.4)

On the other hand, by Lemmas 4.1 and A.5,

|Γn| ≤∑1≤i<j≤k

ki+p−1∑µ=ki

kj+p−1∑ν=kj

|Cov(Znµ, Znν)|

≤

∑1≤i<j≤k

ki+p−1∑µ=ki

kj+p−1∑ν=kj

σ−2n

∣∣∣∣∣∫AµEm(t, s)ds

∫AνEm(t, s)ds

∣∣∣∣∣ · |Cov(ξµ, ξν)|≤ C

k−1∑i=1

ki+p−1∑µ=ki


∣∣∣∣∣ · supj≥1 ∑t:|t−j|≥q

|Cov(ξj, ξt)| ≤ Cu(q). (4.5)

Thus, (4.4) and (4.5) imply the conclusion.

Now, in order to establish the uniformly asymptotic normality, similarly to Yang (2003) we assume that ηnj : j =1, . . . , k are independent random variables, and the distribution of ηnj is the same as that of ynj for j = 1, . . . , k. Let Tn =∑kj=1 ηnj, Bn =

∑kj=1 Var(ηnj). Fn(u),Gn(u) and Gn(u) are the distributions of S

′n, Tn/√Bn and Tn respectively. Clearly Bn = s2n,

Gn(u) = Gn(u/sn).

Lemma 4.4. Under the same conditions as those used in Lemma 4.3, we have

supu|Gn(u)− Φ(u)| ≤ Cγ

δ/22n . (4.6)

Proof. By Lemmas A.1, A.5 and 4.1, we have

k∑j=1

E|ηnj|2+δ ≤ Ck∑j=1

kj+p−1∑i=kj

σ−(2+δ)n


∣∣∣∣2+δ +kj+p−1∑

i=kj

σ−2n

(∫AiEm(t, s)ds

)21+δ/2≤ C(2mn−1)δ/2 + pδ/2(2mn−1)δ/2 ≤ C(2mpn−1)δ/2 = Cγ δ/22n .

Since Bn = s2n → 1 by Lemma 4.3, then B−2n∑kj=1 E|ηnj|

2+δ≤ Cγ δ/22n . Applying the Berry–Esseen theorem, we get (4.6).


As for J2n, J3n, by Lemmas 4.4 and 4.3, respectively, we obtain

J2n = supu|Gn(u/

√Bn)− Φ(u/

√Bn)| = sup

u|Gn(u)− Φ(u)| ≤ Cγ

δ/22n , (4.12)

J3n ≤ C |s2n − 1| ≤ Cγ1/21n + γ

1/22n + u(q)

. (4.13)

Thus, from (4.10)–(4.13) it follows that

supu|Fn(u)− Φ(u)| ≤ C

γ1/21n + γ

1/22n + γ

δ/22n + u

1/3(q). (4.14)

Using Lemma A.4, we get the result of the theorem from (2.2), (4.1) and (4.14).

Proof of Corollary 2.2. If n−12m = O(n−θ ), δ = 1, then taking p = [nρ], q = [n2ρ−1], we have

γ1/21n = O

(n−(θ−ρ)/2

), γ

1/22n = O

(n−(θ−ρ)/2

), u

13 (q) = O

(q−3(θ−ρ)2(2ρ−1)

) 13

= O(n−(θ−ρ)/2

).

These relations imply (2.5) by Theorem 2.2.

5. Proof of the uniformly asymptotic normality for the PA case

Here, similarly to the NA case, we also present several lemmas for the PA case.

Lemma 5.1. Let Assumptions 2.1 and 2.2(i), (ii) be satisfied, then

Var(gn(t)) = O(22mn−1).

Proof. By the definition of Var(g(t)) and Lemma A.5, we have

Var(g(t)) = σ 2n∑i=1

(∫AiEm(t, s)ds

)2+ O(22mn−1)

∑1≤i<j≤n

Cov(εni, εnj)

= O(2mn−1)+ O(22mn−1)n−1∑i=1

n∑j=i−1

Cov(εni, εnj) = O(22mn−1).

This proves Lemma 5.1.

Lemma 5.2. Let Assumptions 2.1 and 2.2(i), (ii), and relation (2.2) be satisfied, then

E(S ′′n )2≤ Cγ3n, E(S ′′′n )

2≤ Cγ4n. (5.1)

P(|S ′′n | ≥ γ1/33n ) ≤ Cγ

1/33n , P(|S

′′′

n | ≥ γ1/34n ) ≤ Cγ

1/34n . (5.2)

Proof. By stationarity and Lemmas 5.1 and A.5, we have

E(S ′′n )2= σ−2n

k∑j=1

lj+q−1∑i=lj

(∫AiEm(t, s)ds

)2E(εni)2

+ 2k∑j=1

∑lj≤i1<i2≤lj+q−1

∣∣∣∣∣∫Ai1

Em(t, s)ds ·∫Ai2

Em(t, s)dsCov(εni1 , εni2)

∣∣∣∣∣+ 2

∑1≤j<s≤k

lj+q−1∑i1=lj

ls+q−1∑i2=ls

∣∣∣∣∣∫Ai1

Em(t, s)ds ·∫Ai2

Em(t, s)dsCov(εni1 , εni2)

∣∣∣∣∣

≤ C

kq+ k∑j=1

q−1∑i=1

(q− i)Cov(εn1, εni+1)+k∑j=1

lj+q−1∑i1=lj

k∑s=j+1

ls+q−1∑i2=ls

Cov(εn1, εni+1)

/n

≤ C [kq+ kqν(1)+ kqν(p)] /n ≤ Ckq/n = Cqp−1 = Cγ3n


E(S ′′′n )2≤ σ−2n

n∑

i=k(p+q)+1

(∫AiEm(t, s)ds

)2E(εni)2

+ 2∑

k(p+q)+1≤i1<i2≤n

∣∣∣∣∣∫Ai1

Em(t, s)ds ·∫Ai2

Em(t, s)ds

∣∣∣∣∣ |Cov(εni1 , εni2)|

≤ C[(n− k(p+ q))+n−k(p+q)−1∑

i=1

Cov(εn1, εni+1)]/n

≤ C[(n/(p+ q)− k)(p+ q)+ ν(1)]/n = C(p+ q)/n = Cpn−1 = Cγ4n.

This completes the proof of (5.1). From (5.1) and the Markov inequality, (5.2) is verified.

Lemma 5.3. Under Assumptions 2.1 and 2.2(i), (ii), we have

|s2n − 1| ≤ C(γ1/23n + γ

1/24n + ν(p)). (5.3)

Proof. As in the proof of (4.4), by Lemmas 5.1, 5.2 and A.5, we have

|E(S ′n)2− 1| ≤ C(γ 1/23n + γ

1/24n ). (5.4)

Similarly to (4.5), it is clear that

|Γn| ≤ Ck∑i=1

ki+p−1∑s=ki

k∑j=i+1

kj+p−1∑t=kj

Cov(εns, εnt)/n ≤ Ckpν(p)/n ≤ Cν(p). (5.5)

From (5.4) and (5.5), we obtain (5.3).

Lemma 5.4. Under the conditions of Theorem 2.3, we have

supu|Gn(u)− Φ(u)| ≤ Cγ

(r−2)/24n . (5.6)

Proof. By Lemmas A.2, A.5 and 5.1, we have

k∑j=1

E|ηnj|r = σ−rnk∑j=1

E

∣∣∣∣∣∣kj+p−1∑i=kj

εni

∫AiEm(t, s)ds

∣∣∣∣∣∣r

≤ Cσ−rnk∑j=1

(2m/n)rpr/2 = Cγ (r−2)/24n . (5.7)

From (5.7) and Lemma 5.3, it follows that B−2n∑kj=1 E|ηnj|

r≤ γ

(r−2)/24n . Thus applying the Berry–Esseen theorem, we get

(5.6).

Lemma 5.5. Under the same assumptions and conditions as those used in Lemma 5.3, we have

supu|Fn(u)− Gn(u)| ≤ C

ν1/3(q)+ γ (r−2)/24n

.

Proof. Adopting the notation of Lemma 4.5, working as in the proof of Lemma 4.5, by using Lemmas A.3, A.5 and 5.2, wehave

|ϕ(t)− ψ(t)| ≤ Ct2ν(q),∫ T

−T

∣∣∣∣ϕ(t)− ψ(t)t

∣∣∣∣ dt ≤ Cν(q)T 2.From Lemma 5.4 and the method of proving (4.9), it can be seen that

T supu

∫|y|≤c/T

∣∣Gn(u+ y)− Gn(u)∣∣ dy ≤ Cγ (r−2)/24n + 1/T ,

and choosing T = ν−1/3(q), using the Esseen inequality again, we have

supu|Fn(u)− Gn(u)| ≤ Cν1/3(q)+ γ

(r−2)/24n .

This completes the proof of the lemma.


Proof of Theorem 2.3. Letting J1n, J2n, J3n be as in the proof of Theorem 2.2, by Lemmas 5.3 and 5.5, as in the proof ofTheorem 2.2, we have

J1n ≤ Cγ(r−2)/24n + ν1/3(q), J2n ≤ Cγ

(r−2)/24n , J3n ≤ C |s2n − 1| ≤ Cγ

1/23n + γ

1/24n + ν

1/3(q).

These imply that

supu|Fn(u)− Φ(u)| ≤ Cγ

1/23n + γ

1/24n + γ

(r−2)/24n + ν1/3(q). (5.8)

Thus, by Lemma A.3, we get the result of Theorem 2.3 from (5.2), (5.3) and (5.8).

Proof of Corollary 2.3. We obtain it by taking p, q in Theorem 2.3 as follows

p = [n3(δ+1)/(6+4δ)], q = [nδ/(3+2δ)].

Acknowledgements

The authors would like to express their thanks to the reviewers whose comments helped in improving a previous versionof this paper.

Appendix

Lemma A.1 (See Yang (2001)). Let Xj : j ≥ 1 be NA random variables, p > 1 with EXj = 0, E|Xj|p <∞, and aj : j ≥ 1 be asequence of real constants. Then there exists a positive constant Cp which only depends on the given number p such that

E

∣∣∣∣∣ n∑j=1

ajXj

∣∣∣∣∣p

≤ Cpn∑j=1

E|ajXj|p, for 1 < p ≤ 2,

and

E|n∑j=1

ajXj|p ≤ Cp

n∑j=1

E|ajXj|p +

(n∑j=1

E(ajXj)2)p/2 , for p > 2.

Lemma A.2 (See Yang (2005)). Let Xj : j ≥ 1 be PA random variables with EXj = 0, E|Xj|2 < ∞, and let aj : j ≥ 1 be asequence of real constants, a := supj |aj| <∞. If supj≥1 E|ξj|2+δ <∞, v(n) = O(n−(r−2)(r+δ)/(2δ)) for some r > 2, and δ > 0.Then

E

∣∣∣∣∣ n∑j=1

ajXj

∣∣∣∣∣r

≤ Carnr/2.

Lemma A.3 (See Yang (2003, 2005)). Let Xj : j ≥ 1 be a sequence of associated (either PA or NA) random variables, and letaj : j ≥ 1 be a real constant sequence, 1 = m0 < m1 < · · · < mk = n. Denote by Yl :=

∑mlj=ml−1+1

ajXj for 1 ≤ l ≤ k. Then∣∣∣∣∣E exp(it

k∑l=1

Yl

)−

k∏l=1

E exp (itYl)

∣∣∣∣∣ ≤ 4t2 ∑1≤s<j≤n

|asaj||Cov(Xs, Xj)|.

Lemma A.4. Suppose that ζn : n ≥ 1, ηn : n ≥ 1 and ξn : n ≥ 1 are three random variable sequences, γn : n ≥ 1 is apositive constant sequence, and γn → 0. If supu

∣∣Fζn(u)− Φ(u)∣∣ ≤ Cγn, then for any ε1 > 0, and ε2 > 0supu

∣∣Fζn+ηn+ξn(u)− Φ(u)∣∣ ≤ Cγn + ε1 + ε2 + P(|ηn| ≥ ε1)+ P(|ξn| ≥ ε2).Proof. This result is an extension of Lemma 3.7 in Yang (2003).

Lemma A.5. Suppose that Assumption 2.2 holds, then(i) sup0≤t,s≤1 |Em(t, s)| = O(2m).(ii) sup0≤t≤1

∫ 10 |Em(t, s)ds| ≤ C,

∫Ai|Em(t, s)|ds = O( 2

m

n ).

(iii)∑ni=1

∫Ai|Em(t, s)|ds ≤ C, and

∑ni=1(

∫AiEm(t, s)ds)2 = O( 2

m

n ).

Proof. The proofs of (i) and (ii) can be found in Antoniadis et al. (1994), and the proof of (iii) follows from (i) and (ii).


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