By Lemma
A.1 with
\(a=\varepsilon_{1}+\varepsilon_{2}\), we have
$$ \begin{aligned}[b] \sup_{-\infty < t< \infty } \bigl\vert P(S_{n} \leq t)-\Phi (t) \bigr\vert &=\sup_{-\infty < t< \infty } \bigl\vert P(S_{n1}+S_{n2}+S_{n3}\leq t)-\Phi (t) \bigr\vert \\ &\leq \sup_{-\infty < t< \infty } \bigl\vert P(S_{n1}\leq t)-\Phi (t) \bigr\vert +\frac{a}{\sqrt{2 \pi }} \\ &\quad {}+P\bigl( \vert S_{n2} \vert \geq \varepsilon_{1} \bigr)+P\bigl( \vert S_{n3} \vert \geq \varepsilon _{2} \bigr). \end{aligned} $$
(3.1)
Firstly, we estimate
\(E(S_{n2})^{2}\) and
\(E(S_{n3})^{2}\). By the condition
\(|X_{i}|\leq d\) and Assumption
(A4), it is easy to find that
\(|Z_{n,i}|\leq C_{1}n^{-1/2}\),
\(|\xi _{j}|\leq C_{1}qn^{-1/2}\),
\(E\xi _{j}^{2}\leq C_{2}qn^{-1}\),
\(j=1,2,\ldots,n\). Combining the definition of LNQD with the definition of
\(\xi_{j}\),
\(j=1,2,\ldots,k\), we can easily prove that
\(\{\xi_{i}\} _{1\leq i\leq k}\) is LNQD. It follows from Lemma
A.2 that
$$\begin{aligned}& E(S_{n2})^{2}=E\Biggl(\sum_{j=1}^{k} \sum_{i=l_{j}}^{l_{j}+q-1}Z_{n,i} \Biggr)^{2} \leq C_{2}kq/n=C_{2}qp^{-1}=C_{2} \gamma_{1n}, \end{aligned}$$
(3.2)
$$\begin{aligned}& E(S_{n3})^{2}=E\Biggl(\sum_{i={k(p+q)+1}}^{n}Z_{n,i} \Biggr)^{2}\leq C_{3}\bigl[n-k(p+q)\bigr]/n \leq C_{3}(p+q)/n=C_{3}\gamma_{2n}. \end{aligned}$$
(3.3)
By (
3.2), (
3.3) and Lemma
A.3, choosing
\(\varepsilon_{1}=M\gamma_{1n} ^{1/2}(\log n)\),
\(\varepsilon_{2}= M\gamma_{2n}^{1/2}(\log n)\) and noting that
\(pq\leq n\), for large enough
M,
n, we have
$$\begin{aligned}& \begin{aligned}[b] P\bigl( \vert S_{n2} \vert > \varepsilon_{1}\bigr) &\leq 2\exp \biggl\{ -\frac{M^{2} \gamma _{1n}\log^{2} n}{2(2 C_{2} \gamma_{1n}+ C_{1} M q/\sqrt{n}\gamma_{1n}^{1/2}\log n)} \biggr\} \\ &\leq 2\exp \biggl\{ -\frac{M^{2}\log^{2} n}{2(2 C_{2}+C_{1} M\log n )} \biggr\} \leq Cn^{-1}, \end{aligned} \end{aligned}$$
(3.4)
$$\begin{aligned}& \begin{aligned}[b] P\bigl( \vert S_{n3} \vert > \varepsilon_{2}\bigr) &\leq 2\exp \biggl\{ -\frac{M^{2}\gamma _{2n}\log^{2} n}{2(2C_{3} \gamma_{2n}+ C_{1} Mn^{-1/2}\gamma_{2n}^{1/2}\log n)} \biggr\} \\ & \leq 2\exp \biggl\{ -\frac{M^{2}\log^{2} n}{2(2C_{3}+ C_{1} M )} \biggr\} \leq Cn^{-1}. \end{aligned} \end{aligned}$$
(3.5)
Secondly, we estimate
\(\sup_{-\infty < t<\infty }|P(S_{n1} \leq t)-\Phi (t)|\). Define
$$ s_{n}^{2}:=\sum_{j=1}^{k} \operatorname{Var}(\eta_{j}),\qquad \Gamma_{n}:=\sum _{1\leq i< j\leq k} \operatorname{Cov}(\eta_{i},\eta_{j}). $$
Clearly
\(s_{n}^{2}=E(S_{n1})^{2}-2\Gamma_{n}\), and since
\(ES_{n}^{2}=1\), by (
3.2) and (
3.3) we get that
$$ \bigl\vert E(S_{n1})^{2}-1 \bigr\vert = \bigl\vert E(S_{n2}+S_{n3})^{2}-2E\bigl[S_{n}(S_{n2}+S_{n3}) \bigr] \bigr\vert \leq C\bigl(\gamma_{1n}^{1/2}+ \gamma_{2n}^{1/2}\bigr). $$
(3.6)
On the other hand, by Assumptions
(A1)(ii),
(A4), and
(A5),
$$ \begin{aligned}[b] \Gamma_{n} &=\sum_{1\leq i< j\leq k} \sum_{s=k_{i}}^{k_{i}+p-1} \sum _{t=k_{j}}^{k_{j}+p-1}\operatorname{Cov}(Z_{n,s},Z _{n,t}) \\ &\leq C n^{-1}\sum_{i=1}^{k-1}\sum _{s=k_{i}} ^{k_{i}+p-1}\sum _{j=q}^{\infty } \bigl\vert \operatorname{Cov}(X_{1},X _{j}) \bigr\vert \\ &\leq C\bigl[kpu(q)\bigr]/n\leq Cu(q). \end{aligned} $$
(3.7)
From (
3.6) and (
3.7), it follows that
$$ \bigl\vert s_{n}^{2}-1 \bigr\vert \leq C\bigl[ \gamma_{1n}^{1/2}+\gamma_{2n}^{1/2}+u(q) \bigr]. $$
(3.8)
We assume that
\(\eta '_{j}\) are the independent random variables and
\(\eta '_{j}\) have the same distribution as
\(\eta_{j}\),
\(j=1,2,\ldots,k\). Let
\(H_{n}:=\sum_{j=1}^{k}\eta '_{j}\). It is easily seen that
$$ \begin{aligned}[b] &\sup_{-\infty < t< \infty } \bigl\vert P(S_{n1} \leq t)-\Phi (t) \bigr\vert \\ &\quad \leq \sup_{-\infty < t< \infty } \bigl\vert P(S_{n1}\leq t)-P(H_{n}\leq t) \bigr\vert \\ &\qquad {}+ \sup_{-\infty < t< \infty } \bigl\vert P(H_{n}\leq t)-\Phi (t/s_{n}) \bigr\vert + \sup_{-\infty < t< \infty } \bigl\vert \Phi (t/s_{n})-\Phi (t) \bigr\vert \\ &\quad :=D_{1}+D _{2}+D_{3}. \end{aligned} $$
Let
\(\phi (t)\) and
\(\varphi (t)\) be the characteristic functions of
\(S_{n1}\) and
\(H_{n}\), respectively. Thus, applying the Esséen inequality(see [
9],Theorem 5.3), for any
\(T>0\),
$$ \begin{aligned} D_{1}&\leq \int_{-T}^{T} \biggl\vert \frac{\phi (t)-\varphi (t)}{t} \biggr\vert \,\mathrm{d}t\\ &\quad {}+T\sup_{-\infty < t< \infty } \int_{ \vert u \vert \leq C/T} \bigl\vert P(H _{n}\leq u+t)-P(H_{n}\leq t) \bigr\vert \,\mathrm{d}u\\ &:=D_{1n}+D_{2n}. \end{aligned} $$
By Assumption
(A1)(ii) and Lemma
A.4, we have that
$$ \begin{aligned}[b] \bigl\vert \phi (t)-\varphi (t) \bigr\vert &= \Biggl\vert E\exp \Biggl( \mathrm{i}t\sum_{j=1}^{k} \eta_{j} \Biggr) -\prod_{j=1}^{k} E\exp {(\mathrm{i}t\eta_{j})} \Biggr\vert \\ &\leq 4t^{2}\sum_{1\leq i< j\leq k}\sum _{s=k_{i}}^{k _{i}+p-1}\sum_{t=k_{j}}^{k_{j}+p-1} \bigl\vert \operatorname{Cov}(Z_{n,s},Z_{n,t}) \bigr\vert \\ &\leq 4Ct^{2}k pn^{-1}\sum_{j=q}^{\infty } \bigl\vert \operatorname{Cov}(X_{1},X_{j}) \bigr\vert \leq Ct^{2}u(q). \end{aligned} $$
Therefore
$$ D_{1n}= \int_{-T}^{T} \biggl\vert \frac{\phi (t)-\varphi (t)}{t} \biggr\vert \,\mathrm{d}t\leq Cu(q)T^{2}. $$
(3.9)
It follows from the Berry–Esséen inequality [[
9], Theorem 5.7] and Lemma
A.2, for
\(r>2\),
$$ \begin{aligned}[b] \sup_{-\infty < t< \infty } \bigl\vert P(H_{n}/s_{n}\leq t)-\Phi (t) \bigr\vert & \leq \frac{C}{s_{n}^{r}}\sum_{j=1}^{k}E \bigl\vert \eta '_{j} \bigr\vert ^{r}= \frac{C}{s _{n}^{r}}\sum_{j=1}^{k}E \vert \eta_{j} \vert ^{r} \\ &\leq \frac{Ck[(p/n)]^{r/2}}{s _{n}^{r}}\leq C\frac{\gamma_{2n}^{(r-2)/2}}{s_{n}^{r}}. \end{aligned} $$
(3.10)
Notice that
\(s_{n}\rightarrow 1\), as
\(n\rightarrow \infty \) by (
3.8). From (
3.10), we get that
$$ \sup_{-\infty < t< \infty } \bigl\vert P(H_{n}/s_{n}\leq t)-\Phi (t) \bigr\vert \leq C \gamma_{2n}^{(r-2)/2}, $$
(3.11)
which implies that
$$ \begin{aligned}[b] &\sup_{-\infty < t< \infty } \bigl\vert P(H_{n} \leq t+u)-P(H_{n}\leq t) \bigr\vert \\ &\quad \leq \sup_{-\infty < t< \infty } \biggl\vert P \biggl( \frac{H_{n}}{s_{n}} \leq \frac{t+u}{s_{n}} \biggr) -\Phi \biggl( \frac{t+u}{s_{n}} \biggr) \biggr\vert \\ &\qquad {}+\sup_{-\infty < t< \infty } \biggl\vert P \biggl( \frac{H_{n}}{s_{n}} \leq \frac{t}{s_{n}} \biggr) -\Phi \biggl( \frac{t}{s_{n}} \biggr) \biggr\vert + \sup_{-\infty < t< \infty } \biggl\vert \Phi \biggl( \frac{t+u}{s_{n}} \biggr) - \Phi \biggl( \frac{t}{s_{n}} \biggr) \biggr\vert \\ &\quad \leq 2\sup_{-\infty < t< \infty } \biggl\vert P \biggl( \frac{H_{n}}{s_{n}} \leq t \biggr) - \Phi (t) \biggr\vert +\sup_{-\infty < t< \infty } \biggl\vert \Phi \biggl( \frac{t+u}{s _{n}} \biggr) -\Phi \biggl( \frac{t}{s_{n}} \biggr) \biggr\vert \\ &\quad \leq C \biggl( \gamma _{2n}^{(r-2)/2}+ \biggl\vert \frac{u}{s_{n}} \biggr\vert \biggr). \end{aligned} $$
(3.12)
By (
3.12), we obtain
$$ D_{2n}=T\sup_{-\infty < t< \infty } \int_{ \vert u \vert \leq C/T} \bigl\vert P(H _{n}\leq t+u)-P(H_{n}\leq t) \bigr\vert \,du\leq C\bigl(\gamma_{2n}^{(r-2)/2}+1/T \bigr). $$
(3.13)
Combining (
3.9) with (
3.13) and choosing
\(T=u^{-1/3}(q)\), we can easily see that
$$ D_{1}\leq C\bigl(u^{1/3}(q)\bigr)+\gamma_{2n}^{(r-2)/2}), $$
(3.14)
and by (
3.11),
$$ D_{2} =\sup_{-\infty < t< \infty } \biggl\vert P \biggl( \frac{H_{n}}{s _{n}}\leq \frac{t}{s_{n}} \biggr) -\Phi \biggl(\frac{t}{s_{n}} \biggr) \biggr\vert \leq C \gamma_{2n}^{(r-2)/2}. $$
(3.15)
On the other hand, from (
3.8) and Lemma 5.2 in [
9], it follows that
$$ \begin{aligned}[b] D_{3}&\leq (2\pi e)^{-1/2}(s_{n}-1)I(s_{n} \geq 1)+(2\pi e)^{-1/2}\bigl(s _{n}^{-1}-1 \bigr)I(0< s_{n}< 1) \\ &\leq C \bigl\vert s^{2}_{n}-1 \bigr\vert \leq C\bigl[ \gamma_{1n} ^{1/2}+\gamma_{2n}^{1/2}+u(q) \bigr]. \end{aligned} $$
(3.16)
Consequently, combining (
3.14), (
3.15) with (
3.16), we can get
$$ \sup_{-\infty < t< \infty } \bigl\vert P(S_{n1}\leq t)-\Phi (t) \bigr\vert \leq C\bigl[ \gamma_{1n}^{1/2}+ \gamma_{2n}^{1/2}+\gamma_{2n}^{(r-2)/2}+u^{1/3}(q) \bigr]. $$
(3.17)
Finally, by (
3.1), (
3.4), (
3.5), and (
3.17), (
2.7) is verified. □