1 Introduction
Suppose that \(\{\xi_{i},-\infty< i<+\infty\}\) is a doubly infinite sequence of identically distributed random variables with zero mean and finite variance and \(\{a_{i},-\infty< i<+\infty\}\) is an absolutely summable sequence of real numbers and is a moving average process based on \(\{\xi_{i},-\infty< i<+\infty\}\). Set \(S_{n}=\sum_{k=1}^{n}X_{k},n\geq1\).
$$ X_{k}=\sum_{i=-\infty}^{+\infty}a_{i} \xi_{i+k},\quad k\geq1, $$
(1)
Under the assumption that \(\{\xi_{i},-\infty< i<+\infty\}\) is a sequence of independent identically distributed (i.i.d.) random variables, many limiting results have been obtained for the moving average process \(\{ X_{k},k\geq1\}\). Burton and Dehling [1] got a large deviation principle; Yang [2] established the central limit theorem and the law of the iterated logarithm; Li et al. [3] and Zhang [4] obtained the complete convergence; and complete moment convergence was proved by Li [5] and Li and Zhang [6].
Anzeige
In this article we will discuss the case of φ-mixing. Suppose that \(\{\xi_{i},-\infty< i<+\infty\}\) is a sequence of identically distributed and φ-mixing random variables with where \(\mathcal{F}_{a}^{b}=\sigma (\xi_{i},a\leq i\leq b )\). By definition, we know that \(\{X_{k},k\geq1 \}\) is φ-mixing if \(\{\xi_{i},-\infty< i<+\infty\}\) is a sequence of φ-mixing distributed random variables.
$$\varphi(m):=\sup_{k\geq1} \bigl\{ \bigl|\mathbb{P} (B|A )-\mathbb {P}(B) \bigr|,A\in\mathcal{F}_{-\infty}^{k},\mathbb{P} (A)\neq0,B\in \mathcal{F}_{k+m}^{\infty}\bigr\} \rightarrow0,\quad m\rightarrow \infty, $$
Since Hsu and Robbins [7] introduced complete convergence, there have been extensions in several directions. Some authors studied the precise asymptotics of complete convergence on moving average. For example, Li and Zhang [8] got precise asymptotics in the law of the iterated logarithm of moving average and Xiao and Yin [9] got moment convergence rates in the law of logarithm for moving average process under dependence, etc.
Xiao and Yin [10] studied precise asymptotics in the law of iterated logarithm for the first moment convergence of i.i.d. random variables. They got the following result.
Theorem 1.1
Let
\(\{Y,Y_{n},n\geq1\}\)
be a sequence of i.i.d.random variables and set
\(U_{n}=\sum_{i=1}^{n}Y_{i}\). \(\mathcal{N}\)
is the standard normal random variable and
\(\mathbb{E}Y=0,\mathbb{E}Y^{2}=\sigma^{2}\), then, for
\(d>0\)
and
\(\beta>0\),
$$ \lim_{\varepsilon\searrow0}\varepsilon^{2\beta/d}\sum _{n=3}^{\infty}\frac { (\log\log n )^{\beta-1}}{n^{3/2}\log n} \mathbb{E} \bigl\{ \vert U_{n}\vert -\varepsilon\sigma\sqrt{n} (\log \log n )^{d/2} \bigr\} _{+}=\frac{d\sigma\mathbb{E} |\mathcal{N}|^{2\beta/d+1}}{\beta(2\beta+d)}. $$
(2)
Anzeige
Inspired by Xiao and Yin [10], we extend this result to moving average processes under φ-mixing random variable. Throughout this article, \(\log x:=\ln(x\vee\mathrm{e})\), and the symbol c denotes a positive constant which may be different in various places, and \([x]\) denotes the largest integer which is not greater than x, and \(\mathcal{N}\) is the standard normal random variable. Now, we state the main result of this article.
Theorem 1.2
Suppose
\(\{X_{i},i\geq1\}\)
is defined as in (1), where
\(\{a_{i},-\infty< i<+\infty\}\)
is a sequence of real numbers with
\(\sum_{i=-\infty}^{+\infty} \vert a_{i}\vert <\infty\), and
\(\{\xi _{i},-\infty< i<+\infty\}\)
is a sequence of identically distributed
φ-mixing random variables with zero mean and finite variance and
\(0<\sigma^{2}:=\mathbb{E}\xi_{1}^{2}+2\sum_{k=2}^{\infty}\mathbb{E} \xi_{1}\xi_{k}<\infty,\sum_{m=1}^{\infty}\varphi^{1/2} (2^{m} )<\infty ,\tau:=\sigma \vert \sum_{i=-\infty}^{+\infty}a_{i}\vert \). Then, for
\(d>0\)
and
\(\beta>0\),
$$ \lim_{\varepsilon\searrow0}\varepsilon^{2\beta/d}\sum _{n=3}^{\infty}\frac { (\log\log n )^{\beta-1}}{n^{3/2}\log n} \mathbb{E} \bigl\{ \vert S_{n}\vert -\varepsilon\tau\sqrt{n} (\log \log n )^{d/2} \bigr\} _{+}= \frac{d\tau\mathbb{E}\vert \mathcal{N}\vert ^{2\beta/d+1}}{\beta (2\beta+d)}. $$
(3)
Remark 1.3
Let \(a_{i}=1\) for \(i=0\) and \(a_{i}=0\) for \(i\neq0\), that is to say, \(X_{k}=\xi _{k}\) with \(\mathbb{E}\xi_{1}=0,\mathbb{E}\xi_{1}^{2}<\infty\) and \(0<\sigma^{2}:=\mathbb{E}\xi_{1}^{2}+2\sum_{k=2}^{\infty}\mathbb{E}\xi_{1}\xi _{k}<\infty\), then, for \(S_{n}=\sum_{k=1}^{n}\xi_{k}\), (3) still holds for \(\tau=\sigma\) when \(\{X_{k};k\geq1\}\) is a sequence of identically distributed φ-mixing random variables. Therefore, this result extends Theorem 1.1.
2 Lemmas
To prove our main result, we need the following lemmas.
Lemma 2.1
(Burton and Dehling [1])
Let
\(\sum_{i=-\infty}^{+\infty}a_{i}\)
be an absolutely convergent series of real numbers with
\(a=\sum_{i=-\infty}^{+\infty}a_{i}\)
and
\(k\geq1\), then
$$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=-\infty}^{+\infty} \Biggl\vert \sum_{j=i+1}^{i+n}a_{j} \Biggr\vert ^{k}=\vert a\vert ^{k}. $$
Lemma 2.2
(Shao [11])
Let
\(\{X_{i},i\geq1\}\)
be a sequence of
φ-mixing random variables with zero means and finite second moments. Set
\(S_{n}=\sum_{i=1}^{n}X_{i}\). Then
If there is some
\(C_{n}\)
such that
\(\displaystyle\max_{1\leq i\leq n}\mathbb{E}S_{i}^{2}\leq C_{n}\), then, for all
\(q\geq2\), there exists a
\(C=C (q,\varphi (\cdot ) )\)
such that
$$ \mathbb{E}S_{n}^{2}\leq8{,}000n \exp \Biggl\{ 6 \sum_{i=1}^{[\log n]}\varphi ^{1/2} \bigl(2^{i}\bigr) \Biggr\} \max_{1\leq i\leq n} \mathbb{E}X_{i}^{2}. $$
(4)
$$ \mathbb{E}\max_{1\leq i\leq n}\vert S_{i} \vert ^{q}\leq C \Bigl(C_{n}^{q/2}+\mathbb{E}\max _{1\leq i\leq n}\vert X_{i}\vert ^{q} \Bigr). $$
(5)
Lemma 2.3
(Lin and Zhou [12])
Let
\(\{X_{i},i\geq1 \}\)
be defined as in (1), and
\(\{\xi_{i},-\infty< i<+\infty\}\)
be a sequence of identically distributed
φ-mixing random variables with zero means and finite variances and
\(0<\sigma^{2}=\mathbb{E}\xi_{1}^{2}+2\sum_{k=2}^{\infty}\mathbb{E} \xi_{1}\xi_{k}<\infty,\sum_{m=1}^{\infty}\varphi^{1/2} (2^{m} )<\infty \), then
where
\(\stackrel{D}{\longrightarrow}\)
denotes convergence in distribution.
$$ \frac{S_{n}}{\tau\sqrt{n}}\stackrel{D}{\longrightarrow} \mathcal {N}(0,1),\quad \tau=\sigma\Biggl\vert \sum_{i=-\infty}^{+\infty}a_{i} \Biggr\vert , $$
(6)
3 Proof of main result
In this section, for \(0<\varepsilon<1\) and \(M>1\), we set
$$b (\varepsilon )= \bigl[\exp \bigl\{ \exp \bigl\{ M\varepsilon ^{-2/d} \bigr\} \bigr\} \bigr]. $$
(7)
Without loss of generality, we assume that \(\tau=1\).
Theorem 1.2 will be proved if we show the following propositions.
Proposition 3.1
Under the conditions of Theorem
1.2, we have
where (9) uniformly holds true with respect to
\(0<\varepsilon<1\).
$$\begin{aligned}& \lim_{\varepsilon\searrow0}\varepsilon^{2\beta/d}\sum _{n=3}^{\infty}\frac { (\log\log n )^{\beta-1}}{n\log n} \mathbb{E} \bigl\{ \vert \mathcal{N}\vert -\varepsilon (\log\log n )^{d/2} \bigr\} _{+}= \frac{d\mathbb{E}\vert \mathcal{N}\vert ^{2\beta/d+1}}{\beta (2\beta+d )}, \end{aligned}$$
(8)
$$\begin{aligned}& \lim_{M\rightarrow\infty}\varepsilon^{2\beta/d}\sum _{n>b(\varepsilon )}\frac{ (\log\log n )^{\beta-1}}{n\log n} \mathbb{E} \bigl\{ \vert \mathcal{N} \vert -\varepsilon (\log\log n )^{d/2} \bigr\} _{+}=0, \end{aligned}$$
(9)
Proof
See the proof of Proposition 1 and Proposition 3 in Xiao and Yin [10]. □
Proposition 3.2
Under the conditions of Theorem
1.2, we have
$$\begin{aligned} &\lim_{\varepsilon\searrow0}\varepsilon^{2\beta/d}\sum _{n\leq b(\varepsilon)}\frac{ (\log\log n )^{\beta-1}}{n\log n} \bigl\vert \mathbb{E} \bigl\{ \vert \mathcal{N}\vert -\varepsilon (\log \log n )^{d/2} \bigr\} _{+} \\ &\quad{}- \mathbb{E} \bigl\{ \vert S_{n}\vert /\sqrt {n}-\varepsilon (\log\log n )^{d/2} \bigr\} _{+}\bigr\vert =0. \end{aligned}$$
(10)
Proof
Let \(\Delta_{n}=\sup_{x\in\mathbb {R}}\vert \mathbb{P} (\vert \mathcal{N}\vert \geq x )-\mathbb{P} (\vert S_{n}\vert /\sqrt{n}\geq x )\vert \). By Lemma 2.3, then we have \(\Delta_{n}\rightarrow0\) as \(n\rightarrow\infty\). We have where \(l(n)= (\log\log n )^{-d/2}\Delta_{n}^{-1/2}\). Hence, for \(\mathrm{I}_{1}\), by applying Lemma 2.3, we have
$$\begin{aligned} &\varepsilon^{2\beta/d}\sum_{n\leq b(\varepsilon)} \frac{ (\log\log n )^{\beta-1}}{n\log n}\bigl\vert \mathbb{E} \bigl\{ \vert \mathcal {N}\vert - \varepsilon (\log\log n )^{d/2} \bigr\} _{+}-\mathbb {E} \bigl\{ \vert S_{n}\vert /\sqrt{n}-\varepsilon (\log\log n )^{d/2} \bigr\} _{+}\bigr\vert \\ &\quad=\varepsilon^{2\beta/d}\sum_{n\leq b(\varepsilon)} \frac{ (\log \log n )^{\beta-1}}{n\log n}\biggl\vert \int_{0}^{\infty}\mathbb{P} \bigl(\vert \mathcal{N} \vert \geq x+\varepsilon (\log\log n )^{d/2} \bigr)\,\mathrm{d}x \\ &\qquad{}- \int_{0}^{\infty}\mathbb{P} \bigl(\vert S_{n}\vert /\sqrt{n}\geq x+\varepsilon (\log\log n )^{d/2} \bigr)\,\mathrm{d}x\biggr\vert \\ &\quad\leq\varepsilon^{2\beta/d}\sum_{n\leq b(\varepsilon)} \frac{ (\log \log n )^{\beta-1+d/2}}{n\log n} \int_{0}^{\infty}\bigl\vert \mathbb{P} \bigl(| \mathcal{N}|\geq (x+\varepsilon ) (\log\log n )^{d/2} \bigr) \\ &\qquad{}- \mathbb{P} \bigl(|S_{n}|/\sqrt{n}\geq (x+\varepsilon ) (\log \log n )^{d/2} \bigr)\bigr\vert \,\mathrm{d}x \\ &\quad\leq\varepsilon^{2\beta/d}\sum_{n\leq b(\varepsilon)} \frac{ (\log \log n )^{\beta-1+d/2}}{n\log n} \int_{0}^{l(n)}\bigl\vert \mathbb{P} \bigl(| \mathcal{N}|\geq (x+\varepsilon ) (\log\log n )^{d/2} \bigr) \\ &\qquad{}- \mathbb{P} \bigl(|S_{n}|/\sqrt{n}\geq (x+\varepsilon ) (\log \log n )^{d/2} \bigr)\bigr\vert \,\mathrm{d}x \\ &\qquad{}+\varepsilon^{2\beta/d}\sum_{n\leq b(\varepsilon)} \frac{ (\log \log n )^{\beta-1+d/2}}{n\log n} \int_{l(n)}^{\infty}\mathbb{P} \bigl(\vert \mathcal{N} \vert \geq (x+\varepsilon ) (\log\log n )^{d/2} \bigr)\,\mathrm{d}x \\ &\qquad{}+\varepsilon^{2\beta/d}\sum_{n\leq b(\varepsilon)} \frac{ (\log \log n )^{\beta-1+d/2}}{n\log n} \int_{l(n)}^{\infty}\mathbb{P} \bigl(\vert S_{n}\vert /\sqrt{n}\geq (x+\varepsilon ) (\log\log n )^{d/2} \bigr)\,\mathrm{d}x \\ &\quad:=\mathrm{I}_{1}+\mathrm{I}_{2}+\mathrm{I}_{3}, \end{aligned}$$
(11)
$$\begin{aligned} \mathrm{I}_{1} \leq&c\varepsilon^{2\beta/d}\sum _{n\leq b(\varepsilon)}\frac { (\log\log n )^{\beta-1+d/2}}{n\log n} \int_{0}^{l(n)}\Delta_{n}\,\mathrm{d}x \leq c\varepsilon^{2\beta/d}\sum_{n\leq b(\varepsilon)} \frac { (\log\log n )^{\beta-1+d/2}}{n\log n}\Delta_{n}l(n) \\ =&c\varepsilon^{2\beta/d}\sum_{n\leq b(\varepsilon)} \frac { (\log\log n )^{\beta-1}}{n\log n}\Delta_{n}^{1/2} \leq\frac{cM^{\beta}}{ (\log\log b (\varepsilon ) )^{\beta}}\sum_{n\leq b(\varepsilon)} \frac{ (\log\log n )^{\beta-1}}{n\log n} \Delta_{n}^{1/2}. \end{aligned}$$
(12)
So, by the Toeplitz lemma, we have
$$ \lim_{\varepsilon\searrow0}\mathrm{I}_{1}=0. $$
(13)
As for \(\mathrm{I}_{2}\), by the Markov inequality, (12), and (13), we have
$$\begin{aligned} \mathrm{I}_{2} \leq&c\varepsilon^{2\beta/d}\sum _{n\leq b(\varepsilon)}\frac { (\log\log n )^{\beta-1+d/2}}{n\log n} \int_{l(n)}^{\infty}\frac{1}{ (x+\varepsilon )^{2} (\log\log n )^{d}}\,\mathrm{d}x \\ \leq&c\varepsilon^{2\beta/d}\sum_{n\leq b(\varepsilon)} \frac{ (\log \log n )^{\beta-1-d/2}}{n\log n}l(n)^{-1} \\ \leq&c\varepsilon^{2\beta/d}\sum_{n\leq b(\varepsilon)} \frac{ (\log \log n )^{\beta-1}}{n\log n}\Delta_{n}^{1/2} \rightarrow0, \quad\mbox{as } \varepsilon\searrow0. \end{aligned}$$
(14)
For \(\mathrm{I}_{3}\), note that \(S_{n}=\sum_{k=1}^{n}X_{k}=\sum_{i=-\infty }^{+\infty}\sum_{k=1}^{n}a_{i}\xi_{i+k}=\sum_{i=-\infty}^{+\infty}a_{i}\sum_{j=i+1}^{i+n}\xi_{j}\). By Lemma 2.1, without loss of generality, we assume \(\sum_{i=-\infty}^{+\infty} \vert a_{i}\vert \leq1\), and set \(S_{n}'=\sum_{i=-\infty}^{+\infty}a_{i}\sum_{j=i+1}^{i+n}\xi_{j}'\), where \(\xi _{j}'=\xi_{j}I (\vert \xi_{j}\vert \leq (x+\varepsilon )\sqrt{n} (\log\log n )^{d/2} )\). By \(\mathbb{E}\xi_{1}=0\), then, for all \(x>0\), we have \(\mathbb{E}\xi_{1}I (|\xi_{1}|\leq x )=-\mathbb{E}\xi_{1}I (|\xi _{1}|>x )\). So, we have thus for \(x\geq l(n)\) and n large enough, we have so we obtain
$$\begin{aligned} \bigl\vert \mathbb{E}S_{n}'\bigr\vert =&\Biggl\vert \sum_{i=-\infty}^{+\infty }a_{i} \mathbb{E}\sum_{j=i+1}^{i+n}\xi_{j}I \bigl( \vert \xi_{j}\vert \leq(x+\varepsilon)\sqrt{n} (\log\log n )^{d/2} \bigr)\Biggr\vert \\ \leq&\sum_{i=-\infty}^{+\infty} \vert a_{i}\vert \Biggl\vert \mathbb{E}\sum_{j=i+1}^{i +n} \xi_{j}I \bigl(\vert \xi_{j}\vert \leq(x+\varepsilon) \sqrt{n} (\log \log n )^{d/2} \bigr)\Biggr\vert \\ =&\sum_{i=-\infty}^{+\infty} \vert a_{i} \vert \Biggl\vert \mathbb{E}\sum_{j=i+1}^{i +n} \xi_{j}I \bigl(\vert \xi_{j}\vert >(x+\varepsilon)\sqrt{n} (\log \log n )^{d/2} \bigr)\Biggr\vert \\ \leq&n\mathbb{E}\vert \xi_{1}\vert I \bigl(\vert \xi_{1}\vert >(x+\varepsilon)\sqrt{n} (\log\log n )^{d/2} \bigr) \\ \leq&\frac{\sqrt{n}\mathbb{E}\xi_{1}^{2}}{(x+\varepsilon) (\log\log n )^{d/2}}, \end{aligned}$$
(15)
$$ \frac{\vert \mathbb{E}S_{n}'\vert }{(x+\varepsilon)\sqrt{n} (\log \log n )^{d/2}}\leq\frac{c\mathbb{E}\xi_{1}^{2}}{ (x+\varepsilon )^{2} (\log\log n )^{d}}\leq\frac{c}{l(n)^{2} (\log\log n )^{d}}\leq c\Delta_{n}< \frac{1}{4}, $$
(16)
$$\begin{aligned} & \int_{l(n)}^{\infty}\mathbb{P} \bigl(\vert S_{n}\vert /\sqrt{n}\geq (x+\varepsilon ) (\log\log n )^{d/2} \bigr)\,\mathrm{d}x \\ &\quad\leq c \int_{l(n)}^{\infty}\mathbb{P} \Biggl(\Biggl\vert \sum _{i=-\infty }^{+\infty}a_{i}\sum _{j=i+1}^{i+n}\xi_{j}I \bigl( \vert \xi_{j}\vert >(x+\varepsilon)\sqrt{n} (\log\log n )^{d/2} \bigr)\Biggr\vert \\ &\qquad{} >\frac{(x+\varepsilon)\sqrt{n} (\log\log n )^{d/2}}{2} \Biggr)\,\mathrm{d}x \\ &\qquad{}+c \int_{l(n)}^{\infty}\mathbb{P} \Biggl(\Biggl\vert \sum _{i=-\infty}^{+\infty }a_{i}\sum _{j=i+1}^{i+n} \bigl( \xi_{j}'- \mathbb{E}\xi_{j}' \bigr)\Biggr\vert >\frac{(x+\varepsilon)\sqrt {n} (\log\log n )^{d/2}}{4} \Biggr)\,\mathrm{d}x \\ &\quad:=\mathrm{I}_{31}+\mathrm{I}_{32}. \end{aligned}$$
(17)
By (11) and (17), we have
$$\mathrm{I}_{3}\leq c\varepsilon^{2\beta/d}\sum _{n\leq b(\varepsilon)}\frac { (\log\log n )^{\beta-1+d/2}}{n\log n} (\mathrm{I}_{31}+ \mathrm{I}_{32} ). $$
On the one hand, by (12) and (13), we obtain
$$\begin{aligned} &\varepsilon^{2\beta/d}\sum_{n\leq b(\varepsilon)} \frac{ (\log \log n )^{\beta-1+d/2}}{n\log n}\mathrm{I}_{31} \\ &\quad\leq c\varepsilon^{2\beta/d}\sum_{n\leq b(\varepsilon)} \frac{ (\log\log n )^{\beta-1+d/2}}{n\log n} \int_{l(n)}^{\infty}\frac {n\mathbb{E}\vert \xi_{1}\vert I (\vert \xi_{1}\vert >(x+\varepsilon )\sqrt{n} ( \log\log n )^{d/2} )}{\sqrt{n}(x+\varepsilon) ( \log\log n )^{d/2}}\,\mathrm{d}x \\ &\quad\leq c\varepsilon^{2\beta/d}\sum_{n\leq b(\varepsilon)} \frac{ (\log\log n )^{\beta-1+d/2}}{n\log n} \int_{l(n)}^{\infty}\frac {\mathbb{E}\xi_{1}^{2}I (\vert \xi_{1}\vert >(x+\varepsilon)\sqrt{n} ( \log\log n )^{d/2} )}{ (x+\varepsilon )^{2} ( \log\log n )^{d}}\,\mathrm{d}x \\ &\quad\leq c\varepsilon^{2\beta/d}\sum_{n\leq b(\varepsilon)} \frac{ (\log\log n )^{\beta-1-d/2}}{n\log n} \int_{l(n)}^{\infty}(x+\varepsilon )^{-2}\, \mathrm{d}x \\ &\quad\leq c\varepsilon^{2\beta/d}\sum_{n\leq b(\varepsilon)} \frac{ (\log\log n )^{\beta-1-d/2}}{n\log n}l(n)^{-1} \\ &\quad\leq c\varepsilon^{2\beta/d}\sum_{n\leq b(\varepsilon)} \frac{ (\log\log n )^{\beta-1}}{n\log n}\Delta_{n}^{1/2} \rightarrow0,\quad \mbox{as } \varepsilon\searrow0. \end{aligned}$$
(18)
On the other hand, by the Markov inequality, the Hölder inequality, and Lemma 2.2, and noting that \(\sum_{m=1}^{\infty }\varphi^{1/2}(2^{m})<\infty\), we have
$$\begin{aligned} & \varepsilon^{2\beta/d}\sum_{n\leq b(\varepsilon)} \frac{ (\log \log n )^{\beta-1+d/2}}{n\log n}\mathrm{I}_{32} \\ &\quad\leq c\varepsilon^{2\beta/d}\sum_{n\leq b(\varepsilon)} \frac{ (\log \log n )^{\beta-1+d/2-dq/2}}{n^{1+q/2}\log n} \int_{l(n)}^{\infty}(x+\varepsilon )^{-q} \mathbb{E}\Biggl\vert \sum_{i=-\infty}^{+\infty }a_{i} \sum_{j=i+1}^{i+n} \bigl( \xi_{j}'- \mathbb{E}\xi_{j}' \bigr)\Biggr\vert ^{q}\, \mathrm{d}x \\ &\quad\leq c\varepsilon^{2\beta/d}\sum_{n\leq b(\varepsilon)} \frac{ (\log \log n )^{\beta-1+d/2-dq/2}}{n^{1+q/2}\log n}\\ &\qquad{}\times \int_{l(n)}^{\infty}(x+\varepsilon )^{-q} \mathbb{E} \Biggl[\sum_{i=-\infty}^{+\infty } \bigl( \vert a_{i}\vert ^{1-\frac{1}{q}} \bigr) \Biggl(\vert a_{i}\vert ^{\frac{1}{q}}\Biggl\vert \sum _{j=i+1}^{i+n} \bigl(\xi_{j}'- \mathbb{E}\xi_{j}' \bigr)\Biggr\vert \Biggr) \Biggr] ^{q}\,\mathrm{d}x \\ &\quad\leq c\varepsilon^{2\beta/d}\sum_{n\leq b(\varepsilon)} \frac{ (\log \log n )^{\beta-1+d/2-dq/2}}{n^{1+q/2}\log n}\\ &\qquad{}\times \int_{l(n)}^{\infty}(x+\varepsilon )^{-q} \mathbb{E} \Biggl[ \Biggl(\sum_{i=-\infty }^{+\infty} \vert a_{i}\vert ^{1-\frac{1}{q} \cdot\frac{1}{1-\frac{1}{q}}} \Biggr)^{1-\frac{1}{q}} \Biggl(\sum _{i=-\infty}^{+\infty} \vert a_{i}\vert ^{\frac{1}{q}\cdot q}\Biggl\vert \sum_{j=i+1}^{i+n} \bigl(\xi_{j}'-\mathbb{E}\xi_{j}' \bigr)\Biggr\vert ^{q} \Biggr)^{\frac{1}{q}} \Biggr]^{q}\, \mathrm{d}x \\ &\quad\leq c\varepsilon^{2\beta/d}\sum_{n\leq b(\varepsilon)} \frac{ (\log \log n )^{\beta-1+d/2-dq/2}}{n^{1+q/2}\log n}\\ &\qquad{}\times \int_{l(n)}^{\infty}(x+\varepsilon )^{-q} \mathbb{E} \Biggl(\sum_{i=-\infty}^{+\infty }|a_{i}| \Biggr)^{ (1-\frac{1}{q} )q} \sum_{i=-\infty}^{+\infty} \vert a_{i}\vert \Biggl\vert \sum_{j=i+1}^{i+n} \bigl(\xi_{j}'-\mathbb{E}\xi_{j}' \bigr)\Biggr\vert ^{q}\,\mathrm{d}x \\ &\quad\leq c\varepsilon^{2\beta/d}\sum_{n\leq b(\varepsilon)} \frac{ (\log \log n )^{\beta-1+d/2-dq/2}}{n^{1+q/2}\log n}\\ &\qquad{}\times \int_{l(n)}^{\infty}(x+\varepsilon )^{-q}\sum _{i=-\infty}^{+\infty} \vert a_{i}\vert \mathbb{E}\Biggl\vert \sum_{j=i+1}^{i+n} \bigl(\xi_{j}'-\mathbb{E}\xi_{j}' \bigr)\Biggr\vert ^{q}\,\mathrm{d}x \\ &\quad\leq c\varepsilon^{2\beta/d}\sum_{n\leq b(\varepsilon)} \frac{ (\log \log n )^{\beta-1+d/2-dq/2}}{n^{1+q/2}\log n} \int_{l(n)}^{\infty}(x+\varepsilon )^{-q} \bigl( \bigl(n\mathbb{E} {\xi'}_{1}^{2} \bigr)^{\frac{q}{2}}+n\mathbb{E} \bigl\vert \xi_{1}' \bigr\vert ^{q} \bigr)\,\mathrm{d}x \\ &\quad:=\mathrm{I}_{321}+\mathrm{I}_{322}. \end{aligned}$$
For \(\mathrm{I}_{321}\), \(q\geq2\). By (12) and (13), we have
$$\begin{aligned} \mathrm{I}_{321} \leq&c\varepsilon^{2\beta/d}\sum _{n\leq b(\varepsilon )}\frac{ (\log\log n )^{\beta-1+d/2-dq/2}}{n\log n} \int _{l(n)}^{\infty}(x+\varepsilon )^{-q}\, \mathrm{d}x \\ \leq&c\varepsilon^{2\beta/d}\sum_{n\leq b(\varepsilon)} \frac{ (\log\log n )^{\beta-1+d/2-dq/2}}{n\log n}l(n)^{1-q} \\ \leq&c\varepsilon^{2\beta/d}\sum_{n\leq b(\varepsilon)} \frac{ (\log\log n )^{\beta-1}}{n\log n}\Delta_{n}^{\frac {q-1}{2}}\rightarrow0, \quad\mbox{as } \varepsilon\searrow0. \end{aligned}$$
(19)
For \(\mathrm{I}_{322}\), by (12) and (13), we have
$$\begin{aligned} \mathrm{I}_{322} \leq&c\varepsilon^{2\beta/d}\sum _{n\leq b(\varepsilon )}\frac{ (\log\log n )^{\beta-1+d/2-dq/2}}{n^{q/2}\log n} \\ &{}\times\int _{l(n)}^{\infty}(x+\varepsilon )^{-q} \mathbb{E}\vert \xi_{1}\vert ^{q}I \bigl(\vert \xi_{1}\vert \leq(x+\varepsilon) \sqrt{n} (\log\log n )^{d/2} \bigr)\,\mathrm{d}x \\ \leq&c\varepsilon^{2\beta/d}\sum_{n\leq b(\varepsilon)} \frac{ (\log \log n )^{\beta-1-d/2}}{n\log n} \\ &{}\times \int_{l(n)}^{\infty}(x+\varepsilon )^{-2} \mathbb{E}\xi_{1}^{2}I \bigl(\vert \xi_{1} \vert \leq (x+\varepsilon) \sqrt{n} (\log\log n )^{d/2} \bigr)\, \mathrm{d}x \\ \leq&c\varepsilon^{2\beta/d}\sum_{n\leq b(\varepsilon)} \frac{ (\log \log n )^{\beta-1-d/2}}{n\log n}l(n)^{-1} \\ =&c\varepsilon^{2\beta/d}\sum_{n\leq b(\varepsilon)} \frac{ (\log \log n )^{\beta-1}}{n\log n}\Delta_{n}^{1/2}\rightarrow0, \quad\mbox{as } \varepsilon\searrow0. \end{aligned}$$
(20)
Proposition 3.3
Under the conditions of Theorem
1.2,
uniformly holds true with respect to
\(0<\varepsilon<1\).
$$ \lim_{M\rightarrow\infty}\varepsilon^{2\beta/d}\sum _{n>b(\varepsilon )}\frac{ (\log\log n )^{\beta-1}}{n\log n} \mathbb{E} \bigl\{ \vert S_{n}\vert /\sqrt{n}-\varepsilon (\log\log n )^{d/2} \bigr\} _{+}=0 $$
(21)
Proof
For \(x\geq0\), M large enough, we have Then
$$\frac{\vert \mathbb{E}S_{n}'\vert }{\sqrt{n}(x+\varepsilon) (\log \log n )^{d/2}}\leq\frac{c}{(x+\varepsilon)^{2} (\log\log n )^{d}}\leq\frac{c}{M^{d}}< \frac{1}{4}. $$
$$\begin{aligned} &\mathbb{P} \bigl(|S_{n}|\geq\sqrt{n}(\log\log n)^{d/2}(x+\varepsilon ) \bigr) \\ &\quad\leq\mathbb{P} \Biggl(\Biggl\vert \sum _{i=-\infty}^{+\infty}a_{i}\sum _{j=i+1}^{i+n}\xi_{j}I \bigl(| \xi_{j}|> (x+\varepsilon)\sqrt{n} (\log\log n )^{d/2} \bigr) \Biggr\vert >\frac{\sqrt{n}(\log\log n)^{d/2}(x+\varepsilon)}{2} \Biggr) \\ &\qquad{}+\mathbb{P} \biggl(\bigl\vert S_{n}'\bigr\vert >\frac{\sqrt{n} (\log\log n )^{d/2}(x+\varepsilon)}{2} \biggr) \\ &\quad\leq\mathbb{P} \Biggl(\Biggl\vert \sum_{i=-\infty}^{+\infty}a_{i} \sum_{j=i+1}^{i+n}\xi_{j}I \bigl(| \xi_{j}|> (x+\varepsilon)\sqrt{n} (\log\log n )^{d/2} \bigr) \Biggr\vert >\frac{\sqrt{n}(\log\log n)^{d/2}(x+\varepsilon)}{2} \Biggr) \\ &\qquad{}+\mathbb{P} \biggl(\bigl\vert S_{n}'- \mathbb{E}S_{n}'\bigr\vert \geq\bigl\vert S_{n}'\bigr\vert -\bigl\vert \mathbb{E}S_{n}' \bigr\vert \\ &\qquad{}>\frac{\sqrt{n} (\log\log n )^{d/2}(x+\varepsilon)}{2}-\frac {\sqrt{n} (\log\log n )^{d/2}(x+\varepsilon)}{4} \biggr) \\ &\quad=\mathbb{P} \Biggl(\Biggl\vert \sum_{i=-\infty}^{+\infty}a_{i} \sum_{j=i+1}^{i+n}\xi_{j}I \bigl(| \xi_{j}|> (x+\varepsilon)\sqrt{n} (\log\log n )^{d/2} \bigr) \Biggr\vert >\frac{\sqrt{n}(\log\log n)^{d/2}(x+\varepsilon)}{2} \Biggr) \\ &\qquad{}+\mathbb{P} \Biggl(\Biggl\vert \sum_{i=-\infty}^{+\infty}a_{i} \sum_{j=i+1}^{i+n} \bigl(\xi_{j}'- \mathbb{E} \xi_{j}' \bigr)\Biggr\vert > \frac{\sqrt{n} (\log\log n )^{d/2}(x+\varepsilon)}{4} \Biggr). \end{aligned}$$
(22)
Therefore, by (22), we obtain
$$\begin{aligned} &\sum_{n>b(\varepsilon)}\frac{ (\log\log n )^{\beta -1}}{n\log n} \mathbb{E} \bigl\{ \vert S_{n}\vert /\sqrt{n}-\varepsilon (\log\log n )^{\frac{d}{2}} \bigr\} _{+} \\ &\quad=\sum_{n>b(\varepsilon)}\frac{ (\log\log n )^{\beta -1}}{n\log n} \int_{0}^{\infty}\mathbb{P} \bigl(\vert S_{n}\vert /\sqrt{n}\geq x+\varepsilon (\log\log n )^{d/2} \bigr)\,\mathrm{d}x \\ &\quad=\sum_{n>b(\varepsilon)}\frac{ (\log\log n )^{\beta-1+\frac {d}{2}}}{n\log n} \int_{0}^{\infty}\mathbb{P} \bigl( \vert S_{n}\vert \geq\sqrt{n} (\log\log n )^{\frac {d}{2}}(x+\varepsilon) \bigr)\,\mathrm{d}x \ \ \bigl(x=y (\log\log n )^{d/2} \bigr) \\ &\quad\leq\sum_{n>b(\varepsilon)}\frac{ (\log\log n )^{\beta -1+\frac{d}{2}}}{n\log n} \int_{0}^{\infty}\mathbb{P} \Biggl( \Biggl\vert \sum _{i=-\infty}^{+\infty}a_{i}\sum _{j=i+1}^{i+n}\xi_{j}I \bigl(\vert \xi_{j}\vert \\ &\qquad{} >(x+\varepsilon) \sqrt{n} (\log\log n )^{\frac{d}{2}} \bigr)\Biggr\vert >\frac {(x+\varepsilon)\sqrt{n} (\log\log n )^{\frac{d}{2}}}{2} \Biggr)\,\mathrm{d}x \\ &\qquad{}+\sum_{n>b(\varepsilon)}\frac{ (\log\log n )^{\beta-1+\frac {d}{2}}}{n\log n} \int_{0}^{\infty}\mathbb{P} \Biggl(\Biggl\vert \sum _{i=-\infty }^{+\infty}a_{i}\sum _{j=i+1}^{i+n} \bigl(\xi_{j}'- \mathbb{E} \xi_{j}' \bigr)\Biggr\vert \\ &\qquad{}> \frac{(x+\varepsilon)\sqrt{n} (\log\log n )^{\frac{d}{2}}}{4} \Biggr)\,\mathrm{d}x \\ &\quad:=J_{1}+J_{2}. \end{aligned}$$
For \(J_{1}\), when \(2\beta< d\), we have thus uniformly holds true with respect to \(0<\varepsilon<1\).
$$\begin{aligned} \varepsilon^{2\beta/d}J_{1} \leq&c\varepsilon^{2\beta/d}\sum _{n>b(\varepsilon)}\frac{ (\log\log n )^{\beta-1+d/2}}{ n\log n} \int_{0}^{\infty}\frac{n\mathbb{E}\vert \xi_{1}\vert I (\vert \xi_{1}\vert >(x+\varepsilon)\sqrt{n} (\log\log n )^{d/2} )}{(x+\varepsilon)\sqrt{n} (\log\log n )^{d/2}}\,\mathrm{d}x \\ \leq&c\varepsilon^{2\beta/d}\sum_{n>b(\varepsilon)} \frac{ (\log \log n )^{\beta-1-d/2}}{ n\log n} \int_{0}^{\infty}\frac{\mathbb{E}\xi_{1}^{2}I (\vert \xi_{1}\vert >(x+\varepsilon)\sqrt{n} (\log\log n )^{d/2} )}{(x+\varepsilon)^{2}}\,\mathrm{d}x \\ \leq&c\varepsilon^{2\beta/d}\sum_{n>b(\varepsilon)} \frac{ (\log \log n )^{\beta-1-d/2}}{ n\log n} \int_{0}^{\infty}(x+\varepsilon)^{-2}\, \mathrm{d}x \\ \leq&c\varepsilon^{2\beta/d-1}\sum_{n>b(\varepsilon)} \frac{ (\log \log n )^{\beta-1-d/2}}{ n\log n} \\ \leq&cM^{\beta-d/2}, \end{aligned}$$
$$ \lim_{M\rightarrow\infty}\varepsilon^{2\beta/d}J_{1}=0 $$
(23)
When \(2\beta\geq d\), we have
$$\begin{aligned} J_{1} \leq&c\sum_{n>b(\varepsilon)}\frac{ (\log\log n )^{\beta-1+d/2}}{ n\log n} \int_{0}^{\infty}\frac{n\mathbb{E}\vert \xi_{1}\vert I (\vert \xi_{1}\vert >(x+\varepsilon)\sqrt{n} (\log\log n )^{d/2} )}{(x+\varepsilon)\sqrt{n} (\log\log n )^{d/2}}\,\mathrm{d}x \\ =&c\sum_{n>b(\varepsilon)}\frac{ (\log\log n )^{\beta-1}}{ \sqrt{n}\log n} \int_{0}^{\infty}\frac{\mathbb{E}\vert \xi_{1}\vert I (\vert \xi_{1}\vert >(x+\varepsilon)\sqrt{n} (\log\log n )^{d/2} )}{x+\varepsilon}\,\mathrm{d}x \\ =&c \int_{0}^{\infty}\frac{1}{x+\varepsilon}\sum _{n>b(\varepsilon)}\frac { (\log\log n )^{\beta-1}}{ \sqrt{n}\log n}\sum_{j=n}^{\infty}\mathbb{E}|\xi_{1}|I \biggl(\sqrt{j} (\log\log j )^{d/2}< \frac{|\xi_{1}|}{x+\varepsilon}\\ &{}\leq\sqrt {j+1} \bigl(\log\log(j+1) \bigr)^{d/2} \biggr)\,\mathrm{d}x \\ =&c \int_{0}^{\infty}\frac{1}{x+\varepsilon}\sum _{j>b(\varepsilon)}\mathbb {E}|\xi_{1}|I \biggl(\sqrt{j} (\log \log j )^{\frac{d}{2}}< \frac {|\xi_{1}|}{x+\varepsilon}\leq\sqrt{j+1} \bigl(\log\log(j+1) \bigr)^{\frac {d}{2}} \biggr)\\ &{}\times\sum_{n=b(\varepsilon)+1}^{j} \frac{ (\log\log n )^{\beta-1}}{ \sqrt{n}\log n}\,\mathrm{d}x. \end{aligned}$$
Since we have \((\log\log n )^{\beta-1} (\log n )^{-1}\rightarrow0\), as \(n>b(\varepsilon)\rightarrow\infty\), then we obtain Hence, when \(2\beta\geq d\), uniformly holds true with respect to \(0<\varepsilon<1\).
$$\begin{aligned} J_{1} \leq&c \int_{0}^{\infty}\frac{1}{x+\varepsilon}\sum _{j>b(\varepsilon )}\sqrt{j}\mathbb{E}|\xi_{1}|I \biggl(\sqrt{j} (\log\log j )^{d/2}< \frac{|\xi_{1}|}{x+\varepsilon}\leq\sqrt{j+1} \bigl(\log\log (j+1) \bigr)^{d/2} \biggr)\,\mathrm{d}x \\ \leq&c \int_{0}^{\infty}\frac{1}{(x+\varepsilon)^{2}}\mathbb{E} \xi_{1}^{2}I \bigl(|\xi_{1}|>(x+\varepsilon) \sqrt{b (\varepsilon )} \bigl(\log\log b(\varepsilon) \bigr)^{d/2} \bigr) \,\mathrm{d}x \\ \leq&c \int_{0}^{\infty}\frac{1}{(x+\varepsilon)^{2}}\mathbb{E} \xi_{1}^{2}I \bigl(|\xi_{1}|>\varepsilon \bigl( \log\log b(\varepsilon) \bigr)^{d/2} \bigr)\,\mathrm{d}x \\ =&c\varepsilon^{-1}\mathbb{E}\xi_{1}^{2}I \bigl(|\xi_{1}|>\varepsilon \bigl(\log\log b(\varepsilon) \bigr)^{d/2}=M^{d/2} \bigr). \end{aligned}$$
$$ \lim_{M\rightarrow\infty}\varepsilon^{2\beta/d}J_{1} \leq\lim_{M\rightarrow\infty}\mathbb{E}\xi_{1}^{2}I \bigl( |\xi_{1}|>M^{d/2} \bigr)=0 $$
(24)
Next we consider \(J_{2}\), similar to the proof of \(\mathrm{I}_{32}\), we have
$$\begin{aligned}[b] \varepsilon^{2\beta/d}J_{2}&\leq c\varepsilon^{2\beta/d}\sum _{n>b(\varepsilon)}\frac{ (\log\log n )^{\beta -1+\frac{d}{2}-\frac{dq}{2}}}{n^{1+\frac{q}{2}}\log n} \int_{0}^{\infty}(x+\varepsilon )^{-q} \bigl( \bigl(n\mathbb{E} {\xi'}_{1}^{2} \bigr)^{q/2}+n\mathbb{E}\bigl\vert \xi_{1}' \bigr\vert ^{q} \bigr)\,\mathrm{d}x \\ &:=J_{21}+J_{22}. \end{aligned} $$
For \(J_{21}\), haven taken \(q>\max \{2\beta/d+1,2 \}\), then we have Thus uniformly holds true with respect to \(0<\varepsilon<1\).
$$\begin{aligned} J_{21} \leq&c\varepsilon^{2\beta/d}\sum _{n>b(\varepsilon)}\frac{ (\log\log n )^{\beta -1+\frac{d}{2}-\frac{dq}{2}}}{n\log n} \int_{0}^{\infty}(x+\varepsilon )^{-q}\, \mathrm{d}x \\ \leq&c\varepsilon^{2\beta/d-q+1}\sum_{n>b(\varepsilon)} \frac{ (\log\log n )^{\beta -1+\frac{d}{2}-\frac{dq}{2}}}{n\log n} \\ \leq&c\varepsilon^{2\beta/d-q+1} \bigl(\log\log b(\varepsilon) \bigr)^{\beta +\frac{d}{2}-\frac{dq}{2}} \\ =&M^{\beta+\frac{d}{2}-\frac{dq}{2}}. \end{aligned}$$
$$ \lim_{M\rightarrow\infty}J_{21}=0 $$
(25)
For \(J_{22}\), if \(2\beta< d\), we have Thus uniformly holds true with respect to \(0<\varepsilon<1\).
$$\begin{aligned} J_{22} \leq&c\varepsilon^{2\beta/d}\sum _{n>b(\varepsilon)}\frac{ (\log\log n )^{\beta -1+\frac{d}{2}-\frac{dq}{2}}}{n^{\frac{q}{2}}\log n}\\ &{}\times \int_{0}^{\infty}(x+\varepsilon )^{-q} \mathbb{E}\vert \xi_{1}\vert ^{q}I \bigl(\vert \xi_{1}\vert \leq(x+\varepsilon)\sqrt{n} (\log\log n )^{d/2} \bigr)\,\mathrm{d}x \\ \leq&c\varepsilon^{2\beta/d}\sum_{n>b(\varepsilon)} \frac{ (\log \log n )^{\beta -1-\frac{d}{2}}}{n\log n}\\ &{}\times\int_{0}^{\infty}(x+\varepsilon )^{-2} \mathbb{E} \xi_{1}^{2}I \bigl(\vert \xi_{1} \vert \leq(x+\varepsilon)\sqrt{n} (\log \log n )^{d/2} \bigr)\, \mathrm{d}x \\ \leq&c\varepsilon^{2\beta/d}\sum_{n>b(\varepsilon)} \frac{ (\log \log n )^{\beta -1-\frac{d}{2}}}{n\log n} \int_{0}^{\infty}(x+\varepsilon )^{-2}\, \mathrm{d}x \\ \leq&c\varepsilon^{2\beta/d-1}\sum_{n>b(\varepsilon)} \frac{ (\log \log n )^{\beta -1-\frac{d}{2}}}{n\log n} \\ =&cM^{\beta-d/2}. \end{aligned}$$
$$ \lim_{M\rightarrow\infty}J_{22}=0 $$
(26)
If \(2\beta\geq d\), we divide \(J_{22}\) into two parts:
$$\begin{aligned} J_{22} \leq&c\varepsilon^{2\beta/d}\sum _{n>b(\varepsilon)}\frac{ (\log\log n )^{\beta -1+\frac{d}{2}-\frac{dq}{2}}}{n^{\frac{q}{2}}\log n}\\ &{}\times \int_{0}^{\infty}(x+\varepsilon )^{-q} \mathbb{E}\vert \xi_{1}\vert ^{q}I \bigl(\vert \xi_{1}\vert \leq(x+\varepsilon)\sqrt{n} (\log\log n )^{d/2} \bigr)\,\mathrm{d}x \\ \leq&c\varepsilon^{2\beta/d}\sum_{n>b(\varepsilon)} \frac{ (\log \log n )^{\beta -1+\frac{d}{2}-\frac{dq}{2}}}{n^{\frac{q}{2}}\log n} \int_{0}^{\infty}(x+\varepsilon )^{-q} \mathbb{E}\vert \xi_{1}\vert ^{q} \bigl[I \bigl(\vert \xi_{1}\vert \leq\sqrt{n} (\log\log n )^{d/2}\varepsilon \bigr) \\ &{}+I \bigl(\sqrt{n} (\log\log n )^{d/2}\varepsilon< \vert \xi _{1}\vert \leq(x+\varepsilon)\sqrt{n} (\log\log n )^{d/2} \bigr) \bigr]\,\mathrm{d}x \\ :=&J_{221}+J_{222}. \end{aligned}$$
For \(J_{221}\), since \(q>2\), so we have Thus uniformly holds true with respect to \(0<\varepsilon<1\).
$$\begin{aligned} J_{221} \leq&c\varepsilon^{2\beta/d-q+1}\sum _{n>b(\varepsilon)}\frac { (\log\log n )^{\beta-1+\frac{d}{2}- \frac{dq}{2}}}{n^{\frac{q}{2}}\log n}\mathbb{E}|\xi_{1}|^{q}I \bigl(|\xi _{1}|\leq\sqrt{b(\varepsilon)} \bigl( \log\log b( \varepsilon) \bigr)^{\frac{d}{2}}\varepsilon \bigr) \\ &{}+c\varepsilon^{2\beta/d-q+1}\sum_{n>b(\varepsilon)} \frac{ (\log \log n )^{\beta-1+\frac{d}{2}- \frac{dq}{2}}}{n^{\frac{q}{2}}\log n}\\ &{}\times\sum_{j=b(\varepsilon)+1}^{n}\mathbb {E}| \xi_{1}|^{q}I \bigl(\sqrt{j-1} \bigl( \log\log (j-1 ) \bigr)^{\frac{d}{2}}\varepsilon< |\xi_{1}|\leq \sqrt{j} ( \log\log j )^{\frac{d}{2}}\varepsilon \bigr) \\ \leq&c\varepsilon^{2\beta/d-1}\frac{ (\log\log b(\varepsilon ) )^{\beta-1-\frac{d}{2}}}{\log b(\varepsilon)} \mathbb{E} \xi_{1}^{2}I \bigl(|\xi_{1}|\leq\sqrt{b( \varepsilon)} \bigl(\log\log b(\varepsilon) \bigr)^{\frac{d}{2}}\varepsilon \bigr) \\ &{}+c\varepsilon^{2\beta/d-q+1}\sum_{j=b(\varepsilon)+1}^{\infty}\mathbb {E}|\xi_{1}|^{q}I \bigl(\sqrt{j-1} \bigl( \log\log (j-1 ) \bigr)^{\frac{d}{2}}\varepsilon< |\xi_{1}|\leq \sqrt{j} ( \log \log j )^{\frac{d}{2}}\varepsilon \bigr)\\ &{}\times\sum_{n=j}^{\infty}\frac { (\log\log n )^{\beta-1+\frac{d}{2}-\frac{dq}{2}}}{n^{\frac {q}{2}}\log n} \\ \leq&c\varepsilon^{2\beta/d-1}\frac{1}{\log\log b(\varepsilon)} \\ &{}+c\varepsilon^{2\beta/d-q+1}\sum_{j=b(\varepsilon)+1}^{\infty}\mathbb {E}|\xi_{1}|^{q}I \bigl(\sqrt{j-1} \bigl( \log\log (j-1 ) \bigr)^{\frac{d}{2}}\varepsilon< |\xi_{1}|\leq \sqrt{j} ( \log \log j )^{\frac{d}{2}}\varepsilon \bigr)j^{1-\frac{q}{2}}\\ &{}\times\frac { (\log\log j )^{\beta-1+\frac{d}{2}-\frac{dq}{2}}}{\log j} \\ \leq&c\varepsilon^{2\beta/d-1+d/2}M^{-1} +c\varepsilon^{2\beta/d-1}\sum_{j=b(\varepsilon)+1}^{\infty}\frac{ (\log\log j )^{\beta-1-\frac{d}{2}}}{\log j}\\ &{}\times\mathbb{E}\xi_{1}^{2}I \bigl(\sqrt{j-1} \bigl( \log\log (j-1 ) \bigr)^{\frac{d}{2}}\varepsilon< |\xi_{1}|\leq \sqrt{j} ( \log\log j )^{\frac{d}{2}}\varepsilon \bigr) \\ \leq&cM^{-1}+c\mathbb{E}\xi_{1}^{2}I \bigl(| \xi_{1}|> \sqrt{b(\varepsilon)} \bigl( \log\log b(\varepsilon) \bigr)^{\frac{d}{2}}\varepsilon>M^{\frac {d}{2}} \bigr). \end{aligned}$$
$$ \lim_{M\rightarrow\infty}J_{221}\leq\lim _{M\rightarrow\infty }M^{-1}+\lim_{M\rightarrow\infty}\mathbb{E} \xi_{1}^{2}I \bigl(|\xi_{1}|>M^{\frac{d}{2}} \bigr)=0 $$
(27)
Finally we consider \(J_{222}\), since \(2\beta\geq d\) and \(q>2\), we have Hence, uniformly holds true with respect to \(0<\varepsilon<1\).
$$\begin{aligned} J_{222} \leq&c\varepsilon^{2\beta/d}\sum _{n>b(\varepsilon)}\frac{ (\log\log n )^{\beta-1+\frac{d}{2}-\frac{dq}{2}}}{n^{\frac {q}{2}}\log n}\mathbb{E}|\xi_{1}|^{q}I \bigl(|\xi_{1}|>\sqrt{n} (\log\log n )^{\frac{d}{2}}\varepsilon \bigr) \\ &{}\times \int_{0}^{\infty}(x+\varepsilon )^{-q} I \bigl(|\xi_{1}|\leq(x+\varepsilon)\sqrt{n} (\log\log n )^{\frac {d}{2}} \bigr)\,\mathrm{d}x \\ \leq&c\varepsilon^{2\beta/d}\sum_{n>b(\varepsilon)} \frac{ (\log \log n )^{\beta-1+\frac{d}{2}-\frac{dq}{2}}}{n^{\frac{q}{2}}\log n}\mathbb{E}|\xi_{1}|^{q}I \bigl(| \xi_{1}|>\sqrt{n} (\log\log n )^{\frac{d}{2}}\varepsilon \bigr)\\ &{}\times \int_{\frac{|\xi_{1}|}{\sqrt{n} (\log \log n )^{\frac{d}{2}}}-\varepsilon}^{\infty}(x+\varepsilon )^{-q}\, \mathrm{d}x \\ \leq&c\varepsilon^{2\beta/d}\sum_{n>b(\varepsilon)} \frac{ (\log \log n )^{\beta-1}}{\sqrt{n}\log n} \mathbb{E}|\xi_{1}|I \bigl(|\xi_{1}|> \sqrt{n} (\log\log n )^{\frac {d}{2}}\varepsilon \bigr) \\ \leq&c\varepsilon^{2\beta/d}\sum_{n>b(\varepsilon)} \frac{ (\log \log n )^{\beta-1}}{\sqrt{n}\log n}\\ &{}\times\sum_{j=n}^{\infty}\mathbb{E}| \xi _{1}|I \bigl(\sqrt{j}(\log\log j)^{\frac{d}{2}}\varepsilon< | \xi_{1}|\leq \sqrt{j+1} \bigl(\log\log(j+1) \bigr)^{\frac{d}{2}} \varepsilon \bigr) \\ \leq&c\varepsilon^{2\beta/d}\sum_{j=b(\varepsilon)}^{\infty}\mathbb {E}|\xi_{1}|I \bigl(\sqrt{j}(\log\log j)^{\frac{d}{2}} \varepsilon< |\xi _{1}|\leq\sqrt{j+1} \bigl(\log\log(j+1) \bigr)^{\frac{d}{2}}\varepsilon \bigr) \\ &{}\times\sum_{n=b(\varepsilon)}^{j} \frac{ (\log\log n )^{\beta-1}}{\sqrt {n}\log n} \\ \leq&c\varepsilon^{2\beta/d}\sum_{j=b(\varepsilon)}^{\infty}\sqrt {j}\mathbb{E}|\xi_{1}|I \bigl(\sqrt{j}(\log\log j)^{\frac{d}{2}} \varepsilon < |\xi_{1}|\leq\sqrt{j+1} \bigl(\log\log(j+1) \bigr)^{\frac {d}{2}}\varepsilon \bigr) \\ \leq&c\varepsilon^{2\beta/d-1}\sum_{j=b(\varepsilon)}^{\infty}\mathbb {E}\xi_{1}^{2}I \bigl(\sqrt{j}(\log\log j)^{\frac{d}{2}}\varepsilon< |\xi _{1}|\leq\sqrt{j+1} \bigl(\log \log(j+1) \bigr)^{\frac{d}{2}}\varepsilon \bigr) \\ \leq&c\varepsilon^{2\beta/d-1}\mathbb{E}\xi_{1}^{2}I \bigl(|\xi_{1}|\geq \varepsilon\sqrt{b(\varepsilon)} \bigl( \log\log b( \varepsilon) \bigr)^{\frac{d}{2}}>M^{\frac{d}{2}} \bigr). \end{aligned}$$
$$ \lim_{M\rightarrow\infty}J_{222}\leq\lim _{M\rightarrow\infty }\varepsilon^{2\beta/d-1}\mathbb{E}\xi_{1}^{2}I \bigl( |\xi_{1}|>M^{d/2} \bigr)\leq\lim_{M\rightarrow\infty} \mathbb{E}\xi _{1}^{2}I \bigl( |\xi_{1}|>M^{d/2} \bigr)=0 $$
(28)
Acknowledgements
The authors would like to thank the referees and the editors for the helpful comments and suggestions. This work was supported by the National Natural Science Foundation of China (11361019, 11661029) and the Guangxi China Science Foundation (2015GXNSFAA139008).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Authors’ information
Qunying Wu is a professor, doctor, working in the field of probability and statistics.