A Berry-Esseen type bound of regression estimator based on linear process errors

J. Korean Math. Soc. 45 (2008), No. 6, pp. 1753–1767

A BERRY-ESSEEN TYPE BOUND OF REGRESSION ESTIMATOR BASED ON LINEAR PROCESS ERRORS

Han-Ying Liang and Yu-Yu Li

Reprinted from the

Journal of the Korean Mathematical Society Vol. 45, No. 6, November 2008

⃝2008 The Korean Mathematical Societyc

A BERRY-ESSEEN TYPE BOUND OF REGRESSION ESTIMATOR BASED ON LINEAR PROCESS ERRORS

Han-Ying Liang and Yu-Yu Li

Abstract. Consider the nonparametric regression model Yni= g(xni) + ϵni (1≤ i ≤ n), where g(·) is an unknown regression function, xni are known fixed design points, and the correlated errors{ϵni, 1≤ i ≤ n} have the same distribution as{Vi, 1≤ i ≤ n}, here Vt=P_∞

j=−∞ψjet−jwith P_∞

j=−∞|ψj| < ∞ and {et} are negatively associated random variables.

Under appropriate conditions, we derive a Berry-Esseen type bound for the estimator of g(·). As corollary, by choice of the weights, the Berry- Esseen type bound can attain O(n^−1/4(log n)^3/4).

1. Introduction Consider the nonparametric regression model

(1.1) Y

= g(x

) + ϵ

, i = 1, . . . , n,

where g is an unknown regression function defined on A, x

are known fixed design points, and ϵ

are random errors. As an estimate of g, we consider the following weighted regression estimator:

(1.2) g

(x) =

∑

w

Y

, x ∈ A, where w

= w

(x) are weight functions.

The above estimator was first proposed by Georgiev [8] and subsequently have been studied by many authors. For instance, when ϵ

are assumed to be independent, consistency and asymptotic normality have been studied by Georgiev and Greblicki [10], Georgiev [9] and M¨ uller [17] among others. Results for the case when ϵ

are dependent have also been studied by various authors.

Fan [7] extended the work of Georgiev [9] and M¨ uller [17] in the estimation of the regression model to the case where {ϵ

} form an L

-mixingale sequence for some 1 ≤ q ≤ 2. Roussas [19] discussed strong consistency and quadratic mean consistency for g

(x) under mixing conditions. Roussas et al. [22] established

Received May 15, 2007; Revised August 27, 2007.

2000 Mathematics Subject Classification. 62G08.

Key words and phrases. nonparametric regression model, negatively associated random variable, Berry-Esseen type bound.

⃝2008 The Korean Mathematical Societyc 1753

asymptotic normality of g

(x) assuming that the errors are from a strictly stationary stochastic process and satisfy the strong mixing condition. Tran et al [26] discussed again asymptotic normality of g

(x) assuming that the errors form a linear time series, more precisely, a weakly stationary linear process based on a martingale difference sequence.

In this paper, we consider the model (1.1) and assume the following form for {ϵ

}:

(A1) For each n, {ϵ

, 1 ≤ i ≤ n} have the same distribution as V

, . . . , V

, where V

= ∑

j=−∞

ψ

e

and {e

} are identically distributed, neg- atively associated random variables with Ee

= 0. Here {ψ

} is a sequence of real numbers with ∑

j=−∞

|ψ

| < ∞.

Here, a finite family of random variables {X

, 1 ≤ i ≤ n} is said to be negatively associated (NA) if, for every pair of disjoint subsets A and B of {1, 2, . . . , n}, we have

Cov(f

(X

, i ∈ A), f

(X

, j ∈ B)) ≤ 0,

whenever f

and f

are coordinatewise increasing provided the covariance ex- ists. An infinite family of random variables is said to be NA, if every finite subfamily is NA.

The notion of negative association was first introduced by Alam and Saxena

[1]. Joag-Dev and Proschan [11] showed that many well known multivari-

ate distributions possess the NA property. Examples include (a) multinomial,

(b) convolution of different multinomials, (c) multivariate hypergeometric, (d)

Dirichlet, (e) Dirichlet compound multinomial, (f) negatively correlated nor-

mal distribution, (g) permutation distribution, (h) random sampling without

replacement, and (i) joint distribution of ranks. The significance of NA, how-

ever, seems to come from the perception that it is an appropriate model when

several species compete for the same limited resources. Because of its wide

applications in multivariate statistical analysis and systems reliability, the no-

tion of NA has recently received considerable attention. We refer to Joag-Dev

and Proschan [11] for fundamental properties, Matula [16] for the three se-

ries theorem, Shao [23] for the Rosenthal-type inequality and the Kolmogorov

exponential inequality, and Su et al. [25] for a moment inequality and weak

convergence, Shao and Su [24] for the law of the iterated logarithm, Liang and

Su [15] and Liang [12] as well as Baek et al. [2] for complete convergence, Liang

and Baek [13] for some strong law, Roussas [20] studied the central limit theo-

rems for weak stationary NA random fields. Asymptotic properties of estimates

related to NA samples have also been studied by some authors. Cai and Rous-

sas [3] gave uniform strong consistency, convergence rates and the asymptotic

distribution of the Kaplan-Meier estimator for observations under randomly

censored failure times. In Cai and Roussas [4], they established Berry-Esseen

bounds for a smooth estimate of the distribution function; Roussas [21] derived

the asymptotic normality of the kernel estimate of the probability density func-

tion; Chen et al. [6] studied strong consistency of estimator in heteroscedastic

regression model under NA samples; Yang [27] studied uniformly asymptotic normality of the regression weighted estimator for NA samples; Liang and Jing [14] discussed the asymptotic properties of g

(x) under NA setting. In par- ticular, Liang and Jing [14] studied the asymptotic normality of g

(x) under assumption (A1) for V

= ∑

j=0

ψ

e

setting.

In this paper, we further investigate model (1.1) and derive a Berry-Esseen type bound for the estimator g

(x) of g( ·) under the errors {ϵ

} satisfy as- sumption (A1). By choice of the weights, the Berry-Esseen type bound can attain O(n

(log n)

).

The layout of the paper is as follows. The main result is presented in Sec- tion 2. In Section 3, some preliminary lemmas are given. The proofs of the main result and preliminary lemmas are provided in Sections 4 and 5, respectively.

In the sequel, let C, c, c

, . . . denote generic finite positive constants, whose values are unimportant and may change from line to line. For random variables X and Y , X = Y means that the distribution of X is the same as that of Y .

All limits are taken as the sample size n tends to ∞, unless specified otherwise.

2. The main result

In order to formulate the main result, we now give the following assumptions.

(A2) The weights satisfy ∑

i=1

|w

(x) | ≤ C, w

:= max

|w

(x) | = O(σ

(x)), where σ

(x) := Var(g

(x)) > 0.

(A3) There exist positive integers p := p(n) and q := q(n) such that for sufficiently large n, p + q ≤ 3n, qp

≤ c < ∞.

Let γ

→ 0 (i = 1, 2, 3) and u(q) → 0, where γ

= nqp

w

, γ

= pw

, γ

= n( ∑

|j|>n

|ψ

|)

and u(q) = sup

∑

|i−j|≥q

|Cov(e

, e

) |.

Our main result is as follows.

Theorem 2.1. Suppose that (A1)-(A3) are satisfied. If E |e

|

< ∞ for some δ > 0, then, for each x ∈ A, we have

sup

u

|P (σ

(x) {g

(x) − Eg

(x) } ≤ u) − Φ(u)| = O(a

),

where a

= (γ

+γ

)(log n)

+γ

+γ

+u

(q)+n

+((nqp

)

+ p

)(log n)

→ 0, Φ(u) represents the standard normal distribution function.

Corollary 2.1. Suppose that (A1)-(A2) are satisfied. Let sup

n≥1

n

(log n)

∑

|j|>n

|ψ

| < ∞

and u(n) = O(n

e

). If E |e

|

< ∞ and w

= O(n

), then for each x ∈ A, we have

sup

u

|P (σ

(x) {g

(x) − Eg

(x) } ≤ u) − Φ(u)| = O(n

(log n)

).

Corollary 2.2. Suppose that (A1) is satisfied, and that spectral density func- tion τ (w) of V

is bounded away from zero and infinity, i.e., 0 < c

≤ τ(w) <

∞, for w ∈ (π, π]. Let sup

n

(log n)

∑

|j|>n

|ψ

| < ∞ and u(n) = O(n

e

). If E |e

|

< ∞, w

= O(n

) and n ∑

i=1

w

(x) ≥ c

> 0, then, for each x ∈ A, we have

sup

u

|P (σ

(x) {g

(x) − Eg

(x) } ≤ u) − Φ(u)| = O(n

(log n)

).

Remark 2.1. (a) In Theorem 2.1, we use u(q) → 0, which is easily satisfied.

For example:

(i) If u(1) < ∞ (cf. Roussas [21]), then u(q) → 0 as q → ∞.

(ii) For stationary NA sequence, Cai and Roussas [4] use the covari- ance coefficient: u

(n) := ∑

j=n

|cov(e

, e

) |

and u

(1) < ∞.

In this case, we have |cov(e

, e

) | = o(j

). Hence

u(q) :=

∑

|cov(e

, e

) | = O( ∑

j=q

|cov(e

, e

) |

j

) = O(q

).

(b) In Roussas et al. [22], the weights are required to satisfy the con- ditions: ∑

i=1

|w

(x) | ≤ C, max

|w

(x) | = O( ∑

i=1

w

(x)),

∑

i=1

w

(x) = O(σ

(x)). Clearly, these conditions imply Assump- tion (A2).

In addition, if we choose

w

(x) = x

− x

h

K( x − x

h

),

where {h

} is a sequence of positive constants tending to 0 and nh

→

∞, and the design points satisfy 0 = x

≤ x

≤ · · · ≤ x

= 1, this weight also was used by Tran et al. [26]. Assume that

(iii) there exist positive constants c

and c

such that c

n

≤ x

− x

≤ c

n

for i = 1, 2, . . . , n,

(iv) K(x) is nonnegative, bounded and continuous almost everywhere on R and has a majorant; that is, K(x) ≤ H(x) all x ∈ R, where H is symmetric, bounded, nonincreasing on [0, ∞) with ∫

H(y)dy <

∞, where the integral is over R.

(v) {ϵ

} are stationary with Eϵ

= 0 and its spectral density func- tion f (w) is bounded away from zero and infinity, i.e., 0 < c

≤ f (w) < ∞ for w ∈ (π, π].

Note that (v) implies that

σ

(x) = E[

∑

w

(x)ϵ

]

=

∫

−π

f (w) |

∑

w

(x)e

|

dw ≥ c

∑

w

(x).

While, by using Lemma 3.1 in Tran et al. [26], (iii) and (iv) im- ply max

|w

(x) | ≤ C/nh

, ∑

i=1

w

(x) ≤ C and 1

h

∑

(x

− x

)K

( x − x

h

) →

∫

−∞

K

(u)du > 0.

Thus σ

(x) ≥ c

∑

i=1

w

(x) ≥

. Hence max

|w

(x) | = o(σ

(x)), which shows that (A2) is mild.

(c) In Roussas et al. [22] (cf. (2.21) there), they assume that nqp

∑

w

(x) → 0, p

∑

w

(x) → 0.

Obviously, the limits above are stronger than γ

→ 0 and γ

→ 0 here, which were also used by Yang [27]. Under some regularity conditions, γ

→ 0 holds with the usual AR, MA and ARMA processes which are extensively used to model serially correlated data.

3. Some preliminary lemmas We write σ

= σ

(x) and

S

:= σ

{g

(x) − Eg

(x) } = σ

∑

w

V

= σ

∑

w

∑

ψ

e

+ σ

∑

w

∑

|j|>n

ψ

e

:= S

+ S

.

(3.3) Note that

S

= σ

∑

w

∑

ψ

e

=

∑

σ

(

∑

i=max{1,l−n}

w

ψ

)

e

:=

∑

σ

t

e

.

Set Z

= σ

t

e

, l = 1 − n, 2 − n, . . . , 2n. Then S

= ∑

l=1−n

Z

. Let k = [

] and

(3.4) S

= S

+ S

+ S

, where

 



 

S

= ∑

m=1

y

, y

= ∑

Z

, S

= ∑

m=1

y

, y

= ∑

Z

, S

= y

, y

= ∑

i=k(p+q)−n+1

Z

,

k

= (m − 1)(p + q) + 1 − n, l

= (m − 1)(p + q) + p + 1 − n, m = 1, . . . , k.

From (3.3) and (3.4) we have

(3.5) S

= S

+ S

+ S

+ S

.

Lemma 3.1. Suppose that (A1)-(A3) are satisfied. If Ee

< ∞, then E(S

)

≤ Cγ

, E(S

)

≤ Cγ

, E(S

)

≤ Cγ

. Lemma 3.2. Suppose that (A1)-(A3) are satisfied. Set s

= ∑

m=1

Var(y

).

If Ee

< ∞, then

|s

− 1| ≤ C(γ

+ γ

+ γ

+ u(q)).

Let η

, m = 1, 2, . . . , k be independent random variables and η

= y

for m = 1, 2, . . . , k. Put T

= ∑

m=1

η

.

Lemma 3.3. Under the assumptions of Theorem 2.1, we have sup

u

|P (T

/s

≤ u) − Φ(u)| ≤ Cγ

. Lemma 3.4. Under the assumptions of Theorem 2.1, we have

sup

u

|P (S

≤ u) − P (T

≤ u)| ≤ C{γ

+ u

(q) }.

Lemma 3.5. Under the assumptions of Theorem 2.1, we have P ( |S

| > cµ

) ≤ C{n

+ (nqp

)

(log n)

}, (3.6)

P ( |S

| > cν

) ≤ C{n

+ p

(log n)

}, (3.7)

P ( |S

| > τ

) ≤ Cγ

, (3.8)

where µ

= γ

(log n)

, ν

= γ

(log n)

, τ

= γ

.

Lemma 3.6. Let X and Y be random variables. Then for any a > 0 sup

u

|P (X + Y ≤ u) − Φ(u)| ≤ sup

u

|P (X ≤ u) − Φ(u)| + a

√ 2π + P ( |Y | > a).

The proof of Lemma 3.6 can be found in Chang and Rao [5].

4. Proof of Theorem 2.1 We observe that

sup

u

|P (S

≤ u) − Φ(u)|

≤ sup

u

|P (S

≤ u) − P (T

≤ u)| + sup

u

|P (T

≤ u) − Φ( u s

) | + sup

u

|Φ( u

s

) − Φ(u))|

:= J

+ J

+ J

.

From Lemma 3.4 we have J

≤ C(γ

+u

(q)), Lemma 3.3 yields that J

≤ Cγ

, according to Lemma 3.2 we obtain that

J

= |Φ(u/s

) − Φ(u))|

≤ C|s

− 1|/s

≤ C|s

− 1| ≤ C(γ

+ γ

+ γ

+ u(q)).

Therefore sup

u

|P (S

≤ u) − Φ(u)| ≤ C{γ

+ γ

+ γ

+ γ

+ u

(q) } and from Lemmas 3.5 and 3.6 we have

sup

u

|P (S

≤ u) − Φ(u)|

≤ sup

u

|P (S

+ S

+ S

+ S

≤ u) − Φ(u)|

≤ sup

u

|P (S

≤ u) − Φ(u)| + (µ

+ ν

+ τ

)/ √ 2π + P ( |S

+ S

+ S

| > µ

+ ν

+ τ

)

≤ C{γ

+ γ

+ γ

+ γ

+ u

(q) } + µ

+ ν

+ τ

+ P ( |S

| > µ

) + P ( |S

| > ν

) + P ( |S

| > τ

)

= O (

(γ

+ γ

)(log n)

+ γ

+ γ

+ u

(q) + n

+ ((nqp

)

+ p

)(log n)

)

= O(a

).

Proof of Corollary 2.1. In Theorem 2.1, taking δ = 1, p = [n

(log n)

], q = [log n], then

u(q) = O(n

(log n)

) by u(n) = O(n

e

);

γ

= n

(log n)

(

n

(log n)

∑

|j|>n

|ψ

| )

= O(n

(log n)

) by sup

n

(log n)

∑

|j|>n

|ψ

| < ∞; γ

= γ

= n

(log n)

by w

= O(n

).

Therefore, the conclusion follows from Theorem 2.1. ¤ Proof of Corollary 2.2. Note that

σ

(x) = E(

∑

w

V

)

=

∫

−π

τ (w) ¯¯ ¯

∑

w

e

¯¯ ¯

dw ≥ C

∑

w

.

Hence, from n ∑

i=1

w

(x) ≥ c

> 0 and w

= O(n

), it follows that w

≤ C

∑

w

(x) = O(σ

(x)) and

∑

|w

(x) | ≤ C,

i.e., (A2) is satisfied. The rest proofs are the same as the proof of Corollary

2.1. ¤

5. Proofs of lemmas

Lemma 5.1. Let {X

, 1 ≤ j ≤ n} be NA random variables with EX

= 0 and E |X

|

< ∞ for some p > 1, and let {a

, j ≥ 1} be a sequence of real numbers.

Then, there exist A

> 0 and B

> 0 such that E max

| ∑

j=1

a

X

|

≤ A

∑

j=1

E |a

X

|

for 1 < p ≤ 2 and E max

1≤k≤n

¯¯ ¯¯ ∑

j=1

a

X

¯¯

¯¯

≤ B

{ ∑

j=1

E |a

X

|

+ ( ∑

j=1

E(a

X

)

)

}

for p > 2.

From Theorem 2 of Shao [23], it is easy to prove Lemma 5.1.

Lemma 5.2 (Yang [27]). Suppose that {X

, j ≥ 1} is a sequence of NA random variables.

(a) Let {a

, j ≥ 1} be a sequence of real numbers, 1 = m

< m

< · · · <

m

= n. Denote by Y

:= ∑

j=m_l−1+1

a

X

for 1 ≤ l ≤ k. Then

¯¯ ¯¯E exp { it

∑

Y

}

−

∏

exp {itY

} ¯¯

¯¯ ≤ 4t

∑

1≤s<j≤n

|a

a

||Cov(X

, X

) |.

(b) Let EX

= 0 and |X

| ≤ b a.s. Set ∆

= ∑

j=1

EX

. Then, for any ϵ > 0,

P ( |

∑

X

| ≥ ϵ) ≤ 2 exp {

− ϵ

2(2∆

+ 2bϵ) }

.

Proof of Lemma 3.1. According to the definition of S

, from Lemma 5.1 and (A1)-(A2) we have

E(S

)

= E(

∑

l_m

∑

Z

)

≤ C

∑

lm

∑

E(Z

)

= C

∑

lm

∑

σ

t

Ee

≤ C

∑

l_m

∑

σ

(

∑

j=max{1, i−n}

w

ψ

)

≤ C

∑

l_m

∑

w

(

∑

j=max{1, i−n}

|ψ

| )

≤ Ckqw

( ∑

j=−∞

|ψ

| )

≤ Cnqp

w

= Cγ

. Similarly,

E(S

)

= E

( ∑

i=k(p+q)−n+1

Z

)

≤ C

∑

σ

t

Ee

≤ C

∑

w

(

∑

j=max{1, i−n}

|ψ

| )

≤ C[3n − k(p + q)]w

( ∑

j=−∞

|ψ

| )

≤ Cpw

= Cγ

. As to S

, (A2) and Ee

< ∞ yield that

ES

= E ¯¯

¯¯σ

∑

i₁=1

w

∑

|j1|>n

ψ

e

¯¯ ¯¯ ¯¯

¯¯σ

∑

i₂=1

w

∑

|j2|>n

ψ

e

¯¯ ¯¯

≤ CE { ∑

i1=1

|w

|

∑

¯¯ ¯¯ ∑

|j1|>n

ψ

e

¯¯ ¯¯ ¯¯

¯¯ ∑

|j2|>n

ψ

e

¯¯ ¯¯ }

≤ Cn ( ∑

|j|>n

|ψ

| )

= Cγ

.

¤ Proof of Lemma 3.2. Let Γ

= ∑

1≤i<j≤k

Cov(y

, y

), then s

= E(S

)

− 2Γ

. Note that ES

= 1, so

E(S

)

= E[S

− (S

+ S

+ S

)]

= 1 + E(S

+ S

+ S

)

− 2ES

(S

+ S

+ S

).

Hence, from Lemma 3.1 we have

(5.9)

|E(S

)

− 1|

≤ E(S

+ S

+ S

)

+ 2 |ES

(S

+ S

+ S

) |

≤ E(S

+ S

+ S

)

+ 2(ES

)

[E(S

+ S

+ S

)

]

≤ {[E(S

)

]

+ [E(S

)

]

+ (ES

)

}

+ 2 {[E(S

)

]

+ [E(S

)

]

+ (ES

)

}

≤ (γ

+ γ

+ γ

)

+ 2(γ

+ γ

+ γ

)

≤ C(γ

+ γ

+ γ

).

On the other hand, (5.10)

|Γ

|

= _| ^∑

1≤i<j≤k

Cov(y

, y

) |

≤ ^∑

1≤i<j≤k ki

∑

s=ki

kj

∑

t=kj

|Cov(Z

, Z

)|

≤

^∑

i=1 ki+p

∑

s=ki

∑

j=i+1 kj

∑

t=kj

min{n,s+n}

∑

u=max{1,s−n}

min{n, t+n}

∑

v=max{1,t−n}

σ

|w

w

||ψ

ψ

||Cov(e

, e

)|

≤ ^C

k

∑

i=1 ki

∑

s=ki

min{n,s+n}

∑

u=max{1,s−n}

|ψ

||w

|

∑

j=i+1 kj

∑

t=kj

| Cov(e

, e

) |

min{n,t+n}

∑

v=max{1,t−n}

|ψ

|

≤ ^C

k

∑

i=1 k_i

∑

s=ki

min{n,s+n}

∑

u=max{1,s−n}

|ψ

||w

| ∑

t:|t−s|≥q

|Cov(e

, e

) |

≤ ^Cu(q)

^∑

i=1 ki

∑

s=ki

∑

u=1

|w

||ψ

|

≤ Cu(q).

Therefore, (5.9) and (5.10) follow that

|s

− 1| ≤ C(γ

+ γ

+ γ

+ u(q)).

¤ Proof of Lemma 3.3. By using Berry-Esseen inequality (cf. Petrov [18], p. 154, Theorem 5.7) we have

sup

u

|P (T

/s

≤ u) − Φ(u)| ≤ C

∑

E |η

|

/s

.

While, according to Lemma 5.1, from (A1) and (A2) it follows that

∑

E |η

|

≤ C

∑

{

∑

j=k_m

¯¯ ¯¯

∑

i=max{1,j−n}

σ

w

ψ

¯¯ ¯¯

E |e

|

+

[

∑

j=k_m

(

∑

i=max{1,j−n}

σ

w

ψ

)

Ee

]

}

≤ Cσ

∑

km

∑

min{n, j+n}

∑

i=max{1, j−n}

|w

||ψ

| · ¯¯

¯¯

∑

i=max{1, j−n}

w

ψ

¯¯ ¯¯

+ C

∑k m=1

[^km∑+p−1 j=km

( min{n,j+n}∑

i=max{1,j−n}

σn⁻¹wniψi−j

)2

·(^km∑^+p−1

j=km

( min{n,j+n}∑

i=max{1,j−n}

σn⁻¹wniψi−j

)2)δ/2]

≤ C {

w

∑

|w

| ( ∑

m=1 km

∑

j=k_m

|ψ

| )

+ (pw

)

∑

m=1

[

∑

(

∑

i=max{1,j−n}

|w

ψ

| )

·

(

∑

i=max{1,j−n}

σ

|w

ψ

| )]}

≤ C{w

+ (pw

)

}

≤ C(pw

)

= Cγ

.

Note that Lemma 3.2 implies s

→ 1 and therefore sup

u

|P (T

/s

≤ u) − Φ(u)| ≤ Cγ

.

¤

Proof of Lemma 3.4. Assume that φ(t) and ψ(t) are the characteristic function of S

and T

, respectively. By Esseen inequality (cf. Petrov [18], p. 146, Theorem 5.3), for any T > 0

(5.11)

sup

u

|P (S

≤ u) − P (T

≤ u)|

≤

∫

−T

| φ(t) − ψ(t)

t |dt

+ T sup

u

∫

|u|≤^C_T

|P (T

≤ u + y) − P (T

≤ u)|dy

:= I

+ I

.

According to Lemma 5.2(a), following the line as in (5.10) we have

¯¯ ¯¯ φ(t) − ψ(t) t

¯¯ ¯¯

= |E exp{itS

} − Π

E exp {ity

}|

≤ 4t

i=1 ki+p−1∑

s=ki

∑k j=i+1

kj∑+p−1 t=kj

min{n, s+n}∑

u=max{1, s−n}

min{n, t+n}∑

v=max{1, t−n}

σ_n⁻²| wnuwnv|| ψu−sψ_v−t|| Cov(es, et)|

≤ Ct

u(q).

Therefore I

≤ Cu(q)T

. On the other hand, from Lemmas 3.2 and 3.3 it follows that

sup

u

|P (T

≤ u + y) − P (T

≤ u)|

≤ sup

y

|P (T

/s

≤ (u + y)/s

) − Φ((u + y)/s

) | + sup

u

|P (T

/s

≤ u/s

) − Φ(u/s

) | + sup

u

|Φ((u + y)/s

) − Φ(u/s

) |

≤ C{γ

+ |y|/s

} ≤ C{γ

+ |y|}.

Therefore I

≤ C(γ

+ 1/T ) and choosing T = u

(q) from (5.11) we obtain that

sup

u

|P (S

≤ u) − P (T

≤ u)| ≤ C(γ

+ u

(q)).

¤ Proof of Lemma 3.5. Note that

S

=

∑

lm

∑

σ

t

e

=

∑

lm

∑

σ

(t

− t

)e

,

where a

= max {0, a}, a

= min {0, −a}, so, without loss of generality, we assume that t

≥ 0 for j ≥ 1 and n ≥ 1. Let

α

= −b

I(Z

< −b

) + Z

I( |Z

| ≤ b

) + b

I(Z

> b

), α

= (Z

+ b

)I(Z

< −b

) + (Z

− b

)I(Z

> b

), β

= α

− Eα

, β

= α

− Eα

,

where b

> 0, which will be specialized later. It is clear that {β

, j ≥ 1} and {β

, j ≥ 1} are mean zero NA sequences from the definition of α

and α

.

Firstly, we prove (3.6). Taking b

= γ

(log n)

. Set

H

=

∑

lm

∑

β

, H

=

∑

lm

∑

β

. Then S

= H

+ H

and

(5.12) P ( |S

| > cµ

) ≤ P (|H

| > cµ

/2) + P ( |H

| > cµ

/2).

To evaluate P ( |H

| > cµ

/2), we shall use Lemma 5.2(b). Note that |β

| ≤ 2b

and the proof of Lemma 3.1 shows that

∆

:=

∑

lm

∑

E(β

) ≤

∑

lm

∑

E(Z

) ≤ Cγ

. Then, by using Lemma 5.2(b), for large c > 0 we have

(5.13) P

( |H

| > cµ

2

) ≤ 2 exp {

− c

µ

8(2∆

+ b

µ

)

}

≤ 2 exp{−cc

log n } ≤ Cn

. We observe that

E(β

)

≤ E(α

)

≤ 2E(Z

+ b

)I( |Z

| > b

)

≤ 4b

E |Z

|

≤ C(σ

|t

|)

b

. Hence, by using Lemma 5.1 we have

P

( |H

| > cµ

2

) ≤ C E(H

)

µ

≤ Cµ

∑

lm

∑

(σ

|t

|)

b

≤ C γ

w

µ

b

≤ C(nqp

)

(log n)

. (5.14)

Therefore, (5.13) and (5.14) follow (3.6).

Next we prove (3.7). Following the line as for (3.6). Choosing b

= γ

(log n)

.

Set Q

= ∑

j=k(p+q)−n+1

β

, Q

= ∑

j=k(p+q)−n+1

β

. Hence S

= Q

+ Q

. Note that

∆

:=

∑

Eβ

≤

∑

EZ

≤ Cγ

. So, by using Lemma 5.2(b), for large c > 0, we have

(5.15) P

( |Q

| > cν

2

) ≤ 2 exp {

− c

ν

8(2∆

+ b

ν

)

}

≤ C exp{−cc

log n } ≤ Cn

. From E(β

)

≤ C(σ

|t

|)

b

, and using Lemma 5.1 we have

P

( |Q

| > cν

2

) ≤ C E(Q

)

ν

≤ Cν

∑

(σ

|t

|)

b

≤ C γ

w

ν

b

≤ Cp

(log n)

. (5.16)

Therefore, (5.15) and (5.16) yield (3.7).

As to (3.8), choosing τ

= γ

= n

( ∑

|j|>n

|ψ

|)

. In view of Lemma 3.1 we have

P ( |S

| > τ

) ≤ E(S

)

/τ

≤ Cγ

.

¤ Acknowledgments. This research was supported by the National Natural Science Foundation of China (10571136, 10871146). The authors are grate- ful to anonymous referee for providing detailed list of comment which greatly improved the presentation of the paper.

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Han-Ying Liang

Department of Mathematics Tongji University

Shanghai 200092, P. R. China E-mail address: [email protected]

Yu-Yu Li

Department of Mathematics Tongji University

Shanghai 200092, P. R. China