Berry-Esseen bounds for wavelet estimator in semiparametric regression model with linear process errors

Chengdong Wei; Yongming Li

doi:10.1186/1029-242X-2012-44

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Berry-Esseen bounds for wavelet estimator in semiparametric regression model with linear process errors

Chengdong Wei^{1, 2} and
Yongming Li³Email author

Journal of Inequalities and Applications20122012:44

https://doi.org/10.1186/1029-242X-2012-44

Received: 25 September 2011

Accepted: 27 February 2012

Published: 27 February 2012

Abstract

Consider the semiparametric regression model Y_i = x_i β + g (t_i ) + ε_i , i = 1, . . . , n, where the linear process errors $ε_{i} = \sum_{j = - \infty}^{\infty} a_{j} e_{i - j}$ with $\sum_{j = - \infty}^{\infty} |a_{j}| < \infty$ , and {e_i } are identically distributed and strong mixing innovations with zero mean. Under appropriate conditions, the Berry-Esseen type bounds of wavelet estimators for β and g(·) are established. Our results obtained generalize the results of nonparametric regression model by Li et al. to semiparametric regression model.

Mathematical Subject Classification: 62G05; 62G08.

Keywords

wavelet estimator semiparametric regression strong mixing Berry-Esseen bound

1 Introduction

Regression analysis is one of the most mature and widely applied branches of statistics. For a long time, however, its main theory has concerned parametric and nonparametric regressions. Recently, semiparametric regressions have received more and more attention. This is mainly because semiparametric regression reduces the high risk of misspecification relating to a fully parametric model and avoids some serious drawbacks of fully nonparametric methods.

In 1986, Engle et al. [1] first introduced the following semiparametric regression model:

Y_{i} = x_{i} β + g (t_{i}) + ε_{i}, i = 1, . . ., n,

(1.1)

where β is an unknown parameter of interest, {(x_i , t_i )} are nonrandom design points, {y_i } are the response variables, g(·) is an unknown function defined on the closed interval [0, 1], and {ε_i } are random errors.

The model (1.1) has been extensively studied. When the errors {ε_i } are independent and identically distributed (i.i.d.) random variables, Chen and Shiah [2], Donald and Dewey [3], and Hamilton and Truong [4] used various estimation methods to obtain estimators of the unknown quantities in (1.1) and discussed the asymptotic properties of these estimators. When {ε_i } are MA (∞) errors with the form $ε_{i} = \sum_{j = 0}^{\infty} a_{j} e_{i - j}$ , where {e_i } are i.i.d. random variables,{a_j } satisfy $\sum_{j = - \infty}^{\infty} |a_{j}| < \infty$ and ${sup}_{n} n \sum_{j = n}^{\infty} |a_{j}| < \infty,$ the law of the iterated logarithm for the semiparametric least square estimator (SLSE) of β and strong convergence rates of the nonparametric estimator of g(·) were discussed by Sun et al. [5]. The Berry-Esseen type bounds for estimators of β and g(·) in model (1) under the linear process errors $ε_{i} = \sum_{j = - \infty}^{\infty} a_{j} e_{i - j}$ with identically distributed and negatively associated random variables {e_i } were derived by Liang and Fan [6].

Let us now recall briefly the definition of strong-mixing dependence. A sequence {e_i , i ∈ Z} is said to be strong mixing (or α-mixing) if α(n) → 0 as n → ∞, where α(n) = sup $\{|P (A B) - P (A) P (B)| : A \in ℱ_{- \infty}^{m}, B \in ℱ_{m + n}^{\infty}\}$ , and $ℱ_{n}^{m}$ denotes the σ-field generated by {e_i : m ≤ i ≤ n}.

For the properties of strong-mixing, one can read the book of Lin and Liu [7]. Recently, Yang and Li [8–10] and Xing et al. [11–13] established moment bounds and maximal moment inequality for partial sums for strong mixing sequences and their application. In this article, we study the Berry-Esseen type bounds for wavelet estimators of β and g(·) in model (1.1) based linear process errors {ε_i } satisfying the following basic assumption (A1). Our results obtained generalize the results in [14] to semiparametric regression model.

(A1) (i) Let

ε_{i} = \sum_{j = - \infty}^{\infty} a_{j} e_{i - j}

, where

\sum_{j = - \infty}^{\infty} |a_{j}| < \infty

, {e_j , j = 0, ± 1, ± 2, . . .} are identically distributed and strong mixing random variables with zero mean.

(ii)

For δ > 0, E|e ₀|^2+δ< ∞ and mixing coefficients α(n) = O(n ^-λ) for λ > (2 + δ)/δ.

Now, we introduce wavelet estimators of β and g for model (1.1). Let β be given, since Ee_i = 0, we have g(t_i ) = E(y_i - x_i β), i = 1, . . . , n. Hence a natural estimator of g(·) is

g_{n} (t, β) = \sum_{j = 1}^{n} (Y_{j} - x_{j} β) \int_{A_{j}} E_{m} (t, s) d s,

where A_j = [s_j-1, s_j] are intervals that partition [0, 1] with t_j ∈ A_j and 0 ≤ t₁ ≤ · · · ≤ t_n ≤ 1, and wavelet kernel E_m (t, s) can be defined by

E_{m} (t, s) = 2^{m} E_{0} (2^{m} t, 2^{m} s), E_{0} (t, s) = \sum_{j \in z} φ (t - j) φ (s - j),

where m = m(n) > 0 is a integer depending only on n, φ(·) is father wavelet with compact support. Set

{\tilde{x}}_{i} = x_{i} - \sum_{j = 1}^{n} x_{j} \int_{A_{j}} E_{m} (t_{i}, s) d s, ỹ_{i} = Y_{i} - \sum_{j = 1}^{n} Y_{j} \int_{A_{j}} E_{m} (t_{i}, s) d s, S_{n}^{2} = \sum_{i = 1}^{n} {\tilde{x}}_{i}^{2} .

(1.2)

In order to estimate β, we seek to minimize

S S (β) = \sum_{i = 1}^{n} {[Y_{i} - x_{i} β - g_{n} (t_{i}, β)]}^{2} = \sum_{i = 1}^{n} {(ỹ_{i} - {\tilde{x}}_{i} β)}^{2} .

(1.3)

The minimizer to (1.3) is found to be

{\hat{β}}_{n} = S_{n}^{- 2} \sum_{i = 1}^{n} {\tilde{x}}_{i} ỹ_{i} .

(1.4)

So, a plug-in estimator of the nonparametric component g(·), based on

{\hat{β}}_{n}

, is given by

ĝ_{n} (t) ≜ ĝ_{n} (t, {\hat{β}}_{n}) = \sum_{i = 1}^{n} (Y_{i} - x_{i} {\hat{β}}_{n}) \int_{A_{i}} E_{m} (t, s) d s .

(1.5)

In the following, the symbols c, C, C₁, C₂, . . . denote positive constants whose values may change from one place to another, b_n = O(a_n ) means b_n ≤ ca_n , [x] denotes the integral part of x, ||e_i ||_r: = (E|e_i |^r)^1/r, Φ(u) represents the standard normal distribution function.

The article is organized as follows. In Section 2, we give some assumptions and main results. Sections 3 and 4 are devoted to the proofs of preliminary results. Proofs of theorems will be provided in Section 5. Some known results used in the proofs of preliminary and main results are appended in Appendix.

2 Assumptions, notations and results

At first we list some assumptions used in this article.

(A2) There exists a function h(·) defined on [0, 1] such that x_i = h(t_i ) + u_i and

(i)

$lim_{n \to \infty} n^{- 1} \sum_{i = 1}^{n} u_{i}^{2} = \sum_{0} (0 < \sum_{0} < \infty),$ (ii) $max_{1 \leq i \leq n} | u_{i} | = O (1) .$
(iii)

For any permutation (j ₁, . . . , j_n ) of the integers (1, . . . , n),
$\underset{n \to \infty}{lim sup} \frac{l}{\sqrt{n} log n} max_{1 \leq m \leq n} |\sum_{i = 1}^{m} u_{j_{i}}| < \infty .$

(A3) The spectral density f (ω) of {ε_i } satisfies 0 < c₁ ≤ f (ω) ≤ c₂< ∞, for ω ∈ (-π, π].

(A4) Let g(·) and h(·) satisfy the Lipschitz condition of order 1 on [0, 1], and h (·) ∈ H^v , $ν > \frac{3}{2}$ , where H^v is the Sobolev space of order v.

(A5) Scaling function φ(·) is γ-regular (γ is a positive integer) and has a compact support, satisfies the Lipschitz condition of order 1 and $| \hat{φ} (ξ) - 1 | = O (ξ)$ as ξ → 0, where $\hat{φ}$ denotes the Fourier transform of φ.

(A6) max_{1 ≤ i ≤ n}|s_i -s_i-1| = O (n^-1).

(A7) There exists a positive constant d₁, such that ${min}_{1 \leq i \leq n} (t_{i} - t_{i - 1}) \geq d_{1} \cdot \frac{1}{n}$ .

For the sake of convenience, we use the following notations. Let p = p(n), q = q(n) denote positive integers such that p + q ≤ 3n and qp^-1 ≤ c < ∞. Set

\begin{gathered} σ_{n 1}^{2} = Var (\sum_{i = 1}^{n} u_{i} ε_{i}), σ_{n 2}^{2} = Var (\sum_{i = 1}^{n} ε_{i} \int_{A_{i}} E_{m} (t, s) d s), u (n) = \sum_{j = n}^{\infty} α^{δ / (2 + δ)} (j); \\ γ_{1 n} = q p^{- 1}, γ_{2 n} = p n^{- 1}, γ_{3 n} = n {(\sum_{| j | > n} | a_{j} |)}^{2}, γ_{4 n} = n p^{- 1} α (q); \\ λ_{1 n} = q p^{- 1} 2^{m}, λ_{2 n} = p n^{- 1} 2^{m}, λ_{3 n} = γ_{3 n}, λ_{4 n} = γ_{4 n}, λ_{5 n} = 2^{- m / 2} + \sqrt{2^{m} / n} log n; \\ μ_{n} (ρ, p) = \sum_{i = 1}^{3} γ_{i n}^{1 / 3} + u (q) + γ_{2 n}^{ρ} + γ_{4 n}^{1 / 4}; \\ v_{n} (m) = 2^{- \frac{2 m}{3}} + {(2^{m} / n)}^{1 / 3} {log}^{2 / 3} n + 2^{- m} log n + n^{1 / 2} 2^{- 2 m} . \end{gathered}

After these assumptions and notations we can formulate the main results as follows:

Theorem 2.1. Suppose that (A1)-(A7) hold. If ρ satisfies

0 < ρ \leq 1 / 2, ρ < min \{\frac{δ}{2}, \frac{δ λ - (2 + δ)}{2 λ + (2 + δ)}\},

(2.1)

then

sup_{u} |P (\frac{S_{n}^{2} ({\hat{β}}_{n} - β)}{σ_{n 1}} \leq u) - Φ (u)| \leq C_{1} (μ_{n} (ρ, p) + v_{n} (m)) .

Corollary 2.1 Under the same conditions as in Theorem 2.1, if ρ = 1/3, 2^m= O(n^2/5),

{sup}_{n \geq 1} n^{7 / 8} {(log n)}^{- 9 / 8} \sum_{| j | > n} | a_{j} | < \infty

and δ > 1/3,

λ \geq max {\frac{2 + δ}{δ}, \frac{7 δ + 14}{6 δ - 2}}

, then for each t ∈ [0, 1], we have

|P (\frac{S_{n}^{2} ({\hat{β}}_{n} - β)}{σ_{n 1}} \leq u) - Φ (u)| \leq C_{2} (n^{- \frac{λ}{6 λ + 7}}) .

(2.2)

Theorem 2.2. Suppose that the conditions in Theorem 2.1 are satisfied. Let n^-12^m→ 0, then for each t ∈ [0, 1]

sup_{u} |P (\frac{ĝ_{n} (t) - E ĝ_{n} (t)}{σ_{n 2}} \leq u) - Φ (u)| \leq C_{3} (\sum_{i = 1}^{3} λ_{in}^{l / 3} + u (q) + λ_{2 n}^{ρ} + λ_{4 n}^{1 / 4} + λ_{5 n}^{(2 + δ) / (3 + δ)}) .

Corollary 2.2. Under the conditions of Theorem 2.2 with ρ = 1/3, δ > 2/3, if n^-12^m= O(n^-θ) with

\frac{λ + 1}{2 λ + 1} < θ \leq 3 / 4

, and

λ > \frac{(2 + δ) (9 θ - 2)}{2 θ (3 δ - 2) + 2}

, then

sup_{u} |P (\frac{ĝ_{n} (t) - E ĝ_{n} (t)}{σ_{n 2}} \leq u) - Φ (u)| \leq C_{4} (n^{- min \{\frac{λ (2 θ - 1) + (θ - 1)}{6 λ + 7}, \frac{4 λ + 4}{22 λ + 11}\}}) .

(2.3)

Remark 2.1. Let $\tilde{h} (t) = h (t) - \sum_{j = 1}^{n} h (t_{j}) \int_{A_{j}} E_{m} (t, s) d s,$ under the assumptions (A4)-(A7) and by the relation (11) of the proof of Theorem 3.2 in [15], we obtain ${sup}_{t} | \tilde{h} (t) | = O (n^{- 1} + 2^{- m})$ . Similarly, let $\tilde{g} (t) = g (t) - \sum_{j = 1}^{n} g (t_{j}) \int_{A_{j}} E_{m} (t, s) d s,$ then ${sup}_{t} | \tilde{g} (t) | = O (n^{- 1} + 2^{- m})$ .

Remark 2.2. (i) By Corollary 2.1, the Berry-Esseen bound of the wavelet estimator

{\hat{β}}_{n}

is near

O (n^{- \frac{1}{6}})

for sufficiently large λ, which is faster than the one in [16]) that can get O(n^-δ/4log n) for δ ≤ 1/2 or O(n^-1/8) for δ > 1/2 for strong mixing sequence, but slower than the one in [6] for weighted estimate that can get O(n^-1/4(log n)^3/4).

(ii)

From Corollary 2.2, the Berry-Esseen bound of the wavelet estimator ĝ_n (·) is near O(n^-1/12) for sufficiently large λ and θ = 3/4.

3 Some preliminary lemmas for ${\hat{β}}_{n}$

From the definition of

{\hat{β}}_{n}

in (1.4), we write

\begin{gathered} S_{n β} : = σ_{n 1}^{- 1} S_{n}^{2} ({\hat{β}}_{n} - β) = σ_{n 1}^{- 1} [\sum_{i = 1}^{n} {\tilde{x}}_{i} ε_{i} - \sum_{i = 1}^{n} {\tilde{x}}_{i} \sum_{j = 1}^{n} ε_{j} \int_{A_{j}} E_{m} (t_{i}, s) d s + \sum_{i = 1}^{n} {\tilde{x}}_{i} {\tilde{g}}_{i}] \\ : = S_{n 1} + S_{n 2} + S_{n 3}, \end{gathered}

(3.1)

where

\begin{gathered} S_{n 1} = σ_{n 1}^{- 1} \sum_{i = 1}^{n} {\tilde{x}}_{i} ε_{i} = σ_{n 1}^{- 1} \sum_{i = 1}^{n} u_{i} ε_{i} + σ_{n 1}^{- 1} \sum_{i = 1}^{n} {\tilde{h}}_{i} ε_{i} - σ_{n 1}^{- 1} \sum_{i = 1}^{n} ε_{i} (\sum_{j = 1}^{n} u_{j} \int_{A_{j}} E_{m} (t_{i}, s) d s) \\ = S_{n 11} + S_{n 12} + S_{n 13}, \end{gathered}

(3.2)

\begin{gathered} | S_{n 2} | = |σ_{n 1}^{- 1} \sum_{i = 1}^{n} {\tilde{x}}_{i} \sum_{j = 1}^{n} ε_{j} \int_{A_{j}} E_{m} (t_{i}, s) d s| \\ \leq |σ_{n 1}^{- 1} \sum_{i = 1}^{n} u_{i} (\sum_{j = 1}^{n} ε_{j} \int_{A j} E_{m} (t_{i}, s) d s)| + |σ_{n 1}^{- 1} \sum_{i = 1}^{n} {\tilde{h}}_{i} (\sum_{j = 1}^{n} ε_{j} \int_{A_{j}} E_{m} (t_{i}, s) d s)| \\ + |σ_{n 1}^{- 1} \sum_{i = 1}^{n} (\sum_{l = 1}^{n} u_{l} \int_{A_{l}} E_{m} (t_{i}, s) d s) (\sum_{j = 1}^{n} ε_{j} \int_{A_{j}} E_{m} (t_{i}, s) d s)| \\ = : S_{n 21} + S_{n 22} + S_{n 23}, \end{gathered}

(3.3)

S_{n 3} = σ_{n 1}^{- 1} \sum_{i = 1}^{n} {\tilde{x}}_{i} {\tilde{g}}_{i} .

(3.4)

For S_{n 11}, we can write

S_{n 11} = σ_{n 1}^{- 1} \sum_{i = 1}^{n} u_{i} ε_{i} = σ_{n 1}^{- 1} \sum_{i = 1}^{n} u_{i} \sum_{j = - n}^{n} a_{j} e_{i - j} + σ_{n 1}^{- 1} \sum_{i = 1}^{n} u_{i} \sum_{| j | > n} a_{j} e_{i - j} : = S_{n 111} + S_{n 112} .

(3.5)

It is not difficult to see that

S_{n 111} = \sum_{l = 1 - n}^{2 n} σ_{n 1}^{- 1} (\sum_{i = max {1, l - n}}^{min {n, l + n}} u_{i} a_{i - l}) e_{l} ≜ \sum_{l = 1 - n}^{2 n} Z_{n l} .

Let k = [3n/(p + q)], then S_{n 111}may be split as

S_{n 111} = S_{n 111}^{'} + S_{n 111}^{″} + S_{n 111}^{‴},

(3.6)

where

\begin{gathered} S_{n 111}^{'} = \sum_{w = 1}^{k} y_{1 n w}, S_{n 111}^{″} = \sum_{w = 1}^{k} y_{1 n w}^{'}, S_{n 111}^{‴} = y_{1 n k + 1}^{'}, \\ y_{1 n w} = \sum_{i = k_{w}}^{k_{w} + p - 1} Z_{n i}, y_{1 n w}^{'} = \sum_{i = l_{w}}^{l_{w} + q - 1} Z_{n i}, y_{1 n k + 1}^{'} = \sum_{i = k (p + q) - n + 1}^{2 n} Z_{n i}, \\ k_{w} = (w - 1) (p + q) + 1 - n, l_{w} = (w - 1) (p + q) + p + 1 - n, w = 1, . . ., k . \end{gathered}

From (3.1) to (3.6), we can write that

S_{n β} = S_{n 111}^{'} + S_{n 111}^{″} + S_{n 111}^{‴} + S_{n 112} + S_{n 12} + S_{n 13} + S_{n 2} + S_{n 3} .

Now, we establish the following lemmas with its proofs.

Lemma 3.1. Suppose that (A1), (A2)(i), and (A3) hold, then

c_{1} π n \leq σ_{n 1}^{2} \leq c_{2} π n, c_{3} n^{- 1} 2^{m} \leq σ_{n 2}^{2} \leq c_{4} n^{- 1} 2^{m} .

Proof. According to the proofs of (3.4) and Theorem 2.3 in [17], for any sequence {γ_l}_l∈N, we have

2 c_{1} π \sum_{l = 1}^{n} γ_{l}^{2} \leq E {(\sum_{l = 1}^{n} γ_{l} ε_{l})}^{2} \leq 2 c_{2} π \sum_{l = 1}^{n} γ_{l}^{2},

which implies the desired results by Lemma A.4 and assumption(A2)(i). ♣

Lemma 3.2. Let assumptions (A1)-(A3), (A5), and (A6) be satisfied, then

E {(S_{n 111}^{″})}^{2} \leq C γ_{1 n}, E {(S_{n 111}^{‴})}^{2} \leq C γ_{2 n}, E {(S_{n 112})}^{2} \leq C γ_{3 n};

(3.7)

P (|S_{n 111}^{″}| \geq γ_{1 n}^{1 / 3}) \leq C γ_{1 n}^{1 / 3}, P (|S_{n 111}^{‴}| \geq γ_{2 n}^{1 / 3}) \leq C γ_{2 n}^{1 / 3}, P (|S_{n 112}| \geq γ_{3 n}^{1 / 3}) \leq C γ_{3 n}^{1 / 3} .

(3.8)

Proof. By Lemmas 3.1 and A.1(i), and assumptions (A1)(i) and (A2)(i), we have

\begin{gathered} E {(S_{n 111}^{″})}^{2} \leq C \sum_{w = 1}^{k} \sum_{i = l_{w}}^{l_{w} + q - 1} σ_{n 1}^{- 2} {(\sum_{j = max \{1, i - n\}}^{min \{n, i + n\}} u_{i} a_{j - i})}^{2} {∥e_{i}∥}_{2 + δ}^{2} \\ \leq C \sum_{w = 1}^{k} \sum_{i = l_{w}}^{l_{w} + q - 1} n^{- 1} {(max_{1 \leq i \leq n} | u_{i} |)}^{2} {(\sum_{j = max \{1, i - n\}}^{min \{n, i + n\}} |a_{j - i}|)}^{2} \\ \leq C k q n^{- 1} {(\sum_{i = - \infty}^{\infty} |a_{j}|)}^{2} \leq C q p^{- 1} = C γ_{1 n}, \end{gathered}

(3.9)

\begin{gathered} E {(S_{n 111}^{‴})}^{2} \leq C \sum_{i = k (p + q) - n + 1}^{2 n} σ_{n 1}^{- 2} {(\sum_{j = max \{1, i - n\}}^{min \{n, i + n\}} u_{i} a_{j - i})}^{2} {∥e_{i}∥}_{2 + δ}^{2} \\ \leq C \sum_{i = k (p + q) - n + 1}^{2 n} n^{- 1} {(max_{1 \leq i \leq n} | u_{i} |)}^{2} {(\sum_{j = max \{1, i - n\}}^{min \{n, i + n\}} |a_{j - i}|)}^{2} \\ \leq C [3 n - k (p + q)] n^{- 1} {(\sum_{j = - \infty}^{\infty} |a_{j}|)}^{2} \leq C p n^{- 1} = C γ_{2 n}, \end{gathered}

(3.10)

and by the Cauchy-inequality

\begin{gathered} E {(S_{n 112})}^{2} \leq C σ_{n 1}^{- 2} (\sum_{i = 1}^{n} {|u_{i}|}^{2}) E ({\sum_{i = 1}^{n} (\sum_{|j| > n} a_{j} e_{i - j})}^{2}) \leq C σ_{n 1}^{- 2} n \sum_{i = 1}^{n} {(\sum_{|j| > n} |a_{j}|)}^{2} {∥e_{i}∥}_{2 + δ}^{2} \\ \leq C n {(\sum_{|j| > n} |a_{j}|)}^{2} = C γ_{3 n} . \end{gathered}

(3.11)

Then, from (3.9) to (3.11), the proof of (3.7) is complete, which implies the desired result (3.8) by the Markov-inequality. ♣

Lemma 3.3. Let assumptions (A1)-(A7) be satisfied, then

\begin{gathered} (a) P \{|S_{n 12}| \geq C {(n^{- 1} + 2^{- m})}^{2 / 3}\} \leq C {(n^{- 1} + 2^{- m})}^{2 / 3} . \\ (b) P \{|S_{n 13}| \geq C {(2^{m} n^{- 1} {log}^{2} n)}^{1 / 3}\} \leq C {(2^{m} n^{- 1} {log}^{2} n)}^{1 / 3} . \\ (c) P \{|S_{n 21}| \geq C {(2^{m} n^{- 1} {log}^{2} n)}^{1 / 3}\} \leq C {(2^{m} n^{- 1} {log}^{2} n)}^{1 / 3} . \\ (d) P \{|S_{n 22}| \geq C {(n^{- 1} + 2^{- m})}^{2 / 3}\} \leq C {(n^{- 1} + 2^{- m})}^{2 / 3} . \\ (e) P \{|S_{n 23}| \geq C {(2^{m} n^{- 1} {log}^{2} n)}^{1 / 3}\} \leq C {(2^{m} n^{- 1} {log}^{2} n)}^{1 / 3} . \\ (f) S_{n 3} \leq C (2^{- m} log n + n^{1 / 2} 2^{- 2 m}) . \end{gathered}

(3.12)

Proof. (a) By assumption (A2), Remark 2.1 and Lemma 3.1, we get

E (S_{n 12}^{2}) \leq c_{2} π σ_{n 1}^{- 2} \sum_{i = 1}^{n} {({\tilde{h}}_{i})}^{2} \leq C {(n^{- 1} + 2^{- m})}^{2},

By this and the Markov inequality, we have

P \{|S_{n 12}| \geq C {(n^{- 1} + 2^{- m})}^{2 / 3}\} \leq C {(n^{- 1} + 2^{- m})}^{2 / 3} .

(3.13)

(b)

Applying Lemmas 3.1, A.4, and A.5, we get that
$\begin{gathered} E S_{n 13}^{2} \leq σ_{n 1}^{- 2} \cdot c_{2} \sum_{i = 1}^{n} {(\sum_{j = 1}^{n} u_{j} \int_{A_{j}} E_{m} (t_{i}, s) d s)}^{2} \\ \leq c_{2} σ_{n 1}^{- 2} \cdot max_{1 \leq i, j \leq n} |\int_{A_{j}} E_{m} (t_{i}, s) d s| \cdot max_{1 \leq j \leq n} \sum_{i = 1}^{n} |\int_{A_{j}} E_{m} (t_{i}, s) d s| \cdot {(max_{1 \leq m \leq n} |\sum_{i = 1}^{m} u_{j_{i}}|)}^{2} \\ \leq C 2^{m} n^{- 1} {log}^{2} n . \end{gathered}$

Therefore

P \{|S_{n 13}| \geq C {(2^{m} n^{- 1} {log}^{2} n)}^{1 / 3}\} \leq C {(2^{m} n^{- 1} {log}^{2} n)}^{1 / 3} .

(3.14)

(c)

Changing the order of summation in {S _{n 21}}, similarly to the calculation for $E S_{n 13}^{2}$ ,
$\begin{gathered} E S_{n 21}^{2} \leq σ_{n 1}^{- 2} \cdot c_{2} \sum_{j = 1}^{n} {(\sum_{i = 1}^{n} u_{i} \int_{A_{j}} E_{m} (t_{i}, s) d s)}^{2} \\ \leq c_{2} σ_{n 1}^{- 2} \cdot max_{1 \leq i, j \leq n} |\int_{A_{j}} E_{m} (t_{i}, s) d s| \cdot max_{1 \leq i \leq n} \sum_{j = 1}^{n} |\int_{A_{j}} E_{m} (t_{i}, s) d s| \cdot {(max_{1 \leq m \leq n} |\sum_{i = 1}^{m} u_{j_{i}}|)}^{2} \\ \leq C 2^{m} n^{- 1} {log}^{2} n . \end{gathered}$

Therefore, we obtain that

P \{|S_{n 21}| \geq C {(2^{m} n^{- 1} {log}^{2} n)}^{1 / 3}\} \leq C {(2^{m} n^{- 1} {log}^{2} n)}^{1 / 3} .

(3.15)

(d)

Similarly, by Lemmas 3.1, A.4, A.5, and Remark 2.1, we get that
$\begin{gathered} E S_{n 22}^{2} \leq c_{2} σ_{n 1}^{- 2} \cdot \sum_{i = 1}^{n} {(\sum_{j = 1}^{n} {\tilde{h}}_{j} \int_{A_{i}} E_{m} (t_{j}, s) d s)}^{2} \\ \leq c_{2} σ_{n 1}^{- 2} \cdot {(sup_{t_{j}} |{\tilde{h}}_{j}|)}^{2} \sum_{i = 1}^{n} (\sum_{j = 1}^{n} |\int_{A_{i}} E_{m} (t_{j}, s) d s|) (\sum_{j = 1}^{n} |\int_{A_{i}} E_{m} (t_{i}, s) d s|) \\ \leq c_{2} σ_{n 1}^{- 2} \cdot {(sup_{t_{j}} |{\tilde{h}}_{j}|)}^{2} \cdot n \leq C {(n^{- 1} + 2^{- m})}^{2} . \end{gathered}$

Thus, we have

P {| S_{n 22} | \geq C {(n^{- 1} + 2^{- m})}^{2 / 3}} \leq C {(n^{- 1} + 2^{- m})}^{2 / 3} .

(3.16)

(e)

We write that
$S_{n 23} = σ_{n 1}^{- 1} \sum_{j = 1}^{n} \{\sum_{i = 1}^{n} \int_{A_{j}} E_{m} (t_{i}, s) d s (\sum_{l = 1}^{n} \int_{A_{l}} E_{m} (t_{i}, s) u_{l} d s)\} ε_{j} .$

Similarly to the calculation for

E S_{n 13}^{2}

by (3.13), Lemmas 3.1, A.4, and A.5, we obtain that

\begin{gathered} E S_{n 23}^{2} \leq c_{2} π σ_{n 1}^{- 2} \sum_{j = 1}^{n} {[\sum_{i = 1}^{n} \int_{A_{j}} E_{m} (t_{i}, s) d s (\sum_{l = 1}^{n} \int_{A_{l}} E_{m} (t_{i}, s) u_{l} d s)]}^{2} \\ \leq c_{2} π σ_{n 1}^{- 2} max_{1 \leq i, j \leq n} \int_{A_{i}} E_{m} (t_{j}, s) d s \cdot max_{1 \leq j \leq n} \sum_{i = 1}^{n} \int_{A_{i}} E_{m} (t_{j}, s) d s \\ \cdot {(max_{1 \leq l \leq n} \sum_{j = 1}^{n} \int_{A_{l}} E_{m} (t_{j}, s) d s \cdot max_{1 \leq m \leq n} |\sum_{i = 1}^{m} u_{j_{i}}|)}^{2} \leq C \cdot 2^{m} n^{- 1} {log}^{2} n . \end{gathered}

Hence, we have

P \{|S_{n 23}| \geq C {(2^{m} n^{- 1} {log}^{2} n)}^{1 / 3}\} \leq C {(2^{m} n^{- 1} {log}^{2} n)}^{1 / 3} .

(3.17)

(f)

By assumption (A2), Remarks 2.1, Lemma A.5, and the Abel inequality, we have
$\begin{gathered} σ_{n 1} S_{n 3} \leq |\sum_{i = 1}^{n} u_{i} {\tilde{g}}_{i}| + |\sum_{i = 1}^{n} {\tilde{h}}_{i} {\tilde{g}}_{i}| + |\sum_{i = 1}^{n} (|\sum_{j = 1}^{n} u_{j}| \int_{A_{j}} E_{m} (t_{i}, s) d s) {\tilde{g}}_{i}| \\ \leq c \{max_{1 \leq i \leq n} | {\tilde{g}}_{i} | max_{1 \leq k \leq n} |\sum_{i = 1}^{k} u_{j i}| + n max_{1 \leq i \leq n} | {\tilde{h}}_{i} | max_{1 \leq i \leq n} | {\tilde{g}}_{i} | \\ + max_{1 \leq i \leq n} | {\tilde{g}}_{i} | max_{1 \leq j \leq n} \sum_{i = 1}^{n} \int_{A_{j}} | E_{m} (t_{i}, s) | d s max_{1 \leq k \leq n} \sum_{i = 1}^{k} | u_{j_{i}} |\} \\ = C_{1} (n^{- 1} + 2^{- m}) \sqrt{n} log n + C_{2} n {(n^{- 1} + 2^{- m})}^{2} . \end{gathered}$

Thus, by Lemma 3.1 we obtain

S_{n 3} \leq C_{1} (n^{- 1} + 2^{- m}) log n + C_{2} \sqrt{n} {(n^{- 1} + 2^{- m})}^{2} \leq C (2^{- m} log n + n^{1 / 2} 2^{- 2 m}) .

(3.18)

Therefore, the desired result (3.12) follows from (3.13)-(3.18) immediately. ♣

Lemma 3.4. Suppose that (A1)-(A3), (A5), and (A6) hold. Set

s_{n}^{2} ≜ \sum_{w = 1}^{k} Var (y_{1 n w}),

then

|s_{n}^{2} - 1| \leq C (γ_{1 n}^{1 / 2} + γ_{2 n}^{1 / 2} + γ_{3 n}^{1 / 2} + u (q)) .

Proof. Let

Γ_{n} = \sum_{1 \leq i < j \leq k}

Cov(y_1ni, y_1nj), then

s_{n}^{2} = E {(S_{n 111}^{'})}^{2} - 2 Γ_{n} .

By (3.5) and (3.6), it is easy to verify that

E {(S_{n 11})}^{2} = 1

, and

E {(S_{n 111}^{'})}^{2} = 1 + E {(S_{n 111}^{″} + S_{n 111}^{‴} + S_{n 112})}^{2} - 2 E [S_{n 11} (S_{n 111}^{″} + S_{n 111}^{‴} + S_{n 112})] .

According to Lemma 3.2, the C_r -inequality and the Cauchy-Schwarz inequality

E {(S_{n 111}^{″} + S_{n 111}^{‴} + S_{n 112})}^{2} \leq C (γ_{1 n} + γ_{2 n} + γ_{3 n}),

and

|E [S_{n 11} (S_{n 111}^{″} + S_{n 111}^{‴} + S_{n 112})]| \leq C (γ_{1 n}^{1 / 2} + γ_{2 n}^{1 / 2} + γ_{3 n}^{1 / 2}) .

Thus, we obtain

| E {(S_{n 111}^{'})}^{2} - 1 | \leq C (γ_{1 n}^{1 / 2} + γ_{2 n}^{1 / 2} + γ_{3 n}^{1 / 2}) .

(3.19)

On the other hand, from Lemma 1.2.4 in [7], Lemmas 3.1 and A.4(iv), we can estimate

\begin{gathered} | Γ_{n} | \leq \sum_{1 \leq i \leq j \leq k} \sum_{s_{1} = k_{i}}^{k_{i} + p - 1} \sum_{t_{1} = k_{j}}^{k_{j} + p - 1} | Cov (Z_{n s_{1}}, Z_{n t_{1}}) | \\ \leq \frac{C}{n} \sum_{1 \leq i \leq j \leq k} \sum_{s_{1} = k_{i}}^{k_{i} + p - 1} \sum_{t_{1} = k_{j}}^{k_{j} + p - 1} \sum_{u = max \{1, s_{1} - n\}}^{min \{n, s_{1} + n\}} \sum_{v = max \{1, t_{1} - n\}}^{min \{n, t_{1} + n\}} | u_{u - s_{1}} u_{v - t_{1}} | | a_{u - s_{1}} a_{v - t_{1}} | | Cov (e_{s_{1}}, e_{t_{1}}) | \\ \leq \frac{C}{n} \sum_{1 \leq i \leq j \leq k} \sum_{s_{1} = k_{i}}^{k_{i} + p - 1} \sum_{t_{1} = k_{j}}^{k_{j} + p - 1} \sum_{u = max \{1, s_{1} - n\}}^{min \{n, s_{1} + n\}} \sum_{v = max \{1, t_{1} - n\}}^{min \{n, t_{1} + n\}} | a_{u - s_{1}} a_{v - t_{1}} | α^{δ / (2 + δ)} (t_{1} - s_{1}) | | e_{t_{1}} | |_{2 + δ} \cdot | | e_{s_{1}} | |_{2 + δ} \\ \leq \frac{C}{n} \sum_{i = 1}^{k - 1} \sum_{s_{1} = k_{i}}^{k_{i} + p - 1} \sum_{u = max \{1, s_{1} - n\}}^{min \{n, s_{1} + n\}} \sum_{j = i + 1}^{k} \sum_{t_{1} = k_{j}}^{k_{j} + p - 1} \sum_{v = max \{1, t_{1} - n\}}^{min \{n, t_{1} + n\}} α^{δ / (2 + δ)} (t_{1} - s_{1}) | a_{u - s 1} a_{v - t_{1}} | \\ \leq \frac{C}{n} \sum_{i = 1}^{k - 1} \sum_{s_{1} = k_{i}}^{k_{i} + p - 1} \sum_{j = i + 1}^{k} \sum_{t_{1} = k_{j}}^{k_{j} + p - 1} α^{δ / (2 + δ)} (t_{1} - s_{1}) \\ \leq \frac{C}{n} \sum_{i = 1}^{k - 1} \sum_{s_{1} = k_{i}}^{k_{i} + p - 1} \sum_{t_{1} : |t_{1} - s_{1}| \geq q} α^{δ / (2 + δ)} (t_{1} - s_{1}) \leq C k p n^{- 1} u (q) \leq C u (q) . \end{gathered}

(3.20)

Therefore, by (3.19) and (3.20), it follows that

|s_{n}^{2} - 1| \leq | E {(S_{1 n}^{'})}^{2} - 1 | + 2 |Γ_{n}| \leq C (γ_{1 n}^{1 / 2} + γ_{2 n}^{1 / 2} + γ_{3 n}^{1 / 2} + u (q)) .

♣

Assume that {η_1nw: w = 1, . . . , k} are independent random variables, and its distribution is the same as that of {y_1nw, w = 1, . . . , k}. Set $T_{n} = \sum_{w = 1}^{k} η_{1 n w}$ , $B_{n 1}^{2} = \sum_{w = 1}^{k} Var (η_{1 n w})$ . Clearly $B_{n 1}^{2} = s_{n 1}^{2} .$ Then, we have the following lemmas:

Lemma 3.5. Let assumptions (A1)-(A3), (A5), (A6), and (2.1) hold, the

sup_{u} |P (T_{n} / B_{n 1} \leq u) - Φ (u)| \leq C γ_{2 n}^{ρ} .

Proof. By the Berry-Esseen inequality (see [18], Theorem 5.7]), we have

sup_{u} | P (T_{n} / B_{n 1} \leq u) - Φ (u) | \leq C \frac{\sum_{w = 1}^{k} E {|y_{1 n w}|}^{r}}{B_{n 1}^{r}}, for 2 < r \leq 3 .

(3.21)

From (2.1), we have 0 < 2ρ ≤ 1, 0 < 2ρ < δ, and, (2 + δ)/δ < (1 + ρ) (2 + δ)/(δ - 2ρ) < λ. Let r = 2(1 + ρ), τ = δ- 2ρ, then r + τ = 2 + δ, and

\frac{r (r + τ)}{2 τ} = \frac{(1 + ρ) (2 + δ)}{δ - 2 ρ} < λ .

According to Lemmas 3.1 and A.1(ii), and the C_r -inequality, taking ε = ρ, we get that

\begin{gathered} \sum_{w = 1}^{k} E| y_{1 n w} |^{r} \leq C \sum_{w = 1}^{k} p^{ρ} \sum_{j = k_{w}}^{k_{w} + p - 1} {|\sum_{i = max \{1, j - n\}}^{min \{n, j + n\}} σ_{n 1}^{- 1} u_{i} a_{i - j}|}^{r} E {|e_{j}|}^{r} \\ + {[{\sum_{j = k_{w}}^{k_{w} + p - 1} (\sum_{i = max \{1, j - n\}}^{min \{n, j + n\}} σ_{n 1}^{- 1} u_{i} a_{i - j})}^{2} {| | e_{i} | |}_{2 + δ}^{2}]}^{r / 2} \\ \leq C σ_{n 1}^{- r} k p^{1 + ρ} \leq C γ_{2 n}^{ρ} . \end{gathered}

(3.22)

Therefore, from Lemma 3.4, relations (3.21) and (3.22), we obtain the result. ♣

Lemma 3.6. Suppose that the conditions in Lemma 3.5 are satisfied, then

sup_{u} | P (S_{n 111}^{'} \leq u) - P (T_{n} \leq u) | \leq C \{γ_{2 n}^{ρ} + γ_{4 n}^{1 / 4}\} .

Proof. Let ϕ₁(t) and ψ₁ (t) be the characteristic functions of $S_{n 111}^{'}$ and T_{n 1}, respectively.

Since

ψ_{1} (t) = E (exp \{i t T_{n 1}\}) = \prod_{w = 1}^{k} E exp \{i t η_{1 n w}\} = \prod_{w = 1}^{k} E exp \{i t y_{1 n w}\},

then from Lemmas A.1(i), A.2, and 3.1, it follows that

\begin{aligned} | ϕ_{1} (t) - ψ_{1} (t) | & \leq C |t| α^{1 / 2} (q) \sum_{w = 1}^{k} | | y_{1 n w} | |_{2} \\ \leq C |t| α^{1 / 2} (q) \sum_{w = 1}^{k} {\{E {[\sum_{i = k_{w}}^{k_{w} + p - 1} σ_{n}^{- 1} (\sum_{j = max \{1, i - n\}}^{min \{n, i + n\}} u_{i} a_{j - i}) e_{i}]}^{2}\}}^{1 / 2} \\ \leq C |t| α^{1 / 2} (q) \sum_{w = 1}^{k} {[\sum_{i = k_{w}}^{k_{w} + p - 1} σ_{n}^{- 2} {(\sum_{j = max \{1, i - n\}}^{min \{n, i + n\}} |u_{i} a_{j - i}|)}^{2} {(E {|e_{i}|}^{2 + δ})}^{2 / 2 + δ}]}^{1 / 2} \\ \leq C |t| α^{1 / 2} (q) {[k \sum_{w = 1}^{k} \sum_{i = k_{w}}^{k_{w} + p - 1} σ_{n}^{- 2}]}^{1 / 2} \leq C |t| α^{1 / 2} (q) k p^{1 / 2} n^{- 1 / 2} \\ \leq C |t| {(k α (q))}^{1 / 2} = C |t| γ_{4 n}^{1 / 2} . \end{aligned}

Therefore

\int_{- T}^{T} |\frac{ϕ_{1} (t) - ψ_{1} (t)}{t}| d t \leq C γ_{4 n}^{1 / 2} T .

(3.23)

As in the calculation of (4.7) in [14], using Lemma 3.5, we have

T sup_{u} \int_{|y| \leq c / T} | P (T_{n} \leq u + y) - P (T_{n} \leq u) | d y \leq C \{γ_{2 n}^{ρ} + 1 / T\} .

(3.24)

Therefore, combining (3.23) and (3.24), choosing

T = γ_{4 n}^{- 1 / 4}

, and using the Esseen inequality (see [[18], Theorem 5.3]), we conclude that

\begin{gathered} sup_{u} | P (S_{n 111}^{'} \leq u) - P (T_{n} \leq u) | \\ \leq \int_{- T}^{T} |\frac{ϕ_{1} (t) - ψ_{1} (t)}{t}| d t + T sup_{u} \int_{|y| \leq c / T} | {\tilde{G}}_{n} (u + y) - {\tilde{G}}_{n} (u) | d y \\ = C \{γ_{2 n}^{ρ} + γ_{4 n}^{1 / 4}\} . \end{gathered}

♣

4 Some preliminary lemmas for ĝ_n(t)

From the definition of ĝ_n (t) in (1.5), We can decompose the sum into three parts:

\begin{gathered} S_{n g} : = σ_{n 2}^{- 1} (ĝ_{n} (t) - E ĝ_{n} (t)) = σ_{n 2}^{- 1} \sum_{i = 1}^{n} ε_{i} \int_{A_{i}} E_{m} (t, s) d s + σ_{n 2}^{- 1} \sum_{i = 1}^{n} x_{i} (β - {\hat{β}}_{n}) \int_{A_{i}} E_{m} (t, s) d s \\ - σ_{n 2}^{- 1} \sum_{i = 1}^{n} x_{i} (β - E {\hat{β}}_{n}) \int_{A_{i}} E_{m} (t, s) d s = : H_{1 n} + H_{2 n} + H_{3 n} . \end{gathered}

Let us decompose the vector H_1ninto two parts:

H_{1 n} = σ_{n 2}^{- 1} \sum_{i = 1}^{n} \int_{A_{i}} E_{m} (t, s) d s (\sum_{j = - n}^{n} a_{j} e_{i - j}) + σ_{n 2}^{- 1} \sum_{i = 1}^{n} \int_{A_{i}} E_{m} (t, s) d s (\sum_{| j | > n} a_{j} e_{i - j}) = : H_{11 n} + H_{12 n},

Where

H_{11 n} = σ_{n 2}^{- 1} \sum_{l = 1 - n}^{2 n} (\sum_{i = max (1, l - n)}^{min (n, n + l)} a_{i - l} \int_{A_{i}} E_{m} (t, s) d s) e_{l} = \sum_{l = 1 - n}^{2 n} M_{n l} .

Similar to S_{n 111}in (3.6), H_11ncan be split as

H_{11 n} = H_{11 n}^{'} + H_{11 n}^{″} + H_{11 n}^{‴},

where

\begin{gathered} H_{11 n}^{'} = \sum_{w = 1}^{k} y_{2 n w}, H_{11 n}^{″} = \sum_{w = 1}^{k} y_{2 n w}^{'}, H_{11 n}^{‴} = y_{2 n k + 1}^{'}, \\ y_{2 n w} = \sum_{i = k_{w}}^{k_{w} + p - 1} M_{n i}, y_{2 n w}^{'} = \sum_{i = l_{w}}^{l_{w} + q - 1} M_{n i}, y_{2 n k + 1}^{'} = \sum_{i = k (p + q) - n + 1}^{2 n} M_{n i} . \end{gathered}

(4.1)

Then

S_{n g} = H_{11 n}^{'} + H_{11 n}^{″} + H_{11 n}^{‴} + H_{2 n} + H_{3 n} .

(4.2)

Set $T_{n 2} = \sum_{w = 1}^{k} η_{2 n w}$ , $B_{n 2}^{2} = \sum_{w = 1}^{k} Var (η_{2 n w})$ . Similarly to Lemmas 3.2-3.6, we have the following lemmas without proofs, except for Lemma 4.2.

Lemma 4.1. Suppose that the conditions in Theorem 2.2 are satisfied, then

\begin{gathered} E {(H_{11 n}^{″})}^{2} \leq C λ_{1 n}, E {(H_{11 n}^{‴})}^{2} \leq C λ_{2 n}, E H_{12 n}^{2} \leq C λ_{3 n}; \\ P (|H_{11 n}^{″}| \geq λ_{1 n}^{1 / 3}) \leq C λ_{1 n}^{1 / 3}, P (|H_{11 n}^{‴}| \geq λ_{2 n}^{1 / 3}) \leq C λ_{2 n}^{1 / 3}, P (|H_{12 n}| \geq λ_{3 n}^{1 / 3}) \leq C λ_{3 n}^{1 / 3} . \end{gathered}

Lemma 4.2. Let assumptions (A1)-(A7) be satisfied, then

E {|H_{2 n}|}^{2 + δ} \leq c λ_{5 n}^{2 + δ}, P (|H_{2 n}| > λ_{5 n}^{(2 + δ) / (3 + δ)}) \leq λ_{5 n}^{(2 + δ) / (3 + δ)}, |H_{3 n}| \leq c λ_{5 n} .

Lemma 4.3. Under the conditions of Theorem 2.2, set

s_{n 2}^{2} = \sum_{w = 1}^{k} Var (y_{2 n w}),

then

|s_{n 2}^{2} - 1| \leq C (λ_{1 n}^{1 / 2} + λ_{2 n}^{1 / 2} + λ_{3 n}^{1 / 2} + u (q)) .

Lemma 4.4. Suppose that the conditions in Theorem 2.2 are satisfied, then

sup_{u} |P (T_{n 2} / B_{n 2} \leq u) - Φ (u)| \leq c λ_{2 n}^{ρ} .

Lemma 4.5. Suppose that the conditions in Theorem 2.2 are satisfied, then

sup_{u} | P (H_{11 n}^{'} \leq u) - P (T_{n 2} \leq u) | \leq C \{λ_{2 n}^{ρ} + γ_{4 n}^{1 / 4}\} .

Proof of Lemma 4.2. Similar to the proof of (A.8) in [6], we first verify that

lim_{n \to \infty} S_{n} / n = lim_{n \to \infty} \frac{1}{n} \sum_{i = 1}^{n} {\tilde{x}}_{i}^{2} = \sum, where 0 < \sum < \infty .

(4.3)

From (1.2), we write

\begin{aligned} \frac{1}{n} \sum_{i = 1}^{n} {\tilde{x}}_{i}^{2} & = \frac{1}{n} \sum_{i = 1}^{n} u_{i}^{2} + \frac{1}{n} \sum_{i = 1}^{n} {\tilde{h}}_{i}^{2} + \frac{1}{n} \sum_{i = 1}^{n} {(\sum_{j = 1}^{n} u_{j} \int_{A_{j}} E_{m} (t_{i}, s) d s)}^{2} + \frac{2}{n} \sum_{i = 1}^{n} u_{i} {\tilde{h}}_{i} \\ - \frac{2}{n} \sum_{i = 1}^{n} u_{i} (\sum_{j = 1}^{n} u_{j} \int_{A_{j}} E_{m} (t_{i}, s) d s) - \frac{2}{n} \sum_{i = 1}^{n} {\tilde{h}}_{i} (\sum_{j = 1}^{n} u_{j} \int_{A_{j}} E_{m} (t_{i}, s) d s) \\ = L_{1 n} + L_{2 n} + L_{3 n} + 2 L_{4 n} - 2 L_{5 n} - 2 L_{6 n} . \end{aligned}

(4.4)

By assumption (A2)(i) and Remark 2.1, we have

L_{1 n} \to \sum, L_{2 n} \leq max_{1 \leq i \leq n} {\tilde{h}}_{i}^{2} = O (n^{- 1} + 2^{- m}) \to 0,

(4.5)

and by assumption (A2)(iii), Lemmas A.4 and A.5, we get that

\begin{aligned} L_{3 n} & \leq \frac{c}{n} max_{1 \leq i, j \leq n} \int_{A_{j}} | E_{m} (t_{i}, s) | d s max_{1 \leq j \leq n} \sum_{i = 1}^{n} | \int_{A_{j}} E_{m} (t_{i}, s) | d s {(max_{1 \leq j \leq n} |\sum_{i = 1}^{l} u_{j_{i}}|)}^{2} \\ = O (\frac{2^{m} {log}^{2} n}{n}) \to 0, \\ |L_{4 n}| & \leq \frac{c}{n} max_{1 \leq i \leq n} |{\tilde{h}}_{i}| \cdot max_{1 \leq l \leq n} |\sum_{i = 1}^{l} u_{j_{i}}| = O (\frac{log n}{2^{m} \sqrt{n}}) \to 0, \\ |L_{5 n}| & \leq \frac{c}{n} max_{1 \leq i, j \leq n} \int_{A_{j}} |E_{m} (t_{i}, s)| d s \cdot max_{1 \leq l \leq n} |\sum_{i^{'} = 1}^{l} u_{j_{i^{'}}}| \cdot max_{1 \leq l \leq n} |\sum_{i = 1}^{l} u_{j_{i}}| \\ = O (\frac{2^{m} {log}^{2} n}{n}) \to 0, \\ |L_{6 n}| & \leq \frac{c}{n} max_{1 \leq i \leq n} |{\tilde{h}}_{i}| \cdot max_{1 \leq j \leq n} \sum_{i = 1}^{n} \int_{A_{j}} | E_{m} (t_{i}, s) | d s \cdot max_{1 \leq l \leq n} |\sum_{i = 1}^{l} u_{j_{i}}| = O (\frac{log n}{2^{m} \sqrt{n}}) \to 0 . \end{aligned}

(4.6)

Therefore, from (4.4) to (4.6), we complete the proof of (4.3).

Recalling the fact that if ξ_n ⇒ ξ ~ N (0, 1) then

E |ξ_{n}| \to E |ξ| = \sqrt{2 / π}

and E|ξ_n |^2+δ→ E|ξ|^2+δ< ∞, by Theorem 2.1, Lemma 3.1 and relation (4.3), we deduce that

| β - E {\hat{β}}_{n} | \leq E | β - {\hat{β}}_{n} | \leq O (σ_{n 1} / S_{n}^{2}) = O (n^{- 1 / 2}),

and

E {| {\hat{β}}_{n} - β |}^{2 + δ} \leq O ({(σ_{n 1} / S_{n}^{2})}^{2 + δ}) = O (n^{- (1 + δ / 2)}) .

Therefore, applying the Abel Inequality, by (A2)(iii) and (A4)(i), we get that

\begin{aligned} |H_{3 n}| & = σ_{n 2}^{- 1} |β - E \hat{β}| \end{aligned}

Berry-Esseen bounds for wavelet estimator in semiparametric regression model with linear process errors

Abstract

Keywords

1 Introduction

2 Assumptions, notations and results

3 Some preliminary lemmas for β ^ n

4 Some preliminary lemmas for ĝ n (t)

3 Some preliminary lemmas for ${\hat{β}}_{n}$

4 Some preliminary lemmas for ĝ_n(t)