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

## Abstract

Consider the semiparametric regression model Y i = x i β + g (t i ) + ε i , i = 1, . . . , n, where the linear process errors ${\epsilon }_{i}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{\sum }_{j=-\infty }^{\infty }{a}_{j}{e}_{i-j}$ with ${\sum }_{j=-\infty }^{\infty }\left|{a}_{j}\right|\phantom{\rule{0.3em}{0ex}}<\phantom{\rule{0.3em}{0ex}}\infty$, 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.

## 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.  first introduced the following semiparametric regression model:

${Y}_{i}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{x}_{i}\beta +g\left({t}_{i}\right)+{\epsilon }_{i},\phantom{\rule{1em}{0ex}}i\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}1,...,\phantom{\rule{0.3em}{0ex}}n,$
(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 , Donald and Dewey , and Hamilton and Truong  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 ${\epsilon }_{i}={\sum }_{j=0}^{\infty }{a}_{j}{e}_{i-j}$, where {e i } are i.i.d. random variables,{a j } satisfy ${\sum }_{j=-\infty }^{\infty }\left|{a}_{j}\right|\phantom{\rule{0.3em}{0ex}}<\phantom{\rule{0.3em}{0ex}}\infty$ and ${sup}_{n}n{\sum }_{j=n}^{\infty }\left|{a}_{j}\right|<\infty ,$ 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. . The Berry-Esseen type bounds for estimators of β and g(·) in model (1) under the linear process errors ${\epsilon }_{i}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{\sum }_{j=-\infty }^{\infty }{a}_{j}{e}_{i-j}$ with identically distributed and negatively associated random variables {e i } were derived by Liang and Fan .

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 $\left\{\left|\mathsf{\text{P}}\left(AB\right)-\mathsf{\text{P}}\left(A\right)\phantom{\rule{0.3em}{0ex}}\mathsf{\text{P}}\phantom{\rule{0.3em}{0ex}}\left(B\right)\right|:A\in {ℱ}_{-\infty }^{m},B\in {ℱ}_{m+n}^{\infty }\right\}$, and ${ℱ}_{n}^{m}$ denotes the σ-field generated by {e i : min}.

For the properties of strong-mixing, one can read the book of Lin and Liu . Recently, Yang and Li  and Xing et al.  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  to semiparametric regression model.

(A1) (i) Let ${\epsilon }_{i}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{\sum }_{j=-\infty }^{\infty }{a}_{j}{e}_{i-j}$, where $\sum _{j=-\infty }^{\infty }\left|{a}_{j}\right|<\infty$, {e j , j = 0, ± 1, ± 2, . . .} are identically distributed and strong mixing random variables with zero mean.

1. (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

${g}_{n}\left(t,\beta \right)=\sum _{j=1}^{n}\left({Y}_{j}\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}{x}_{j}\beta \right)\underset{{A}_{j}}{\int }{E}_{m}\left(t,s\right)\phantom{\rule{0.3em}{0ex}}ds,$

where A j = [sj-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

${E}_{m}\left(t,s\right)={2}^{m}{E}_{0}\left({2}^{m}t,\phantom{\rule{0.3em}{0ex}}{2}^{m}s\right),\phantom{\rule{2.77695pt}{0ex}}{E}_{0}\left(t,s\right)=\sum _{j\in z}\phi \phantom{\rule{0.3em}{0ex}}\left(t-j\right)\phantom{\rule{0.3em}{0ex}}\phi \phantom{\rule{0.3em}{0ex}}\left(s-j\right),$

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

${\stackrel{̃}{x}}_{i}={x}_{i}-\sum _{j=1}^{n}{x}_{j}\underset{{A}_{j}}{\int }{E}_{m}\left({t}_{i},s\right)\phantom{\rule{0.3em}{0ex}}ds,{ỹ}_{i}={Y}_{i}-\sum _{j=1}^{n}{Y}_{j}\underset{{A}_{j}}{\int }{E}_{m}\left({t}_{i},s\right)\phantom{\rule{0.3em}{0ex}}ds,{S}_{n}^{2}=\sum _{i=1}^{n}{\stackrel{̃}{x}}_{i}^{2}.$
(1.2)

In order to estimate β, we seek to minimize

$SS\left(\beta \right)\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\sum _{i=1}^{n}{\left[{Y}_{i}-{x}_{i}\beta -{g}_{n}\left({t}_{i},\beta \right)\right]}^{2}=\sum _{i=1}^{n}{\left({ỹ}_{i}-{\stackrel{̃}{x}}_{i}\beta \right)}^{2}.$
(1.3)

The minimizer to (1.3) is found to be

${\stackrel{^}{\beta }}_{n}={S}_{n}^{-2}\sum _{i=1}^{n}{\stackrel{̃}{x}}_{i}{ỹ}_{i}.$
(1.4)

So, a plug-in estimator of the nonparametric component g(·), based on ${\stackrel{^}{\beta }}_{n}$, is given by

${\mathit{ĝ}}_{n}\left(t\right)\triangleq {\mathit{ĝ}}_{n}\left(t,{\stackrel{^}{\beta }}_{n}\right)\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\sum _{i=1}^{n}\left({Y}_{i}-{x}_{i}{\stackrel{^}{\beta }}_{n}\right)\phantom{\rule{2.77695pt}{0ex}}\underset{{A}_{i}}{\int }{E}_{m}\left(t,s\right)\phantom{\rule{0.3em}{0ex}}ds.$
(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

(A2) There exists a function h(·) defined on [0, 1] such that x i = h(t i ) + u i and

1. (i)

$\underset{n\to \infty }{lim}{n}^{-1}{\sum }_{i=1}^{n}{u}_{i}^{2}={\sum }_{0}\phantom{\rule{1em}{0ex}}\left(0<{\sum }_{0}<\infty \right),$ (ii) $\underset{1\le i\le n}{max}|{u}_{i}|\phantom{\rule{2.77695pt}{0ex}}=O\phantom{\rule{0.3em}{0ex}}\left(1\right).$

2. (iii)

For any permutation (j 1, . . . , j n ) of the integers (1, . . . , n),

$\underset{n\to \infty }{lim sup}\frac{\mathsf{\text{l}}}{\sqrt{n}log\phantom{\rule{0.2em}{0ex}}n}\phantom{\rule{2.77695pt}{0ex}}\underset{1\le m\le n}{max}\left|\sum _{i=1}^{m}{u}_{{j}_{i}}\right|<\infty .$

(A3) The spectral density f (ω) of {ε i } satisfies 0 < c1f (ω) ≤ c2< ∞, for ω (-π, π].

(A4) Let g(·) and h(·) satisfy the Lipschitz condition of order 1 on [0, 1], and h (·) Hv , $\nu >\frac{3}{2}$, where Hv 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 $|\stackrel{^}{\phi }\phantom{\rule{0.3em}{0ex}}\left(\xi \right)-1|\phantom{\rule{2.77695pt}{0ex}}=O\left(\xi \right)$ as ξ → 0, where $\stackrel{^}{\phi }$ denotes the Fourier transform of φ.

(A6) max1 ≤ in|s i -si-1| = O (n-1).

(A7) There exists a positive constant d1, such that ${min}_{1\le i\le n}\left({t}_{i}-{t}_{i-1}\right)\ge {d}_{1}\cdot \frac{1}{n}$.

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-1c < ∞. Set

$\begin{array}{c}{\sigma }_{n1}^{2}=\mathsf{\text{Var}}\left(\sum _{i=1}^{n}{u}_{i}{\epsilon }_{i}\right),\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{\sigma }_{n2}^{2}=\mathsf{\text{Var}}\left(\sum _{i=1}^{n}{\epsilon }_{i}\underset{{A}_{i}}{\int }{E}_{m}\left(t,s\right)ds\right),\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}u\left(n\right)=\sum _{j=n}^{\infty }{\alpha }^{\delta /\left(2+\delta \right)}\left(j\right);\\ {\gamma }_{1n}=q{p}^{-1},\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{\gamma }_{2n}=p{n}^{-1},\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{\gamma }_{3n}=n{\left(\sum _{|j|>n}|{a}_{j}|\right)}^{2},\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{\gamma }_{4n}=n{p}^{-1}\alpha \left(q\right);\\ {\lambda }_{1n}=q{p}^{-1}{2}^{m},\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{\lambda }_{2n}=p{n}^{-1}{2}^{m},\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{\lambda }_{3n}={\gamma }_{3n},\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{\lambda }_{4n}={\gamma }_{4n},\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{\lambda }_{5n}={2}^{-m/2}+\sqrt{{2}^{m}/n}logn;\\ {\mu }_{n}\left(\rho ,p\right)=\sum _{i=1}^{3}{\gamma }_{in}^{1/3}+u\phantom{\rule{0.3em}{0ex}}\left(q\right)+{\gamma }_{2n}^{\rho }+{\gamma }_{4n}^{1/4};\\ {v}_{n}\left(m\right)={2}^{-\frac{2m}{3}}+{\left({2}^{m}/n\right)}^{1/3}{log}^{2/3}n+{2}^{-m}logn+{n}^{1/2}{2}^{-2m}.\end{array}$

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<\rho \le 1/2,\phantom{\rule{1em}{0ex}}\rho
(2.1)

then

$\underset{u}{sup}\left|P\left(\frac{{S}_{n}^{2}\left({\stackrel{^}{\beta }}_{n}-\beta \right)}{{\sigma }_{n1}}\le u\right)-\Phi \left(u\right)\right|\le {C}_{1}\left({\mu }_{n}\left(\rho ,p\right)+{v}_{n}\left(m\right)\right).$

Corollary 2.1 Under the same conditions as in Theorem 2.1, if ρ = 1/3, 2m= O(n2/5), ${sup}_{n\ge 1}{n}^{7/8}{\left(log\phantom{\rule{0.3em}{0ex}}n\right)}^{-9/8}{\sum }_{|j|>n}|{a}_{j}|<\infty$ and δ > 1/3, $\lambda \ge max\left\{\frac{2+\delta }{\delta },\frac{7\delta +14}{6\delta -2}\right\}$, then for each t [0, 1], we have

$\left|P\left(\frac{{S}_{n}^{2}\left({\stackrel{^}{\beta }}_{n}-\beta \right)}{{\sigma }_{n1}}\le u\right)-\Phi \left(u\right)\right|\le {C}_{2}\left({n}^{-\frac{\lambda }{6\lambda +7}}\right).$
(2.2)

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

$\underset{u}{sup}\left|P\left(\frac{{\mathit{ĝ}}_{n}\left(t\right)-\mathsf{\text{E}}{\mathit{ĝ}}_{n}\left(t\right)}{{\sigma }_{n2}}\le u\right)-\Phi \left(u\right)\right|\le {C}_{3}\left(\sum _{i=1}^{3}{\lambda }_{\mathsf{\text{in}}}^{\mathsf{\text{l}}/3}+u\left(q\right)+{\lambda }_{2n}^{\rho }+{\lambda }_{4n}^{1/4}+{\lambda }_{5n}^{\left(2+\delta \right)/\left(3+\delta \right)}\right).$

Corollary 2.2. Under the conditions of Theorem 2.2 with ρ = 1/3, δ > 2/3, if n-12m= O(n-θ) with $\frac{\lambda +1}{2\lambda +1}<\theta \le 3/4$, and $\lambda >\frac{\left(2+\delta \right)\left(9\theta -2\right)}{2\theta \left(3\delta -2\right)+2}$, then

$\underset{u}{sup}\left|P\phantom{\rule{0.3em}{0ex}}\left(\frac{{\mathit{ĝ}}_{n}\left(t\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}\mathsf{\text{E}}{\mathit{ĝ}}_{n}\left(t\right)\phantom{\rule{0.3em}{0ex}}}{{\sigma }_{n2}}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}u\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}\Phi \left(u\right)\right|\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}{C}_{4}\phantom{\rule{0.3em}{0ex}}\left({n}^{-min\left\{\frac{\lambda \left(2\theta -1\right)\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}\left(\theta -1\right)}{6\lambda +7},\frac{4\lambda +4}{22\lambda +11}\phantom{\rule{0.3em}{0ex}}\right\}}\right).$
(2.3)

Remark 2.1. Let $\stackrel{̃}{h}\left(t\right)=h\left(t\right)-{\sum }_{j=1}^{n}h\left({t}_{j}\right){\int }_{{A}_{j}}{E}_{m}\left(t,\phantom{\rule{2.77695pt}{0ex}}s\right)ds,$ under the assumptions (A4)-(A7) and by the relation (11) of the proof of Theorem 3.2 in , we obtain ${sup}_{t}|\stackrel{̃}{h}\left(t\right)|\phantom{\rule{2.77695pt}{0ex}}=O\left({n}^{-1}+{2}^{-m}\right)$. Similarly, let $\stackrel{̃}{g}\left(t\right)=g\left(t\right)-{\sum }_{j=1}^{n}g\left({t}_{j}\right){\int }_{{A}_{j}}{E}_{m}\left(t,\phantom{\rule{2.77695pt}{0ex}}s\right)ds,$ then ${sup}_{t}|\stackrel{̃}{g}\left(t\right)|\phantom{\rule{2.77695pt}{0ex}}=O\left({n}^{-1}+{2}^{-m}\right)$.

Remark 2.2. (i) By Corollary 2.1, the Berry-Esseen bound of the wavelet estimator ${\stackrel{^}{\beta }}_{n}$ is near $O\phantom{\rule{0.3em}{0ex}}\left({n}^{-\frac{1}{6}}\right)$ for sufficiently large λ, which is faster than the one in ) 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  for weighted estimate that can get O(n-1/4(log n)3/4).

1. (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 ${\stackrel{^}{\mathbit{\beta }}}_{\mathbit{n}}$

From the definition of ${\stackrel{^}{\beta }}_{n}$ in (1.4), we write

$\begin{array}{c}{S}_{n\beta }:={\sigma }_{n1}^{-1}{S}_{n}^{2}\left({\stackrel{^}{\beta }}_{n}-\beta \right)={\sigma }_{n1}^{-1}\left[\sum _{i=1}^{n}{\stackrel{̃}{x}}_{i}{\epsilon }_{i}-\sum _{i=1}^{n}{\stackrel{̃}{x}}_{i}\sum _{j=1}^{n}{\epsilon }_{j}{\int }_{{A}_{j}}{E}_{m}\left({t}_{i},s\right)\phantom{\rule{0.3em}{0ex}}ds+\sum _{i=1}^{n}{\stackrel{̃}{x}}_{i}{\stackrel{̃}{g}}_{i}\right]\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{0.3em}{0ex}}:={S}_{n1}+{S}_{n2}+{S}_{n3},\hfill \end{array}$
(3.1)

where

$\begin{array}{c}{S}_{n1}={\sigma }_{n1}^{-1}\sum _{i=1}^{n}{\stackrel{̃}{x}}_{i}{\epsilon }_{i}={\sigma }_{n1}^{-1}\sum _{i=1}^{n}{u}_{i}{\epsilon }_{i}+{\sigma }_{n1}^{-1}\sum _{i=1}^{n}{\stackrel{̃}{h}}_{i}{\epsilon }_{i}-{\sigma }_{n1}^{-1}\sum _{i=1}^{n}{\epsilon }_{i}\left(\sum _{j=1}^{n}{u}_{j}{\int }_{{A}_{j}}{E}_{m}\left({t}_{i},\phantom{\rule{2.77695pt}{0ex}}s\right)ds\right)\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{0.3em}{0ex}}={S}_{n11}+{S}_{n12}+{S}_{n13},\hfill \end{array}$
(3.2)
$\begin{array}{c}\phantom{\rule{1em}{0ex}}|{S}_{n2}|\phantom{\rule{2.77695pt}{0ex}}=\left|{\sigma }_{n1}^{-1}\sum _{i=1}^{n}{\stackrel{̃}{x}}_{i}\sum _{j=1}^{n}{\epsilon }_{j}{\int }_{{A}_{j}}{E}_{m}\left({t}_{i},s\right)ds\right|\hfill \\ \le \left|{\sigma }_{n1}^{-1}\sum _{i=1}^{n}{u}_{i}\left(\sum _{j=1}^{n}{\epsilon }_{j}{\int }_{Aj}{E}_{m}\left({t}_{i},\phantom{\rule{2.77695pt}{0ex}}s\right)ds\right)\phantom{\rule{0.3em}{0ex}}\right|+\left|{\sigma }_{n1}^{-1}\sum _{i=1}^{n}{\stackrel{̃}{h}}_{i}\left(\sum _{j=1}^{n}{\epsilon }_{j}{\int }_{{A}_{j}}{E}_{m}\left({t}_{i},\phantom{\rule{2.77695pt}{0ex}}s\right)ds\right)\phantom{\rule{0.3em}{0ex}}\right|\hfill \\ \phantom{\rule{1em}{0ex}}+\left|{\sigma }_{n1}^{-1}\sum _{i=1}^{n}\left(\sum _{l=1}^{n}{u}_{l}{\int }_{{A}_{l}}{E}_{m}\left({t}_{i},\phantom{\rule{2.77695pt}{0ex}}s\right)ds\right)\left(\sum _{j=1}^{n}{\epsilon }_{j}{\int }_{{A}_{j}}{E}_{m}\left({t}_{i},\phantom{\rule{2.77695pt}{0ex}}s\right)ds\right)\right|\hfill \\ =:{S}_{n21}+{S}_{n22}+{S}_{n23},\hfill \end{array}$
(3.3)
${S}_{n3}={\sigma }_{n1}^{-1}\sum _{i=1}^{n}{\stackrel{̃}{x}}_{i}{\stackrel{̃}{g}}_{i}.$
(3.4)

For Sn 11, we can write

${S}_{n11}={\sigma }_{n1}^{-1}\sum _{i=1}^{n}{u}_{i}{\epsilon }_{i}={\sigma }_{n1}^{-1}\sum _{i=1}^{n}{u}_{i}\sum _{j=-n}^{n}{a}_{j}{e}_{i-j}+{\sigma }_{n1}^{-1}\sum _{i=1}^{n}{u}_{i}\sum _{|j|>n}{a}_{j}{e}_{i-j}:={S}_{n111}+{S}_{n112}.$
(3.5)

It is not difficult to see that

${S}_{n111}=\sum _{l=1-n}^{2n}{\sigma }_{n1}^{-1}\left(\sum _{i=max\left\{1,l-n\right\}}^{min\left\{n,l+n\right\}}{u}_{i}{a}_{i-l}\right){e}_{l}\triangleq \sum _{l=1-n}^{2n}{Z}_{nl}.$

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

${S}_{n111}={S}_{n111}^{\prime }+{S}_{n111}^{″}+{S}_{n111}^{‴},$
(3.6)

where

$\begin{array}{c}{S}_{n111}^{\prime }={\sum }_{w=1}^{k}{y}_{1nw},\phantom{\rule{1em}{0ex}}{S}_{n111}^{″}=\sum _{w=1}^{k}{y}_{1nw}^{\prime },\phantom{\rule{1em}{0ex}}{S}_{n111}^{‴}={y}_{1nk+1}^{\prime },\hfill \\ {y}_{1nw}=\sum _{i={k}_{w}}^{{k}_{w}+p-1}{Z}_{ni},\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{y}_{1nw}^{\prime }=\sum _{i={l}_{w}}^{{l}_{w}+q-1}{Z}_{ni},\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{y}_{1nk+1}^{\prime }=\sum _{i=k\left(p+q\right)-n+1}^{2n}{Z}_{ni},\hfill \\ {k}_{w}=\left(w-1\right)\phantom{\rule{0.3em}{0ex}}\left(p+q\right)+1-n,{l}_{w}=\left(w-1\right)\left(p+q\right)+p+1-n,w=1,...,k.\hfill \end{array}$

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

${S}_{n\beta }\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{S}_{n111}^{\prime }\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{S}_{n111}^{″}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{S}_{n111}^{‴}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{S}_{n112}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{S}_{n12}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{S}_{n13}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{S}_{n2}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{S}_{n3}.$

Now, we establish the following lemmas with its proofs.

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

${c}_{1}\pi n\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}{\sigma }_{n1}^{2}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}{c}_{2}\pi n,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{1em}{0ex}}{c}_{3}{n}^{-1}{2}^{m}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}{\sigma }_{n2}^{2}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}{c}_{4}{n}^{-1}{2}^{m}.$

Proof. According to the proofs of (3.4) and Theorem 2.3 in , for any sequence {γ l }lN, we have

$2{c}_{1}\pi \phantom{\rule{0.3em}{0ex}}\sum _{l=1}^{n}{\gamma }_{l}^{2}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}E\phantom{\rule{0.3em}{0ex}}{\left(\sum _{l=1}^{n}{\gamma }_{l}{\epsilon }_{l}\right)}^{2}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}2{c}_{2}\pi \phantom{\rule{0.3em}{0ex}}\sum _{l=1}^{n}{\gamma }_{l}^{2},$

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

$\mathsf{\text{E}}{\left({S}_{n111}^{″}\right)}^{2}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C{\gamma }_{1n},\phantom{\rule{0.3em}{0ex}}\mathsf{\text{E}}{\left({S}_{n111}^{‴}\right)}^{2}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C{\gamma }_{2n},\phantom{\rule{0.3em}{0ex}}\mathsf{\text{E}}{\left({S}_{n112}\right)}^{2}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C{\gamma }_{3n};$
(3.7)
$P\left(\left|{S}_{n111}^{″}\right|\phantom{\rule{0.3em}{0ex}}\ge \phantom{\rule{0.3em}{0ex}}{\gamma }_{1n}^{1/3}\right)\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C{\gamma }_{1n}^{1/3},\phantom{\rule{0.3em}{0ex}}P\phantom{\rule{0.3em}{0ex}}\left(\left|{S}_{n111}^{‴}\right|\phantom{\rule{0.3em}{0ex}}\ge \phantom{\rule{0.3em}{0ex}}{\gamma }_{2n}^{1/3}\right)\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C{\gamma }_{2n}^{1/3},\phantom{\rule{0.3em}{0ex}}P\left(\left|{S}_{n112}\right|\phantom{\rule{0.3em}{0ex}}\ge \phantom{\rule{0.3em}{0ex}}{\gamma }_{3n}^{1/3}\right)\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C{\gamma }_{3n}^{1/3}.$
(3.8)

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

$\begin{array}{c}\mathsf{\text{E}}{\left({S}_{n111}^{″}\right)}^{2}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C\phantom{\rule{0.3em}{0ex}}\sum _{w=1}^{k}\sum _{i={l}_{w}}^{{l}_{w}+q-1}{\sigma }_{n1}^{-2}\phantom{\rule{0.3em}{0ex}}{\left(\sum _{j=max\left\{1,i-n\right\}}^{min\left\{n,i+n\right\}}{u}_{i}{a}_{j-i}\right)}^{2}\phantom{\rule{0.3em}{0ex}}{∥{e}_{i}∥}_{2+\delta }^{2}\hfill \\ \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C\phantom{\rule{0.3em}{0ex}}\sum _{w=1}^{k}\sum _{i={l}_{w}}^{{l}_{w}+q-1}\phantom{\rule{0.3em}{0ex}}{n}^{-1}\phantom{\rule{0.3em}{0ex}}{\left(\underset{1\le i\le n}{max}|{u}_{i}|\right)}^{2}\phantom{\rule{0.3em}{0ex}}{\left(\sum _{j=max\left\{1,i-n\right\}}^{min\left\{n,i+n\right\}}\left|{a}_{j-i}\right|\right)}^{2}\hfill \\ \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}Ckq{n}^{-1}\phantom{\rule{0.3em}{0ex}}{\left(\sum _{i=-\infty }^{\infty }\left|{a}_{j}\right|\right)}^{2}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}Cq{p}^{-1}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}C{\gamma }_{1n},\hfill \end{array}$
(3.9)
$\begin{array}{c}\mathsf{\text{E}}{\left({S}_{n111}^{‴}\right)}^{2}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C\phantom{\rule{0.3em}{0ex}}\sum _{i=k\left(p+q\right)-n+1}^{2n}{\sigma }_{n1}^{-2}\phantom{\rule{0.3em}{0ex}}{\left(\sum _{j=max\left\{1,i-n\right\}}^{min\left\{n,i+n\right\}}{u}_{i}{a}_{j-i}\right)}^{2}\phantom{\rule{0.3em}{0ex}}{∥{e}_{i}∥}_{2+\delta }^{2}\hfill \\ \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\le C\sum _{i=k\left(p+q\right)-n+1}^{2n}{n}^{-1}{\left(\underset{1\le i\le n}{max}|{u}_{i}|\right)}^{2}\phantom{\rule{0.3em}{0ex}}{\left(\sum _{j=max\left\{1,i-n\right\}}^{min\left\{n,i+n\right\}}\left|{a}_{j-i}\right|\right)}^{2}\hfill \\ \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}\phantom{\rule{1em}{0ex}}C\left[3n\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}k\left(p\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}q\right)\right]{n}^{-1}\phantom{\rule{0.3em}{0ex}}{\left(\sum _{j=-\infty }^{\infty }\left|{a}_{j}\right|\right)}^{2}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}Cp{n}^{-1}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}C{\gamma }_{2n},\hfill \end{array}$
(3.10)

and by the Cauchy-inequality

$\begin{array}{c}\mathsf{\text{E}}{\left({S}_{n112}\right)}^{2}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C{\sigma }_{n1}^{-2}\phantom{\rule{0.3em}{0ex}}\left(\sum _{i=1}^{n}{\left|{u}_{i}\right|}^{2}\right)E\left({\sum _{i=1}^{n}\left(\sum _{\left|j\right|\phantom{\rule{0.3em}{0ex}}>n}{a}_{j}{e}_{i-j}\right)}^{2}\right)\phantom{\rule{0.3em}{0ex}}\le C{\sigma }_{n1}^{-2}n\sum _{i=1}^{n}{\left(\sum _{\left|j\right|\phantom{\rule{0.3em}{0ex}}>n}\left|{a}_{j}\right|\right)}^{2}\phantom{\rule{0.3em}{0ex}}{∥{e}_{i}∥}_{2+\delta }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2.2em}{0ex}}\le \phantom{\rule{2.77695pt}{0ex}}Cn\phantom{\rule{0.3em}{0ex}}{\left(\sum _{\left|j\right|\phantom{\rule{0.3em}{0ex}}>n}\left|{a}_{j}\right|\right)}^{2}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}C{\gamma }_{3n}.\hfill \end{array}$
(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{array}{c}\left(a\right)\phantom{\rule{0.3em}{0ex}}P\left\{\left|{S}_{n12}\right|\phantom{\rule{0.3em}{0ex}}\ge C{\left({n}^{-1}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{2}^{-m}\right)}^{2/3}\right\}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C{\left({n}^{-1}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{2}^{-m}\right)}^{2/3}.\hfill \\ \left(b\right)\phantom{\rule{0.3em}{0ex}}P\phantom{\rule{0.3em}{0ex}}\left\{\left|{S}_{n13}\right|\ge C{\left({2}^{m}{n}^{-1}\phantom{\rule{0.3em}{0ex}}{log}^{2}\phantom{\rule{0.3em}{0ex}}n\right)}^{1/3}\right\}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C{\left({2}^{m}{n}^{-1}\phantom{\rule{0.3em}{0ex}}{log}^{2}\phantom{\rule{0.3em}{0ex}}n\right)}^{1/3}.\hfill \\ \left(c\right)\phantom{\rule{0.3em}{0ex}}P\phantom{\rule{0.3em}{0ex}}\left\{\left|{S}_{n21}\right|\ge C{\left({2}^{m}{n}^{-1}\phantom{\rule{0.3em}{0ex}}{log}^{2}\phantom{\rule{0.3em}{0ex}}n\right)}^{1/3}\right\}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C{\left({2}^{m}{n}^{-1}\phantom{\rule{0.3em}{0ex}}{log}^{2}\phantom{\rule{0.3em}{0ex}}n\right)}^{1/3}.\hfill \\ \left(d\right)\phantom{\rule{0.3em}{0ex}}P\left\{\left|{S}_{n22}\right|\phantom{\rule{0.2em}{0ex}}\ge C{\left({n}^{-1}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{2}^{-m}\right)}^{2/3}\right\}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C{\left({n}^{-1}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{2}^{-m}\right)}^{2/3}.\hfill \\ \left(e\right)\phantom{\rule{0.3em}{0ex}}P\phantom{\rule{0.3em}{0ex}}\left\{\left|{S}_{n23}\right|\ge C{\left({2}^{m}{n}^{-1}\phantom{\rule{0.3em}{0ex}}{log}^{2}\phantom{\rule{0.3em}{0ex}}n\right)}^{1/3}\right\}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C{\left({2}^{m}{n}^{-1}\phantom{\rule{0.3em}{0ex}}{log}^{2}\phantom{\rule{0.3em}{0ex}}n\right)}^{1/3}.\hfill \\ \left(f\right)\phantom{\rule{0.3em}{0ex}}{S}_{n3}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C\left({2}^{-m}\phantom{\rule{0.3em}{0ex}}log\phantom{\rule{0.3em}{0ex}}n\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{n}^{1/2}{2}^{-2m}\right).\hfill \end{array}$
(3.12)

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

$\mathsf{\text{E}}\left({S}_{n12}^{2}\right)\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}{c}_{2}\pi {\sigma }_{n1}^{-2}\phantom{\rule{0.3em}{0ex}}\sum _{i=1}^{n}{\left({\stackrel{̃}{h}}_{i}\right)}^{2}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C{\left({n}^{-1}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{2}^{-m}\right)}^{2},$

By this and the Markov inequality, we have

$P\left\{\left|{S}_{n12}\right|\phantom{\rule{0.3em}{0ex}}\ge \phantom{\rule{0.3em}{0ex}}C{\left({n}^{-1}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{2}^{-m}\right)}^{2/3}\right\}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C{\left({n}^{-1}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{2}^{-m}\right)}^{2/3}.$
(3.13)
1. (b)

Applying Lemmas 3.1, A.4, and A.5, we get that

$\begin{array}{c}\mathsf{\text{E}}{S}_{n13}^{2}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{\sigma }_{n1}^{-2}\phantom{\rule{0.3em}{0ex}}\cdot \phantom{\rule{0.3em}{0ex}}{c}_{2\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}}\sum _{i=1}^{n}{\left(\sum _{j=1}^{n}{u}_{j}\phantom{\rule{0.3em}{0ex}}\underset{{A}_{j}}{\int }{E}_{m}\left({t}_{i},\phantom{\rule{0.3em}{0ex}}s\right)ds\right)}^{2}\hfill \\ \phantom{\rule{0.4em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}{c}_{2}{\sigma }_{n1}^{-2}\phantom{\rule{0.3em}{0ex}}\cdot \underset{1\le i,j\le n}{max}\phantom{\rule{0.3em}{0ex}}\left|\underset{{A}_{j}}{\int }{E}_{m}\left({t}_{i},\phantom{\rule{0.3em}{0ex}}s\right)ds\right|\phantom{\rule{0.3em}{0ex}}\cdot \underset{1\le j\le n}{max}\phantom{\rule{0.3em}{0ex}}\sum _{i=1}^{n}\left|\underset{{A}_{j}}{\int }{E}_{m}\left({t}_{i},\phantom{\rule{0.3em}{0ex}}s\right)\phantom{\rule{0.3em}{0ex}}ds\right|\phantom{\rule{0.3em}{0ex}}\cdot {\left(\underset{1\le m\le n}{max}\left|\sum _{i=1}^{m}{u}_{{j}_{i}}\right|\right)}^{2}\hfill \\ \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C{2}^{m}{n}^{-1}\phantom{\rule{0.3em}{0ex}}{log}^{2}\phantom{\rule{0.3em}{0ex}}n.\hfill \end{array}$

Therefore

$P\left\{\left|{S}_{n13}\right|\phantom{\rule{0.3em}{0ex}}\ge \phantom{\rule{0.3em}{0ex}}C{\left({2}^{m}{n}^{-1}\phantom{\rule{0.3em}{0ex}}{log}^{2}n\right)}^{1/3}\right\}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C{\left({2}^{m}{n}^{-1}\phantom{\rule{0.3em}{0ex}}{log}^{2}\phantom{\rule{0.3em}{0ex}}n\right)}^{1/3}.$
(3.14)
1. (c)

Changing the order of summation in {S n 21}, similarly to the calculation for $\mathsf{\text{E}}{S}_{n13}^{2}$,

$\begin{array}{c}\mathsf{\text{E}}{S}_{n21}^{2}\phantom{\rule{0.3em}{0ex}}\le {\sigma }_{n1}^{-2}\phantom{\rule{0.3em}{0ex}}\cdot \phantom{\rule{0.3em}{0ex}}{c}_{2}\phantom{\rule{0.3em}{0ex}}\sum _{j=1}^{n}{\left(\sum _{i=1}^{n}{u}_{i}\phantom{\rule{0.3em}{0ex}}\underset{{A}_{j}}{\int }{E}_{m}\left({t}_{i},\phantom{\rule{0.3em}{0ex}}s\right)ds\right)}^{2}\hfill \\ \phantom{\rule{2.8em}{0ex}}\le \phantom{\rule{2.77695pt}{0ex}}{c}_{2}{\sigma }_{n1}^{-2}\phantom{\rule{2.77695pt}{0ex}}\cdot \underset{1\le i,j\le n}{max}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\left|\underset{{A}_{j}}{\int }{E}_{m}\left({t}_{i},\phantom{\rule{0.3em}{0ex}}s\right)ds\right|\phantom{\rule{0.3em}{0ex}}\cdot \phantom{\rule{0.3em}{0ex}}\underset{1\le i\le n}{max}\phantom{\rule{0.3em}{0ex}}\sum _{j=1}^{n}\phantom{\rule{0.3em}{0ex}}\left|\underset{{A}_{j}}{\int }{E}_{m}\left({t}_{i},\phantom{\rule{0.3em}{0ex}}s\right)ds\right|\phantom{\rule{0.3em}{0ex}}\cdot {\left(\underset{1\le m\le n}{max}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\left|\sum _{i=1}^{m}{u}_{{j}_{i}}\right|\right)}^{2}\hfill \\ \phantom{\rule{2.8em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C{2}^{m}{n}^{-1}\phantom{\rule{0.3em}{0ex}}{log}^{2}\phantom{\rule{0.3em}{0ex}}n.\hfill \end{array}$

Therefore, we obtain that

$P\phantom{\rule{0.3em}{0ex}}\left\{\left|{S}_{n21}\right|\phantom{\rule{0.3em}{0ex}}\ge \phantom{\rule{0.3em}{0ex}}C{\left({2}^{m}{n}^{-1}\phantom{\rule{0.3em}{0ex}}{log}^{2}\phantom{\rule{0.3em}{0ex}}n\right)}^{1/3}\right\}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C{\left({2}^{m}{n}^{-1}\phantom{\rule{0.3em}{0ex}}{log}^{2}\phantom{\rule{0.3em}{0ex}}n\right)}^{1/3}.$
(3.15)
1. (d)

Similarly, by Lemmas 3.1, A.4, A.5, and Remark 2.1, we get that

$\begin{array}{c}\mathsf{\text{E}}{S}_{n22}^{2}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}{c}_{2}{\sigma }_{n1}^{-2}\phantom{\rule{0.3em}{0ex}}\cdot \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\sum _{i=1}^{n}{\left(\sum _{j=1}^{n}{\stackrel{̃}{h}}_{j}\phantom{\rule{0.3em}{0ex}}\underset{{A}_{i}}{\int }{E}_{m}\left({t}_{j},\phantom{\rule{0.3em}{0ex}}s\right)ds\right)}^{2}\hfill \\ \phantom{\rule{2.8em}{0ex}}\le \phantom{\rule{2.77695pt}{0ex}}{c}_{2}{\sigma }_{n1}^{-2}\phantom{\rule{2.77695pt}{0ex}}\cdot {\left(\underset{{t}_{j}}{sup}\phantom{\rule{0.3em}{0ex}}\left|{\stackrel{̃}{h}}_{j}\right|\right)}^{2}\phantom{\rule{0.3em}{0ex}}\sum _{i=1}^{n}\left(\phantom{\rule{0.3em}{0ex}}\sum _{j=1}^{n}\left|\underset{{A}_{i}}{\int }{E}_{m}\left({t}_{j},\phantom{\rule{0.3em}{0ex}}s\right)ds\right|\phantom{\rule{0.3em}{0ex}}\right)\phantom{\rule{0.3em}{0ex}}\left(\phantom{\rule{0.3em}{0ex}}\sum _{j=1}^{n}\left|\underset{{A}_{i}}{\int }{E}_{m}\left({t}_{i},\phantom{\rule{0.3em}{0ex}}s\right)ds\right|\phantom{\rule{0.3em}{0ex}}\right)\hfill \\ \phantom{\rule{2.8em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}{c}_{2}{\sigma }_{n1}^{-2}\phantom{\rule{0.3em}{0ex}}\cdot {\left(\underset{{t}_{j}}{sup}\phantom{\rule{0.3em}{0ex}}\left|{\stackrel{̃}{h}}_{j}\right|\right)}^{2}\phantom{\rule{0.3em}{0ex}}\cdot \phantom{\rule{0.3em}{0ex}}n\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C{\left({n}^{-1}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{2}^{-m}\right)}^{2}.\hfill \end{array}$

Thus, we have

$\phantom{\rule{0.3em}{0ex}}P\left\{|{S}_{n22}|\phantom{\rule{0.3em}{0ex}}\ge \phantom{\rule{0.3em}{0ex}}C{\left({n}^{-1}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{2}^{-m}\right)}^{2/3}\right\}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C{\left({n}^{-1}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{2}^{-m}\right)}^{2/3}.$
(3.16)
1. (e)

We write that

$\phantom{\rule{0.3em}{0ex}}{S}_{n23}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{\sigma }_{n1}^{-1}\phantom{\rule{0.3em}{0ex}}\sum _{j=1}^{n}\left\{\sum _{i=1}^{n}\underset{{A}_{j}}{\int }{E}_{m}\left({t}_{i},\phantom{\rule{0.3em}{0ex}}s\right)ds\phantom{\rule{0.3em}{0ex}}\left(\sum _{l=1}^{n}\underset{{A}_{l}}{\int }{E}_{m}\left({t}_{i},\phantom{\rule{0.3em}{0ex}}s\right)\phantom{\rule{0.3em}{0ex}}{u}_{l}ds\phantom{\rule{0.3em}{0ex}}\right)\right\}{\epsilon }_{j}.$

Similarly to the calculation for $\mathsf{\text{E}}{S}_{n13}^{2}$ by (3.13), Lemmas 3.1, A.4, and A.5, we obtain that

$\begin{array}{c}\mathsf{\text{E}}{S}_{n23}^{2}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{c}_{2}\pi {\sigma }_{n1}^{-2}\phantom{\rule{0.3em}{0ex}}\sum _{j=1}^{n}{\left[\sum _{i=1}^{n}\underset{{A}_{j}}{\int }{E}_{m}\left({t}_{i},\phantom{\rule{0.3em}{0ex}}s\right)ds\phantom{\rule{0.3em}{0ex}}\left(\sum _{l=1}^{n}\underset{{A}_{l}}{\int }{E}_{m}\left({t}_{i},\phantom{\rule{0.3em}{0ex}}s\right)\phantom{\rule{0.3em}{0ex}}{u}_{l}ds\phantom{\rule{0.3em}{0ex}}\right)\right]}^{2}\hfill \\ \phantom{\rule{2.6em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}{c}_{2}\pi {\sigma }_{n1}^{-2}\phantom{\rule{0.3em}{0ex}}\underset{1\le i,j\le n}{max}\phantom{\rule{0.3em}{0ex}}\underset{{A}_{i}}{\int }{E}_{m}\left({t}_{j},\phantom{\rule{0.3em}{0ex}}s\right)ds\cdot \phantom{\rule{0.3em}{0ex}}\underset{1\le j\le n}{max}\phantom{\rule{0.3em}{0ex}}\sum _{i=1}^{n}\underset{{A}_{i}}{\int }{E}_{m}\left({t}_{j},\phantom{\rule{0.3em}{0ex}}s\right)ds\hfill \\ \phantom{\rule{2.5em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\cdot \phantom{\rule{0.3em}{0ex}}{\left(\underset{1\le l\le n}{max}\phantom{\rule{0.3em}{0ex}}\sum _{j=1}^{n}\underset{{A}_{l}}{\int }{E}_{m}\left({t}_{j},\phantom{\rule{0.3em}{0ex}}s\right)ds\phantom{\rule{0.3em}{0ex}}\cdot \underset{1\le m\le n}{max}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\left|\sum _{i=1}^{m}{u}_{{j}_{i}}\right|\right)}^{2}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C\phantom{\rule{0.3em}{0ex}}\cdot {2}^{m}{n}^{-1}\phantom{\rule{0.3em}{0ex}}{log}^{2}\phantom{\rule{0.3em}{0ex}}n.\hfill \end{array}$

Hence, we have

$P\phantom{\rule{0.3em}{0ex}}\left\{\left|{S}_{n23}\right|\phantom{\rule{0.3em}{0ex}}\ge \phantom{\rule{0.3em}{0ex}}C{\left({2}^{m}{n}^{-1}\phantom{\rule{0.3em}{0ex}}{log}^{2}\phantom{\rule{0.3em}{0ex}}n\right)}^{1/3}\right\}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C{\left({2}^{m}{n}^{-1}\phantom{\rule{0.3em}{0ex}}{log}^{2}\phantom{\rule{0.3em}{0ex}}n\right)}^{1/3}.$
(3.17)
1. (f)

By assumption (A2), Remarks 2.1, Lemma A.5, and the Abel inequality, we have

Thus, by Lemma 3.1 we obtain

${S}_{n3}\phantom{\rule{0.3em}{0ex}}\le {C}_{1}\left({n}^{-1}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{2}^{-m}\phantom{\rule{0.3em}{0ex}}\right)\phantom{\rule{0.3em}{0ex}}log\phantom{\rule{0.3em}{0ex}}n\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{C}_{2}\sqrt{n}{\left({n}^{-1}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{2}^{-m}\right)}^{2}\phantom{\rule{0.3em}{0ex}}\le C\left({2}^{-m}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}log\phantom{\rule{0.3em}{0ex}}n\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{n}^{1/2}{2}^{-2m}\right).\phantom{\rule{0.3em}{0ex}}$
(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 ${s}_{n}^{2}\phantom{\rule{0.3em}{0ex}}\triangleq \phantom{\rule{0.3em}{0ex}}{\sum }_{w=1}^{k}\mathsf{\text{Var}}\left({y}_{1nw}\right),$ then

$\left|{s}_{n}^{2}\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}1\right|\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C\phantom{\rule{0.3em}{0ex}}\left({\gamma }_{1n}^{1/2}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{\gamma }_{2n}^{1/2}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{\gamma }_{3n}^{1/2}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}u\left(q\right)\phantom{\rule{0.3em}{0ex}}\right).$

Proof. Let ${\Gamma }_{n}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{\sum }_{1\le iCov(y1ni, y1nj), then ${s}_{n}^{2}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\mathsf{\text{E}}{\left({S}_{n111}^{\prime }\right)}^{2}\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}2{\Gamma }_{n}.$ By (3.5) and (3.6), it is easy to verify that $\mathsf{\text{E}}{\left({S}_{n11}\right)}^{2}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}1$, and

$\mathsf{\text{E}}\phantom{\rule{0.3em}{0ex}}{\left({S}_{n111}^{\prime }\right)}^{2}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}1\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}\mathsf{\text{E}}{\left({S}_{n111}^{″}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{S}_{n111}^{‴}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{S}_{n112}\right)}^{2}\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}2\mathsf{\text{E}}\phantom{\rule{0.3em}{0ex}}\left[{S}_{n11}\left({S}_{n111}^{″}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{S}_{n111}^{‴}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{S}_{n112}\right)\right].$

According to Lemma 3.2, the C r -inequality and the Cauchy-Schwarz inequality

$\phantom{\rule{0.3em}{0ex}}\mathsf{\text{E}}{\left({S}_{n111}^{″}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{S}_{n111}^{‴}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{S}_{n112}\right)}^{2}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C\phantom{\rule{0.3em}{0ex}}\left({\gamma }_{1n}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{\gamma }_{2n}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{\gamma }_{3n}\right),$

and

$\left|E\left[{S}_{n11}\left({S}_{n111}^{″}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{S}_{n111}^{‴}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{S}_{n112}\right)\right]\phantom{\rule{0.3em}{0ex}}\right|\le \phantom{\rule{0.3em}{0ex}}C\phantom{\rule{0.3em}{0ex}}\left({\gamma }_{1n}^{1/2}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{\gamma }_{2n}^{1/2}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{\gamma }_{3n}^{1/2}\right).$

Thus, we obtain

$|E{\left({S}_{n111}^{\prime }\right)}^{2}\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}1|\le \phantom{\rule{0.3em}{0ex}}C\phantom{\rule{0.3em}{0ex}}\left({\gamma }_{1n}^{1/2}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{\gamma }_{2n}^{1/2}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{\gamma }_{3n}^{1/2}\right).$
(3.19)

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

$\begin{array}{c}\phantom{\rule{1.5em}{0ex}}|{\Gamma }_{n}|\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}\sum _{1\le i\le j\le k}\sum _{{s}_{1}={k}_{i}}^{{k}_{i}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}p-1}\sum _{{t}_{1}={k}_{j}}^{{k}_{j}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}p-1}\phantom{\rule{0.3em}{0ex}}|\mathsf{\text{Cov}}\left({Z}_{n{s}_{1}},\phantom{\rule{0.3em}{0ex}}{Z}_{n{t}_{1}}\right)|\hfill \\ \le \phantom{\rule{0.3em}{0ex}}\frac{C}{n}\sum _{1\le i\le j\le k}\sum _{{s}_{1}={k}_{i}}^{{k}_{i}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}p-1}\sum _{{t}_{1}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{k}_{j}}^{{k}_{j}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}p-1}\sum _{u=max\left\{1,{s}_{1}\phantom{\rule{0.3em}{0ex}}-n\right\}}^{min\left\{n,{s}_{1}\phantom{\rule{0.3em}{0ex}}+n\right\}}\phantom{\rule{0.3em}{0ex}}\sum _{v=max\left\{1,{t}_{1}\phantom{\rule{0.3em}{0ex}}-n\right\}}^{min\left\{n,{t}_{1\phantom{\rule{0.3em}{0ex}}}+n\right\}}|{u}_{u-{s}_{1}\phantom{\rule{0.3em}{0ex}}}{u}_{v-{t}_{1}}||{a}_{u-{s}_{1}}\phantom{\rule{0.3em}{0ex}}{a}_{v-{t}_{1}}||\phantom{\rule{0.3em}{0ex}}\mathsf{\text{Cov}}\left({e}_{{s}_{1}},\phantom{\rule{0.3em}{0ex}}{e}_{{t}_{1}}\right)|\phantom{\rule{0.3em}{0ex}}\hfill \\ \le \phantom{\rule{0.3em}{0ex}}\frac{C}{n}\phantom{\rule{0.3em}{0ex}}\sum _{1\le i\le j\le k}\sum _{{s}_{1}={k}_{i}}^{{k}_{i}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}p-1}\sum _{{t}_{1}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{k}_{j}}^{{k}_{j}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}p-1}\sum _{u=max\left\{1,{s}_{1}\phantom{\rule{0.3em}{0ex}}-n\right\}}^{min\left\{n,{s}_{1}\phantom{\rule{0.3em}{0ex}}+n\right\}}\phantom{\rule{0.3em}{0ex}}\sum _{v=max\left\{1,{t}_{1}\phantom{\rule{0.3em}{0ex}}-n\right\}}^{min\left\{n,{t}_{1\phantom{\rule{0.3em}{0ex}}}+n\right\}}|{a}_{u-{s}_{1}}\phantom{\rule{0.3em}{0ex}}{a}_{v-{t}_{1}}|\phantom{\rule{0.3em}{0ex}}{\alpha }^{\delta /\left(2+\delta \right)}\phantom{\rule{0.3em}{0ex}}\left({t}_{1}\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}{s}_{1}\right)||\phantom{\rule{0.3em}{0ex}}{e}_{{t}_{1}}||{}_{2+\delta }\phantom{\rule{0.3em}{0ex}}\cdot \phantom{\rule{0.3em}{0ex}}||{e}_{{s}_{1}}||{}_{2+\delta }\hfill \\ \le \phantom{\rule{0.3em}{0ex}}\frac{C}{n}\phantom{\rule{0.3em}{0ex}}\sum _{i=1}^{k-1}\sum _{{s}_{1}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{k}_{i}}^{{k}_{i}+p-1}\sum _{u=max\left\{1,{s}_{1}-n\right\}}^{min\phantom{\rule{0.3em}{0ex}}\left\{n,{s}_{1}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}n\right\}}\sum _{j=i+1}^{k}\sum _{{t}_{1}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{k}_{j}}^{{k}_{j}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}p-1}\sum _{v=max\left\{1,{t}_{1}-n\right\}}^{min\left\{n,{t}_{1}+n\right\}}{\alpha }^{\delta /\left(2+\delta \right)}\phantom{\rule{0.3em}{0ex}}\left({t}_{1}\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}{s}_{1}\right)|\phantom{\rule{0.3em}{0ex}}{a}_{u-s1}\phantom{\rule{0.3em}{0ex}}{a}_{v-{t}_{1}}|\hfill \\ \le \phantom{\rule{0.3em}{0ex}}\frac{C}{n}\phantom{\rule{0.3em}{0ex}}\sum _{i=1}^{k-1}\sum _{{s}_{1}\phantom{\rule{0.3em}{0ex}}={k}_{i}}^{{k}_{i}+p-1}\sum _{j\phantom{\rule{0.3em}{0ex}}=i+1}^{k}\sum _{{t}_{1}\phantom{\rule{0.3em}{0ex}}={k}_{j}}^{{k}_{j}\phantom{\rule{0.3em}{0ex}}+p-1}{\alpha }^{\delta /\left(2+\delta \right)}\phantom{\rule{0.3em}{0ex}}\left({t}_{1}\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}{s}_{1}\right)\hfill \\ \le \phantom{\rule{0.3em}{0ex}}\frac{C}{n}\phantom{\rule{0.3em}{0ex}}\sum _{i\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}1}^{k-1}\sum _{{s}_{1}={k}_{i}}^{{k}_{i}+p-1}\sum _{{t}_{1}:\left|{t}_{1}\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}{s}_{1}\right|\phantom{\rule{0.3em}{0ex}}\ge q}{\alpha }^{\delta /\left(2+\delta \right)}\phantom{\rule{0.3em}{0ex}}\left({t}_{1}\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}{s}_{1}\right)\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}Ckp{n}^{-1}u\left(q\right)\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}Cu\left(q\right).\hfill \end{array}$
(3.20)

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

$\left|{s}_{n}^{2}\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}1\right|\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}|E{\left({S}_{1n}^{\prime }\right)}^{2}\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}1|\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}2\left|{\Gamma }_{n}\right|\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C\phantom{\rule{0.3em}{0ex}}\left({\gamma }_{1n}^{1/2}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{\gamma }_{2n}^{1/2}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{\gamma }_{3n}^{1/2}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}u\left(q\right)\right).\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}$

♣

Assume that {η1nw: w = 1, . . . , k} are independent random variables, and its distribution is the same as that of {y1nw, w = 1, . . . , k}. Set ${T}_{n}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{\sum }_{w\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}1}^{k}{\eta }_{1nw}$, ${B}_{n1}^{2}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{\sum }_{w=1}^{k}\mathsf{\text{Var}}\left({\eta }_{1nw}\right)$. Clearly ${B}_{n1}^{2}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{s}_{n1}^{2}.$ Then, we have the following lemmas:

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

$\underset{u}{sup}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\left|P\phantom{\rule{0.3em}{0ex}}\left({T}_{n}/\phantom{\rule{0.3em}{0ex}}{B}_{n1}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}u\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}\Phi \phantom{\rule{0.3em}{0ex}}\left(u\right)\right|\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C{\gamma }_{2n}^{\rho }.\phantom{\rule{0.3em}{0ex}}$

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

$\underset{u}{sup\phantom{\rule{0.3em}{0ex}}}\phantom{\rule{0.3em}{0ex}}|P\phantom{\rule{0.3em}{0ex}}\left({T}_{n}\phantom{\rule{0.3em}{0ex}}/\phantom{\rule{0.3em}{0ex}}{B}_{n1}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}u\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}\Phi \phantom{\rule{0.3em}{0ex}}\left(u\right)\phantom{\rule{0.3em}{0ex}}|\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C\phantom{\rule{0.3em}{0ex}}\frac{{\sum }_{w=1}^{k}\mathsf{\text{E}}{\left|{y}_{1nw}\right|}^{r}}{{B}_{n1}^{r}},\phantom{\rule{1em}{0ex}}\mathsf{\text{for}}\phantom{\rule{0.3em}{0ex}}2
(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 $\frac{r\left(r+\tau \right)}{2\tau }\phantom{\rule{0.3em}{0ex}}=\frac{\left(1+\rho \right)\left(2+\delta \right)}{\delta -2\rho }\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}<\phantom{\rule{0.3em}{0ex}}\lambda .$ According to Lemmas 3.1 and A.1(ii), and the C r -inequality, taking ε = ρ, we get that

$\begin{array}{c}\sum _{w=1}^{k}\mathsf{\text{E|}}{y}_{1nw}{|}^{r}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C\phantom{\rule{0.3em}{0ex}}\sum _{w=1}^{k}{p}^{\rho }\phantom{\rule{0.3em}{0ex}}\sum _{j={k}_{w}}^{{k}_{w}+p-1}{\left|\sum _{i=max\left\{1,j-n\right\}}^{min\left\{n,j+n\right\}}{\sigma }_{n1}^{-1}{u}_{i}{a}_{i-j}\right|}^{r}\phantom{\rule{0.3em}{0ex}}\mathsf{\text{E}}{\left|{e}_{j}\right|}^{r}\hfill \\ \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{\left[{\sum _{j={k}_{w}}^{{k}_{w}+p-1}\left(\sum _{i=max\left\{1,j-n\right\}}^{min\left\{n,j+n\right\}}{\sigma }_{n1}^{-1}{u}_{i}{a}_{i-j}\right)}^{2}\phantom{\rule{0.3em}{0ex}}{||{e}_{i}||}_{2+\delta }^{2}\right]}^{r/2}\hfill \\ \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C{\sigma }_{n1}^{-r}\phantom{\rule{0.3em}{0ex}}k{p}^{1+\rho }\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C{\gamma }_{2n}^{\rho }.\hfill \end{array}$
(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

$\underset{u}{sup}\phantom{\rule{0.3em}{0ex}}|P\left({S}_{n111}^{\prime }\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}u\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}P\left({T}_{n}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}u\right)|\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C\phantom{\rule{0.3em}{0ex}}\left\{{\gamma }_{2n}^{\rho }\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{\gamma }_{4n}^{1/4}\right\}.$

Proof. Let ϕ1(t) and ψ1 (t) be the characteristic functions of ${S}_{n111}^{\prime }$and Tn 1, respectively.

Since

${\psi }_{1}\left(t\right)\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\mathsf{\text{E}}\left(exp\left\{\mathbf{i}t{T}_{n1}\right\}\right)\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\prod _{w=1}^{k}\mathsf{\text{E}}exp\left\{\mathbf{i}t{\eta }_{1nw}\right\}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\prod _{w=1}^{k}\mathsf{\text{E}}exp\left\{\mathbf{i}t{y}_{1nw}\right\},$

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

$\begin{array}{cc}\hfill |{\varphi }_{1}\left(t\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}{\psi }_{1}\left(t\right)|\phantom{\rule{0.3em}{0ex}}& \le \phantom{\rule{0.3em}{0ex}}C\left|t\right|{\alpha }^{1/2}\left(q\right)\sum _{w=1}^{k}||{y}_{1nw}|{|}_{2}\hfill \\ \le \phantom{\rule{2.77695pt}{0ex}}C\left|t\right|{\alpha }^{1/2}\left(q\right)\sum _{w=1}^{k}{\left\{\mathsf{\text{E}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{\left[\sum _{i={k}_{w}}^{{k}_{w}+p-1}{\sigma }_{n}^{-1}\left(\sum _{j=max\left\{1,i-n\right\}}^{min\left\{n,i+n\right\}}{u}_{i}{a}_{j-i}\right){e}_{i}\right]}^{2}\right\}}^{1/2}\hfill \\ \le \phantom{\rule{2.77695pt}{0ex}}C\left|t\right|{\alpha }^{1/2}\left(q\right)\sum _{w=1}^{k}\phantom{\rule{0.3em}{0ex}}{\left[\sum _{i={k}_{w}}^{{k}_{w}+p-1}{\sigma }_{n}^{-2}{\left(\sum _{j=max\left\{1,i-n\right\}\phantom{\rule{0.3em}{0ex}}}^{min\left\{n,i+n\right\}}\left|{u}_{i}{a}_{j-i}\right|\right)}^{2}\phantom{\rule{0.3em}{0ex}}{\left(\mathsf{\text{E}}{\left|{e}_{i}\right|}^{2+\delta }\right)}^{2/2+\delta }\right]}^{1/2}\phantom{\rule{0.3em}{0ex}}\hfill \\ \le \phantom{\rule{2.77695pt}{0ex}}C\left|t\right|{\alpha }^{1/2}\left(q\right){\left[k\sum _{w=1}^{k}\sum _{i={k}_{w}}^{{k}_{w}+p-1}{\sigma }_{n}^{-2}\right]}^{1/2}\phantom{\rule{0.3em}{0ex}}\le C\left|t\right|{\alpha }^{1/2}\left(q\right)k{p}^{1/2}{n}^{-1/2}\hfill \\ \le \phantom{\rule{2.77695pt}{0ex}}C\left|t\right|{\left(k\alpha \left(q\right)\right)}^{1/2}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{2.77695pt}{0ex}}C\left|t\right|{\gamma }_{4n}^{1/2}.\hfill \end{array}$

Therefore

$\underset{-T}{\overset{T}{\int }}\left|\frac{{\varphi }_{1}\left(t\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}{\psi }_{1}\left(t\right)}{t}\right|dt\phantom{\rule{0.3em}{0ex}}\le C{\gamma }_{4n}^{1/2}T.$
(3.23)

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

$T\underset{u}{sup}\phantom{\rule{0.3em}{0ex}}\underset{\left|y\right|\le c/T}{\int }|P\left({T}_{n}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}u+y\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}P\left({T}_{n}\le u\right)|dy\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C\phantom{\rule{0.3em}{0ex}}\left\{{\gamma }_{2n}^{\rho }+1/T\right\}.$
(3.24)

Therefore, combining (3.23) and (3.24), choosing $T\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{\gamma }_{4n}^{-1/4}$, and using the Esseen inequality (see [, Theorem 5.3]), we conclude that

$\begin{array}{c}\phantom{\rule{5em}{0ex}}\underset{u}{sup}\phantom{\rule{0.3em}{0ex}}|\phantom{\rule{0.3em}{0ex}}P\left({S}_{n111}^{\prime }\phantom{\rule{0.3em}{0ex}}\le u\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}P\left({T}_{n}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}u\right)|\hfill \\ \phantom{\rule{3.5em}{0ex}}\le \underset{-T}{\overset{T}{\int }}\left|\frac{{\varphi }_{1}\left(t\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}{\psi }_{1}\left(t\right)}{t}\right|dt+T\underset{u}{sup}\phantom{\rule{0.3em}{0ex}}\underset{\left|y\right|\phantom{\rule{0.3em}{0ex}}\le c/T}{\int }|{\stackrel{̃}{G}}_{n}\left(u+y\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{2.77695pt}{0ex}}{\stackrel{̃}{G}}_{n}\left(u\right)|dy\hfill \\ \phantom{\rule{3.5em}{0ex}}=\phantom{\rule{0.3em}{0ex}}C\phantom{\rule{0.3em}{0ex}}\left\{{\gamma }_{2n}^{\rho }+{\gamma }_{4n}^{1/4}\right\}.\hfill \end{array}$

♣

## 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:

$\begin{array}{c}\hfill {S}_{ng}\phantom{\rule{0.3em}{0ex}}:\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{\sigma }_{n2}^{-1}\left({\mathit{ĝ}}_{n}\left(t\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}\mathsf{\text{E}}{\mathit{ĝ}}_{n}\left(t\right)\right)\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{\sigma }_{n2}^{-1}\sum _{i=1}^{n}{\epsilon }_{i}{\int }_{{A}_{i}}{E}_{m}\left(t,s\right)ds+{\sigma }_{n2}^{-1}\sum _{i=1}^{n}{x}_{i}\left(\beta \phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}{\stackrel{^}{\beta }}_{n}\right)\phantom{\rule{0.3em}{0ex}}{\int }_{{A}_{i}}{E}_{m}\left(t,s\right)ds\\ \phantom{\rule{4em}{0ex}}-{\sigma }_{n2}^{-1}\sum _{i=1}^{n}{x}_{i}\left(\beta \phantom{\rule{0.3em}{0ex}}-\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{E}}{\stackrel{^}{\beta }}_{n}\right)\phantom{\rule{0.3em}{0ex}}{\int }_{{A}_{i}}{E}_{m}\left(t,s\right)ds\phantom{\rule{0.3em}{0ex}}=:\phantom{\rule{0.3em}{0ex}}{H}_{1n}+{H}_{2n}+{H}_{3n}.\hfill \end{array}$

Let us decompose the vector H1ninto two parts:

${H}_{1n}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{\sigma }_{n2}^{-1}\phantom{\rule{0.3em}{0ex}}\sum _{i=1}^{n}\underset{{A}_{i}}{\int }{E}_{m}\left(t,s\right)ds\left(\sum _{j=-n}^{n}{a}_{j}{e}_{i-j}\right)+\phantom{\rule{0.3em}{0ex}}{\sigma }_{n2}^{-1}\phantom{\rule{0.3em}{0ex}}\sum _{i=1}^{n}\underset{{A}_{i}}{\int }{E}_{m}\left(t,s\right)ds\phantom{\rule{0.3em}{0ex}}\left(\sum _{|j|>n}{a}_{j}{e}_{i-j}\right)\phantom{\rule{0.3em}{0ex}}=:{H}_{11n}+{H}_{12n},\phantom{\rule{0.3em}{0ex}}$

Where

${H}_{11n\phantom{\rule{0.3em}{0ex}}}=\phantom{\rule{2.77695pt}{0ex}}{\sigma }_{n2}^{-1}\sum _{l=1-n}^{2n}\left(\sum _{i=max\left(1,l-n\right)}^{min\left(n,n+l\right)}{a}_{i-l}\phantom{\rule{0.3em}{0ex}}\underset{{A}_{i}}{\int }{E}_{m}\left(t,s\right)ds\right)\phantom{\rule{0.3em}{0ex}}{e}_{l}\phantom{\rule{2.77695pt}{0ex}}=\sum _{l=1-n}^{2n}{M}_{nl}.$

Similar to Sn 111in (3.6), H11ncan be split as ${H}_{11n}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{H}_{11n}^{\prime }\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{H}_{11n}^{″}\phantom{\rule{0.3em}{0ex}}+{H}_{11n}^{‴},$ where

$\begin{array}{c}{H}_{11n}^{\prime }\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{2.77695pt}{0ex}}{\sum }_{w=1}^{k}{y}_{2nw},\phantom{\rule{2.77695pt}{0ex}}{H}_{11n}^{″}\phantom{\rule{2.77695pt}{0ex}}=\phantom{\rule{0.3em}{0ex}}\sum _{w=1}^{k}{y}_{2nw}^{\prime },\phantom{\rule{2.77695pt}{0ex}}{H}_{11n}^{‴}\phantom{\rule{2.77695pt}{0ex}}=\phantom{\rule{0.3em}{0ex}}{y}_{2nk+1}^{\prime },\phantom{\rule{0.3em}{0ex}}\hfill \\ {y}_{2nw}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\sum _{i={k}_{w}}^{{k}_{w}+p-1}{M}_{ni},\phantom{\rule{0.3em}{0ex}}{y}_{2nw}^{\prime }\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\sum _{i={l}_{w}}^{{l}_{w}+q-1}{M}_{ni},\phantom{\rule{2.77695pt}{0ex}}{y}_{2nk+1}^{\prime }\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\sum _{i=k\left(p+q\right)-n+1}^{2n}{M}_{ni}.\hfill \end{array}$
(4.1)

Then

${S}_{ng}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{H}_{11n}^{\prime }\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{H}_{11n}^{″}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{H}_{11n}^{‴}\phantom{\rule{0.3em}{0ex}}+{H}_{2n}\phantom{\rule{0.3em}{0ex}}+{H}_{3n}.$
(4.2)

Set ${T}_{n2}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{\sum }_{w=1}^{k}{\eta }_{2nw}$, ${B}_{n2}^{2}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{\sum }_{w=1}^{k}\mathsf{\text{Var}}\left({\eta }_{2nw}\right)$. 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

$\begin{array}{c}\phantom{\rule{2em}{0ex}}\mathsf{\text{E}}{\left({H}_{11n}^{″}\right)}^{2}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C{\lambda }_{1n},\phantom{\rule{0.3em}{0ex}}\mathsf{\text{E}}{\left({H}_{11n}^{‴}\right)}^{2}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C{\lambda }_{2n},\phantom{\rule{0.3em}{0ex}}\mathsf{\text{E}}{H}_{12n}^{2}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C{\lambda }_{3n};\\ P\left(\left|{H}_{11n}^{″}\right|\ge \phantom{\rule{0.3em}{0ex}}{\lambda }_{1n}^{1/3}\right)\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C{\lambda }_{1n}^{1/3},\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}P\left(\left|{H}_{11n}^{‴}\right|\ge \phantom{\rule{0.3em}{0ex}}{\lambda }_{2n}^{1/3}\right)\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C{\lambda }_{2n}^{1/3},\phantom{\rule{0.3em}{0ex}}P\left(\left|{H}_{12n}\right|\ge \phantom{\rule{0.3em}{0ex}}{\lambda }_{3n}^{1/3}\right)\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C{\lambda }_{3n}^{1/3}.\end{array}$

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

$E{\left|{H}_{2n}\right|}^{2+\delta }\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}c{\lambda }_{5n}^{2+\delta },\phantom{\rule{0.3em}{0ex}}P\left(\left|{H}_{2n}\right|\phantom{\rule{0.3em}{0ex}}>\phantom{\rule{0.3em}{0ex}}{\lambda }_{5n}^{\left(2+\delta \right)\phantom{\rule{0.3em}{0ex}}/\phantom{\rule{0.3em}{0ex}}\left(3+\delta \right)}\right)\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}{\lambda }_{5n}^{\left(2+\delta \right)\phantom{\rule{0.3em}{0ex}}/\phantom{\rule{0.3em}{0ex}}\left(3+\delta \right)},\phantom{\rule{0.3em}{0ex}}\left|{H}_{3n}\right|\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}c{\lambda }_{5n}.$

Lemma 4.3. Under the conditions of Theorem 2.2, set ${s}_{n2}^{2}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{\sum }_{w=1}^{k}\mathsf{\text{Var}}\left({y}_{2nw}\right),$ then

$\left|{s}_{n2}^{2}\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}1\right|\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C\phantom{\rule{0.3em}{0ex}}\left({\lambda }_{1n}^{1/2}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{\lambda }_{2n}^{1/2}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{\lambda }_{3n}^{1/2}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}u\left(q\right)\right).$

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

$\underset{u}{sup}\left|P\left({T}_{n2}/{B}_{n2}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}u\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}\Phi \phantom{\rule{0.3em}{0ex}}\left(u\right)\right|\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}c{\lambda }_{2n}^{\rho }.$

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

$\underset{u}{sup}|P\left({H}_{11n}^{\prime }\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}u\right)-\phantom{\rule{0.3em}{0ex}}P\left({T}_{n2}\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}u\right)|\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}C\phantom{\rule{0.3em}{0ex}}\left\{{\lambda }_{2n}^{\rho }\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{\gamma }_{4n}^{1/4}\right\}.$

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

$\underset{n\to \infty }{lim}\phantom{\rule{0.3em}{0ex}}{S}_{n}/n\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\underset{n\to \infty }{lim}\phantom{\rule{0.3em}{0ex}}\frac{1}{n}\phantom{\rule{0.3em}{0ex}}\sum _{i=1}^{n}{\stackrel{̃}{x}}_{i}^{2}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\sum ,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\mathsf{\text{where}}\phantom{\rule{0.3em}{0ex}}0\phantom{\rule{0.3em}{0ex}}<\phantom{\rule{0.3em}{0ex}}\sum <\phantom{\rule{0.3em}{0ex}}\infty .$
(4.3)

From (1.2), we write

$\begin{array}{cc}\hfill \frac{1}{n}\phantom{\rule{0.3em}{0ex}}\sum _{i=1}^{n}{\stackrel{̃}{x}}_{i}^{2}\phantom{\rule{0.3em}{0ex}}& =\phantom{\rule{0.3em}{0ex}}\frac{1}{n}\phantom{\rule{0.3em}{0ex}}\sum _{i=1}^{n}{u}_{i}^{2}+\phantom{\rule{0.3em}{0ex}}\frac{1}{n}\phantom{\rule{0.3em}{0ex}}\sum _{i=1}^{n}{\stackrel{̃}{h}}_{i}^{2}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}\frac{1}{n}\phantom{\rule{0.3em}{0ex}}\sum _{i=1}^{n}\phantom{\rule{0.3em}{0ex}}{\left(\sum _{j=1}^{n}\phantom{\rule{0.3em}{0ex}}{u}_{j}\phantom{\rule{0.3em}{0ex}}{\int }_{{A}_{j}}\phantom{\rule{0.3em}{0ex}}{E}_{m}\phantom{\rule{0.3em}{0ex}}\left({t}_{i},s\right)ds\right)}^{2}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}\frac{2}{n}\sum _{i=1}^{n}\phantom{\rule{0.3em}{0ex}}{u}_{i}{\stackrel{̃}{h}}_{i}\phantom{\rule{0.3em}{0ex}}\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}-\frac{2}{n}\sum _{i=1}^{n}{u}_{i}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\left(\sum _{j=1}^{n}\phantom{\rule{0.3em}{0ex}}{u}_{j}\phantom{\rule{0.3em}{0ex}}{\int }_{{A}_{j}}\phantom{\rule{0.3em}{0ex}}{E}_{m}\phantom{\rule{0.3em}{0ex}}\left({t}_{i},s\right)ds\right)\phantom{\rule{0.3em}{0ex}}-\frac{2}{n}\sum _{i=1}^{n}{\stackrel{̃}{h}}_{i}\phantom{\rule{0.3em}{0ex}}\left(\underset{}{\overset{}{}}\right)\hfill \end{array}$