Proof of Theorem 2.1
Write
$$\begin{aligned}& Y_{j}'=-n^{1/p}I\bigl(Y_{j}< -n^{1/p} \bigr)+Y_{j}I\bigl(|Y_{j}|\leq n^{1/p} \bigr)+n^{1/p}I\bigl(Y_{j}>n^{1/p}\bigr), \\& Y_{j}''=Y_{j}-Y_{j}'= \bigl(Y_{j}+n^{1/p}\bigr)I\bigl(Y_{j}< -n^{1/p} \bigr)+\bigl(Y_{j}-n^{1/p}\bigr)I\bigl(Y_{j}>n^{1/p} \bigr). \end{aligned}$$
Note that
$$\sum_{k=1}^{n}X_{k}=\sum _{k=1}^{n}\sum_{j=-\infty}^{+\infty }a_{j}Y_{k+j}= \sum_{j=-\infty}^{+\infty}a_{j}\sum _{i=j+1}^{j+n}Y_{i}. $$
Since \(\sum_{j=-\infty}^{+\infty}|a_{j}|<\infty\), \(EY_{i}=0\), it follows from Lemma 2.3 that
$$\begin{aligned}& n^{-1/p} \Biggl\vert E\max_{1\leq k\leq n}\sum _{j=-\infty}^{+\infty}a_{j}\sum _{i=j+1}^{j+k}Y_{i}' \Biggr\vert \quad (\text{since } EY_{i}=0) \\& \quad \leq n^{-1/p}\sum_{j=-\infty}^{+\infty} \vert a_{j} \vert \sum_{i=j+1}^{j+n} \bigl(n^{1/p}P\bigl( \vert Y_{i} \vert >n^{1/p} \bigr)+E \vert Y_{i} \vert I\bigl( \vert Y_{i} \vert >n^{1/p}\bigr)\bigr) \\& \quad \leq Cn^{1-1/p}E \vert Y \vert I\bigl( \vert Y \vert >n^{1/p}\bigr) \\& \quad \leq CE \vert Y \vert ^{p}I\bigl( \vert Y \vert >n^{1/p}\bigr)\rightarrow0,\quad n\rightarrow\infty. \end{aligned}$$
Therefore, for large enough n, we obtain
$$ n^{-1/p} \Biggl\vert E\max_{1\leq k\leq n}\sum _{j=-\infty}^{+\infty}a_{j}\sum _{i=j+1}^{j+k}Y_{i}' \Biggr\vert < \varepsilon/4. $$
(3.1)
By (3.1), we have
$$\begin{aligned}& \sum_{n=1}^{\infty} n^{\alpha-2}P\Bigl\{ \max_{1\leq k\leq n} \vert S_{k} \vert \geq \varepsilon n^{1/p}\Bigr\} \\& \quad = \sum_{n=1}^{\infty} n^{\alpha-2}P \Biggl\{ \max_{1\leq k\leq n} \Biggl\vert \sum _{j=-\infty}^{+\infty}a_{j}\sum _{i=j+1}^{j+k}Y_{i} \Biggr\vert \geq \varepsilon n^{1/p}\Biggr\} \\& \quad \leq \sum_{n=1}^{\infty} n^{\alpha-2}P\Biggl\{ \max_{1\leq k\leq n} \Biggl\vert \sum _{j=-\infty}^{+\infty}a_{j}\sum _{i=j+1}^{j+k}Y''_{i} \Biggr\vert \geq \varepsilon n^{1/p}/2\Biggr\} \\& \qquad {} + \sum_{n=1}^{\infty} n^{\alpha-2}P\Biggl\{ \max_{1\leq k\leq n} \Biggl\vert \sum _{j=-\infty}^{+\infty}a_{j}\sum _{i=j+1}^{j+k}\bigl(Y'_{i}-EY'_{i} \bigr) \Biggr\vert \geq\varepsilon n^{1/p}/4\Biggr\} \\& \quad =: I_{1}+I_{2}. \end{aligned}$$
For \(I_{1}\), noting \(\alpha p>1\), it follows from Lemma 2.3 and the Markov inequality that
$$\begin{aligned} I_{1} \leq& C\sum_{n=1}^{\infty} n^{\alpha-2-1/p}E\max_{1\leq k\leq n} \Biggl\vert \sum _{j=-\infty}^{\infty}a_{j}\sum _{i=j+1}^{j+k}Y''_{i} \Biggr\vert \\ \leq& C\sum_{n=1}^{\infty} n^{\alpha-2-1/p} \sum_{j=-\infty}^{+\infty } \vert a_{j} \vert \sum_{i=j+1}^{j+n}E \vert Y_{i} \vert I\bigl( \vert Y_{i} \vert >n^{1/p}\bigr) \\ \leq& C\sum_{n=1}^{\infty} n^{\alpha-1-1/p}E \vert Y \vert I\bigl( \vert Y \vert >n^{1/p}\bigr) \\ =& C\sum_{n=1}^{\infty} n^{\alpha-1-1/p}\sum _{l=n}^{\infty }E \vert Y \vert I \bigl(l^{1/p}< \vert Y \vert \leq(l+1)^{1/p}\bigr) \\ \leq& C\sum_{l=1}^{\infty}E \vert Y \vert I\bigl(l^{1/p}< \vert Y \vert \leq(l+1)^{1/p}\bigr)\sum _{n=1}^{l} n^{\alpha-1-1/p} \\ \leq& C\sum_{l=1}^{\infty}l^{\alpha-1/p}E \vert Y \vert I\bigl(l^{1/p}< \vert Y \vert \leq(l+1)^{1/p} \bigr) \leq E \vert Y \vert ^{\alpha p}< \infty. \end{aligned}$$
(3.2)
For \(I_{2}\), from Lemma 2.2, we see that \(\{Y'_{j}-EY'_{j}\}\) is still a \(\rho^{-}\)-mixing random variable sequence. By Lemma 2.1, Markov and Hölder inequalities, we have that for any \(v\geq2\),
$$\begin{aligned} I_{2} \leq&{C}\sum_{n=1}^{\infty} n^{\alpha-2-v/p}E\Biggl\{ \max_{1\leq k\leq n} \Biggl\vert \sum _{j=-\infty}^{+\infty}a_{j}\Biggl(\sum _{i=j+1}^{j+k}\bigl(Y'_{i}-EY'_{i} \bigr)\Biggr) \Biggr\vert \Biggr\} ^{v} \\ \leq&{C}\sum_{n=1}^{\infty}n^{\alpha-2-v/p}E \Biggl\{ \sum_{j=-\infty }^{+\infty} \vert a_{j} \vert \max_{1\leq k\leq n} \Biggl\vert \sum _{i=j+1}^{j+k}\bigl(Y'_{i}-EY'_{i} \bigr) \Biggr\vert \Biggr\} ^{v} \\ =&{C}\sum_{n=1}^{\infty}n^{\alpha-2-v/p}E \Biggl\{ \sum_{j=-\infty}^{+\infty } \vert a_{j} \vert ^{1-1/v}\Biggl( \vert a_{j} \vert ^{1/v}\max_{1\leq k\leq n} \Biggl\vert \sum _{i=j+1}^{j+k}\bigl(Y'_{i}-EY'_{i} \bigr) \Biggr\vert \Biggr)\Biggr\} ^{v} \\ \leq&{C}\sum_{n=1}^{\infty} n^{\alpha-2-v/p} \Biggl(\sum_{j=-\infty }^{+\infty} \vert a_{j} \vert \Biggr)^{v-1}\Biggl(\sum _{j=-\infty}^{+\infty} \vert a_{j} \vert E\Biggl( \max_{1\leq k\leq n} \Biggl\vert \sum_{i=j+1}^{j+k} \bigl(Y'_{i}-EY'_{i}\bigr) \Biggr\vert ^{v}\Biggr)\Biggr) \\ \leq& C\sum_{n=1}^{\infty} n^{\alpha-2-v/p} \sum_{j=-\infty}^{+\infty } \vert a_{j} \vert \Biggl\{ \sum_{i=j+1}^{j+n}E \bigl\vert Y'_{i} \bigr\vert ^{v}+\Biggl(\sum _{i=j+1}^{j+n}E \bigl\vert Y'_{i} \bigr\vert ^{2} \Biggr)^{v/2}\Biggr\} \\ =& C\sum_{n=1}^{\infty} n^{\alpha-2-v/p}\sum _{j=-\infty}^{+\infty } \vert a_{j} \vert \Biggl\{ \sum_{i=j+1}^{j+n}\bigl\{ E \vert Y_{i} \vert ^{v}I\bigl( \vert Y_{i} \vert \leq n^{1/p}\bigr)+n^{v/p}P\bigl( \vert Y_{i} \vert >n^{1/p}\bigr)\bigr\} \Biggr\} \\ &{}+ C\sum_{n=1}^{\infty} n^{\alpha-2-v/p}\sum _{j=-\infty}^{+\infty } \vert a_{j} \vert \Biggl\{ \sum_{i=j+1}^{j+n}\bigl\{ E \vert Y_{i} \vert ^{2}I\bigl( \vert Y_{i} \vert \leq n^{1/p}\bigr)+n^{2/p}P\bigl( \vert Y_{i} \vert >n^{1/p}\bigr)\bigr\} ^{v/2}\Biggr\} \\ \leq& C\sum_{n=1}^{\infty} n^{\alpha-1-v/p} \bigl\{ E \vert Y \vert ^{v}I\bigl( \vert Y \vert \leq n^{1/p}\bigr)+n^{v/p}P\bigl( \vert Y \vert >n^{1/p}\bigr)\bigr\} \\ &+ C\sum_{n=1}^{\infty} n^{\alpha-2-v/p+v/2}\bigl\{ E \vert Y \vert ^{2}I\bigl( \vert Y \vert \leq n^{1/p}\bigr)+n^{2/p}P\bigl( \vert Y \vert >n^{1/p}\bigr)\bigr\} ^{v/2} \\ =:&I_{21}+I_{22}. \end{aligned}$$
For \(I_{21}\), taking \(v\geq\alpha p\), we have
$$\begin{aligned} I_{21} =&C\sum_{n=1}^{\infty} n^{\alpha-1-v/p}E \vert Y \vert ^{v}I\bigl( \vert Y \vert \leq n^{1/p}\bigr)+C\sum_{n=1}^{\infty} n^{\alpha-1}P\bigl( \vert Y \vert >n^{1/p}\bigr) \\ =&C\sum_{n=1}^{\infty} n^{\alpha-1-v/p}\sum _{l=1}^{n}E \vert Y \vert ^{v}I\bigl((l-1)^{{1/p}}< \vert Y \vert \leq l^{1/p}\bigr) \\ &{}+C\sum_{n=1}^{\infty }n^{\alpha-1} \sum_{l=n}^{\infty}P\bigl(l^{1/p}< \vert Y \vert \leq(l+1)^{1/p}\bigr) \\ =&C\sum_{l=1}^{\infty}E \vert Y \vert ^{v}I\bigl((l-1)^{{1/p}}< \vert Y \vert \leq l^{1/p}\bigr)\sum_{n=l}^{\infty} n^{\alpha-1-v/p} \\ &{}+C\sum_{l=1}^{\infty}l^{\alpha} P\bigl(l^{1/p}< \vert Y \vert \leq(l+1)^{1/p}\bigr) \\ \leq& C\sum_{l=1}^{\infty}l^{\alpha -v/p}E \vert Y \vert ^{v}I\bigl((l-1)^{{1/p}}< \vert Y \vert \leq l^{1/p}\bigr) +CE \vert Y \vert ^{\alpha p} \\ \leq& CE \vert Y \vert ^{\alpha p}< \infty. \end{aligned}$$
(3.3)
For \(I_{22}\), we will prove the claim based on two cases.
If \(\alpha p<2\), setting \(v=2\), similar to the proof of \(I_{21}\), we obtain
$$ I_{22}=C\sum_{n=1}^{\infty} n^{\alpha-1-2/p}\bigl\{ E \vert Y \vert ^{2}I\bigl( \vert Y \vert \leq n^{1/p}\bigr)+n^{2/p}P\bigl( \vert Y \vert >n^{1/p}\bigr)\bigr\} < \infty. $$
(3.4)
If \(\alpha p\geq2\), noting that \(E|Y|^{2}<\infty\), choosing \(v>\max\{ 2,(\alpha-1)2p/(2-p)\}\), we have that \(\alpha-v/p+v/2<1\), and therefore
$$ I_{22}\leq C\sum_{n=1}^{\infty} n^{\alpha-2-v/p+v/2}\bigl\{ E|Y|^{2}\bigr\} ^{v/2}< \infty. $$
(3.5)
Thus, from (3.2)–(3.5), we see that (2.2) is satisfied. □
Next, we prove Theorem 2.2.
Proof of Theorem 2.2
From the proof of Theorem 2.1, we only need to prove \(I_{1}<\infty\), \(I_{2}<\infty\).
For \(I_{1}\), noting \(\theta\in(0,1)\) if \(p=1\), and \(\theta=1\) if \(1< p<2\), by the Markov and \(C_{r}\)-inequalities, we have
$$\begin{aligned} I_{1} \leq& C\sum_{n=1}^{\infty} n^{-1-\theta/p}E\max_{1\leq k\leq n} \Biggl\vert \sum _{j=-\infty}^{+\infty}a_{j}\sum _{i=j+1}^{j+k}Y''_{i} \Biggr\vert ^{\theta} \\ \leq& C\sum_{n=1}^{\infty} n^{-1-\theta/p} \sum_{j=-\infty}^{+\infty } \vert a_{j} \vert ^{\theta}\sum_{i=j+1}^{j+n}E \vert Y_{i} \vert ^{\theta }I\bigl( \vert Y_{i} \vert >n^{1/p}\bigr) \\ \leq& C\sum_{n=1}^{\infty} n^{-\theta/p}E \vert Y \vert ^{\theta}I\bigl( \vert Y \vert >n^{1/p}\bigr) \\ =&C\sum_{n=1}^{\infty} n^{-\theta/p}\sum _{l=n}^{\infty}E \vert Y \vert ^{\theta }I\bigl(l^{1/p}< \vert Y \vert \leq(l+1)^{1/p} \bigr) \\ =& C\sum_{l=1}^{\infty}E \vert Y \vert ^{\theta}I\bigl(l^{1/p}< \vert Y \vert \leq (l+1)^{1/p}\bigr)\sum_{n=1}^{l}n^{-\theta/p} \\ \leq& C\sum_{l=1}^{\infty}l^{1-\theta/p}E \vert Y \vert ^{\theta }I\bigl(l^{1/p}< \vert Y \vert \leq(l+1)^{1/p}\bigr) < CE \vert Y \vert ^{p}< \infty. \end{aligned}$$
(3.6)
For \(I_{2}\), by Markov and Hölder inequalities, as well as Lemmas 2.1–2.3, we obtain
$$\begin{aligned} I_{2} =&\sum_{n=1}^{\infty} n^{-1-2/p}E\Biggl\{ \max_{1\leq k\leq n} \Biggl\vert \sum _{j=-\infty}^{+\infty}a_{j}\Biggl(\sum _{i=j+1}^{j+k}\bigl(Y'_{i}-EY'_{i} \bigr)\Biggr) \Biggr\vert ^{2}\Biggr\} \\ =&\sum_{n=1}^{\infty}n^{-1-2/p}E\Biggl\{ \sum_{j=-\infty}^{+\infty } \vert a_{j} \vert ^{1/2}\Biggl( \vert a_{j} \vert ^{1/2} \max_{1\leq k\leq n} \Biggl\vert \sum_{i=j+1}^{j+k} \bigl(Y'_{i}-EY'_{i}\bigr) \Biggr\vert \Biggr)\Biggr\} ^{2} \\ \leq&\sum_{n=1}^{\infty} n^{-1-2/p} \Biggl(\sum_{j=-\infty}^{+\infty } \vert a_{j} \vert \Biggr) \Biggl(\sum_{j=1}^{\infty} \vert a_{j} \vert E\Biggl(\max_{1\leq k\leq n} \Biggl\vert \sum_{i=j+1}^{j+k}\bigl(Y'_{i}-EY'_{i} \bigr) \Biggr\vert ^{2}\Biggr)\Biggr) \\ \leq& C\sum_{n=1}^{\infty} n^{-1-2/p} \sum_{j=-\infty}^{+\infty } \vert a_{j} \vert \sum_{i=j+1}^{j+n}E \bigl\vert Y'_{i} \bigr\vert ^{2} \\ \leq& C\sum_{n=1}^{\infty} n^{-1-2/p} \sum_{j=-\infty}^{+\infty } \vert a_{j} \vert \Biggl\{ \sum_{i=j+1}^{j+n}\bigl\{ E \vert Y_{i} \vert ^{2}I\bigl( \vert Y_{i} \vert \leq n^{1/p}\bigr)+n^{2/p}P\bigl( \vert Y_{i} \vert >n^{1/p}\bigr)\bigr\} \Biggr\} \\ \leq& C\sum_{n=1}^{\infty} n^{-2/p} \bigl\{ E \vert Y \vert ^{2}I\bigl( \vert Y \vert \leq n^{1/p}\bigr)+n^{2/p}P\bigl( \vert Y \vert >n^{1/p}\bigr)\bigr\} \\ \leq& C\sum_{n=1}^{\infty} n^{-2/p} \sum_{l=1}^{n}E \vert Y \vert ^{2}I\bigl((l-1)^{1/p}< \vert Y \vert \leq l^{1/p}\bigr)+CE \vert Y \vert ^{p} \\ \leq& C\sum_{l=1}^{\infty} l^{1-2/p}E \vert Y \vert ^{2}I\bigl((l-1)^{1/p}< \vert Y \vert < l^{1/p}\bigr)+CE \vert Y \vert ^{p} \\ \leq& CE \vert Y \vert ^{p}< \infty. \end{aligned}$$
(3.7)
Combing (3.6) and (3.7), we see that Theorem 2.2 holds. □
Proof of Theorem 2.3
Let \(x>n^{\gamma/p}\) and consider
$$\begin{aligned}& Y_{j}'=-x^{1/\gamma}I\bigl(Y_{j}< -x^{1/\gamma} \bigr)+Y_{j}I\bigl( \vert Y_{j} \vert \leq x^{1/\gamma}\bigr)+x^{1/\gamma}I\bigl(Y_{j}>x^{1/\gamma} \bigr), \\& Y_{j}''=Y_{j}-Y_{j}'= \bigl(Y_{j}+x^{1/\gamma}\bigr)I\bigl(Y_{j}< -x^{1/\gamma } \bigr)+\bigl(Y_{j}-x^{1/\gamma}\bigr)I\bigl(Y_{j}>x^{1/\gamma} \bigr). \end{aligned}$$
Similarly to the proof of (3.1), for large enough x, we get
$$ x^{-1/\gamma} \Biggl\vert E\max_{1\leq k\leq n}\sum _{j=-\infty}^{+\infty }a_{j}\sum _{i=j+1}^{j+k}Y_{i}' \Biggr\vert < \varepsilon/4. $$
(3.8)
Then, we have
$$\begin{aligned}& \sum_{n=1}^{\infty} n^{\alpha-2-\gamma/p}E\Bigl\{ \max_{1\leq k\leq n} \vert S_{k} \vert -\varepsilon n^{1/p}\Bigr\} _{+}^{\gamma} \\& \quad = \sum_{n=1}^{\infty} n^{\alpha-2-\gamma/p} \int_{0}^{\infty}P\Bigl(\max_{1\leq k\leq n} \vert S_{k} \vert >\varepsilon n^{1/p}+x^{1/\gamma} \Bigr)\,dx \\& \quad \leq \sum_{n=1}^{\infty} n^{\alpha-2-\gamma/p} \int_{0}^{n^{\gamma /p}}P\Bigl(\max_{1\leq k\leq n} \vert S_{k} \vert >\varepsilon n^{1/p}\Bigr)\,dx \\& \qquad {} +\sum_{n=1}^{\infty} n^{\alpha-2-\gamma/p} \int_{n^{\gamma /p}}^{\infty}P\Bigl(\max_{1\leq k\leq n} \vert S_{k} \vert >x^{1/\gamma}\Bigr)\,dx \\& \quad \leq \sum_{n=1}^{\infty} n^{\alpha-2}P\Bigl(\max_{1\leq k\leq n} \vert S_{k} \vert >\varepsilon n^{1/p}\Bigr) \\& \qquad {} +\sum_{n=1}^{\infty} n^{\alpha-2-\gamma/p} \int_{n^{\gamma /p}}^{\infty}P\Bigl(\max_{1\leq k\leq n} \vert S_{k} \vert >x^{1/\gamma}\Bigr)\,dx. \end{aligned}$$
It follows from Theorem 2.1 that
$$ \sum_{n=1}^{\infty} n^{\alpha-2}P\Bigl(\max _{1\leq k\leq n} \vert S_{k} \vert >\varepsilon n^{1/p}\Bigr)< \infty. $$
(3.9)
To prove (2.4) of Theorem 2.3, we only need to prove
$$I_{3}=:\sum_{n=1}^{\infty} n^{\alpha-2-\gamma/p} \int_{n^{\gamma /p}}^{\infty}P\Bigl(\max_{1\leq k\leq n}|S_{k}|>x^{1/\gamma} \Bigr)\,dx< \infty. $$
By (3.8), we get
$$\begin{aligned} I_{3} \leq& C\sum_{n=1}^{\infty} n^{\alpha-2-\gamma/p} \int_{n^{\gamma /p}}^{\infty}P\Biggl\{ \max_{1\leq k\leq n} \Biggl\vert \sum_{j=-\infty}^{+\infty }a_{j} \sum_{i=j+1}^{j+k}Y''_{i} \Biggr\vert \geq x^{1/\gamma}/2\Biggr\} \,dx \\ &{}+C\sum_{n=1}^{\infty} n^{\alpha-2-\gamma/p} \int_{n^{\gamma /p}}^{\infty}P\Biggl\{ \max_{1\leq k\leq n} \Biggl\vert \sum_{j=-\infty}^{+\infty }a_{j} \sum_{i=j+1}^{j+k}\bigl(Y'_{i}-EY'_{i} \bigr) \Biggr\vert \geq x^{1/\gamma}/4\Biggr\} \,dx \\ =:&I_{31}+I_{32}. \end{aligned}$$
For \(I_{31}\), by Markov inequality and Lemma 2.3, we get
$$\begin{aligned} I_{31} \leq& C\sum_{n=1}^{\infty} n^{\alpha-2-\gamma/p} \int_{n^{\gamma /p}}^{\infty}x^{-1/\gamma}E\max _{1\leq k\leq n} \Biggl\vert \sum_{j=-\infty }^{+\infty}a_{j} \sum_{i=j+1}^{j+k}Y''_{i} \Biggr\vert \,dx \\ \leq& C\sum_{n=1}^{\infty} n^{\alpha-2-\gamma/p} \int_{n^{\gamma /p}}^{\infty}x^{-1/\gamma}\sum _{j=-\infty}^{+\infty} \vert a_{j} \vert \sum _{i=j+1}^{j+n}E \bigl\vert Y''_{i} \bigr\vert \,dx \\ \leq& C\sum_{n=1}^{\infty} n^{\alpha-2-\gamma/p} \int_{n^{\gamma /p}}^{\infty}x^{-1/\gamma}\sum _{j=-\infty}^{+\infty} \vert a_{j} \vert \sum _{i=j+1}^{j+n}E \vert Y_{i} \vert I\bigl( \vert Y_{i} \vert >x^{1/\gamma}\bigr)\,dx \\ \leq& C\sum_{n=1}^{\infty} n^{\alpha-1-\gamma/p} \int_{n^{\gamma /p}}^{\infty}x^{-1/\gamma}E \vert Y \vert I \bigl( \vert Y \vert >x^{1/\gamma}\bigr)\,dx \\ =& C\sum_{n=1}^{\infty} n^{\alpha-1-\gamma/p}\sum _{l=n}^{\infty} \int _{l^{\gamma/p}}^{(l+1)^{\gamma/p}}x^{-1/\gamma}E \vert Y \vert I \bigl( \vert Y \vert >x^{1/\gamma }\bigr)\,dx \\ \leq& C\sum_{n=1}^{\infty} n^{\alpha-1-\gamma/p} \sum_{l=n}^{\infty }l^{\gamma/p(1-1/\gamma)-1}E \vert Y \vert I\bigl( \vert Y \vert >{l^{1/p}}\bigr) \\ =& C\sum_{l=1}^{\infty}l^{\gamma/p-1/p-1}E \vert Y \vert I\bigl( \vert Y \vert >l^{1/p}\bigr)\sum _{n=1}^{l} n^{\alpha-1-\gamma/p} \\ \leq& \textstyle\begin{cases} C\sum_{l=1}^{\infty}l^{\gamma/p-1/p-1}E|Y|I(|Y|>l^{1/p})l^{\alpha -\gamma/p}, & \text{if } \gamma< \alpha p, \\ C\sum_{l=1}^{\infty}l^{\gamma/p-1/p-1}E|Y|I(|Y|>l^{1/p})\log(l+1), & \text{if } \gamma= \alpha p, \\ C\sum_{l=1}^{\infty}l^{\gamma/p-1/p-1}E|Y|I(|Y|>l^{1/p}), & \text{if } \gamma> \alpha p \end{cases}\displaystyle \\ \leq& \textstyle\begin{cases} C\sum_{l=1}^{\infty}l^{\alpha-1/p}E|Y|I(|Y|>l^{1/p}), & \text{if } \gamma< \alpha p, \\ C\sum_{l=1}^{\infty}l^{\alpha-1/p}\log(l+1)E|Y|I(|Y|>l^{1/p}),& \text{if } \gamma=\alpha p, \\ C\sum_{l=1}^{\infty}l^{\gamma/p-1/p}E|Y|I(|Y|>l^{1/p}), & \text{if } \gamma> \alpha p \end{cases}\displaystyle \\ \leq& \textstyle\begin{cases} CE|Y|^{\alpha p}< \infty, & \text{if } \gamma< \alpha p, \\ CE|Y|^{\alpha p}\log(|Y|+1)< \infty, & \text{if } \gamma=\alpha p, \\ CE|Y| ^{\gamma}< \infty, &\text{if } \gamma>\alpha p. \end{cases}\displaystyle \end{aligned}$$
(3.10)
For \(I_{32}\), from Lemmas 2.1–2.3, Markov and Hölder inequalities, we have that for any \(v\geq2\),
$$\begin{aligned} I_{32} \leq& C\sum_{n=1}^{\infty} n^{\alpha-2-\gamma/p} \int_{n^{\gamma /p}}^{\infty}x^{-v/\gamma}E\Biggl\{ \max _{1\leq k\leq n} \Biggl\vert \sum_{j=-\infty }^{+\infty}a_{j} \sum_{i=j+1}^{j+k}\bigl(Y'_{i}-EY'_{i} \bigr) \Biggr\vert \Biggr\} ^{v}\,dx \\ \leq& C\sum_{n=1}^{\infty} n^{\alpha-2-\gamma/p} \int_{n^{\gamma /p}}^{\infty}x^{-v/\gamma}E\Biggl\{ \sum _{j=-\infty}^{+\infty} \vert a_{j} \vert \max _{1\leq k\leq n} \Biggl\vert \sum_{i=j+1}^{j+k} \bigl(Y'_{i}-EY'_{i}\bigr) \Biggr\vert \Biggr\} ^{v}\,dx \\ \leq& C\sum_{n=1}^{\infty} n^{\alpha-2-\gamma/p} \int_{n^{\gamma /p}}^{\infty}x^{-v/\gamma}E\Biggl\{ \sum _{j=-\infty}^{+\infty } \vert a_{j} \vert ^{1-1/v}\Biggl( \vert a_{j} \vert ^{1/v}\max _{1\leq k\leq n} \Biggl\vert \sum_{i=j+1}^{j+k} \bigl(Y'_{i}-EY'_{i}\bigr) \Biggr\vert \Biggr)\Biggr\} ^{v}\,dx \\ \leq& C\sum_{n=1}^{\infty} n^{\alpha-2-\gamma/p} \int_{n^{\gamma /p}}^{\infty}x^{-v/\gamma}\Biggl(\sum _{j=-\infty}^{+\infty } \vert a_{j} \vert \Biggr)^{v-1} \\ &{}\times\Biggl(\sum_{j=-\infty}^{+\infty} \vert a_{j} \vert E\Biggl(\max_{1\leq k\leq n} \Biggl\vert \sum_{i=j+1}^{j+k}\bigl(Y'_{i}-EY'_{i} \bigr) \Biggr\vert ^{v}\Biggr)\Biggr)\,dx \\ \leq& C\sum_{n=1}^{\infty} n^{\alpha-2-\gamma/p} \int_{n^{\gamma /p}}^{\infty}x^{-v/\gamma}\sum _{j=-\infty}^{+\infty} \vert a_{j} \vert \Biggl\{ \sum_{i=j+1}^{j+n}E \bigl\vert Y'_{i} \bigr\vert ^{v}+\Biggl(\sum _{i=j+1}^{j+n}E \bigl\vert Y'_{j} \bigr\vert ^{2} \Biggr)^{v/2}\Biggr\} \,dx \\ \leq& C\sum_{n=1}^{\infty} n^{\alpha-2-\gamma/p} \int_{n^{\gamma /p}}^{\infty}x^{-v/\gamma}\sum _{j=-\infty}^{+\infty} \vert a_{j} \vert \\ &{}\times\Biggl\{ \sum _{i=j+1}^{j+n}\bigl\{ E \vert Y_{i} \vert ^{v}I\bigl( \vert Y_{i} \vert \leq x^{1/\gamma}\bigr)+x^{v/\gamma }P\bigl( \vert Y_{i} \vert >x^{1/\gamma}\bigr)\bigr\} \Biggr\} \,dx \\ &{} + C\sum_{n=1}^{+\infty} n^{\alpha-2-\gamma/p} \int_{n^{\gamma /p}}^{\infty}x^{-v/\gamma}\sum _{j=-\infty}^{+\infty} \vert a_{j} \vert \\ &{}\times \Biggl\{ \sum _{i=j+1}^{j+n}\bigl\{ E \vert Y_{i} \vert ^{2}I\bigl( \vert Y_{i} \vert \leq x^{1/\gamma}\bigr)+x^{2/\gamma }P\bigl( \vert Y_{i} \vert >x^{1/\gamma}\bigr)\bigr\} \Biggr\} ^{v/2}\,dx \\ \leq& C\sum_{n=1}^{\infty} n^{\alpha-1-\gamma/p} \int_{n^{\gamma /p}}^{\infty}x^{-v/\gamma}\bigl\{ E \vert Y \vert ^{v}I\bigl( \vert Y \vert \leq x^{1/\gamma} \bigr)+x^{v/\gamma }P\bigl( \vert Y \vert >x^{1/\gamma}\bigr)\bigr\} \,dx \\ &{}+C\sum_{n=1}^{\infty} n^{\alpha-2-\gamma/p+v/2} \int_{n^{\gamma /p}}^{\infty}x^{-v/\gamma}\bigl\{ E \vert Y \vert ^{2}I\bigl( \vert Y \vert \leq x^{1/\gamma} \bigr)+x^{2/\gamma }P\bigl( \vert Y \vert >x^{1/\gamma}\bigr)\bigr\} ^{v/2}\,dx \\ =:&I_{321}+I_{322}. \end{aligned}$$
For \(I_{321}\), taking \(v> \alpha p\), we have
$$\begin{aligned} I_{321} =&C\sum_{n=1}^{\infty} n^{\alpha-1-\gamma/p}\sum_{l=n}^{\infty } \int_{l^{\gamma/p}}^{(l+1)^{\gamma/p}}\bigl[x^{-v/\gamma}E \vert Y \vert ^{v}I\bigl( \vert Y \vert \leq x^{1/\gamma}\bigr)+P \bigl( \vert Y \vert >x^{1/\gamma}\bigr)\bigr]\,dx \\ \leq& C\sum_{n=1}^{\infty} n^{\alpha-1-\gamma/p} \sum_{l=n}^{\infty }\bigl[l^{(\gamma/p(1-v/\gamma)-1)}E \vert Y \vert ^{v}I\bigl( \vert Y \vert \leq(l+1)^{1/p} \bigr)+l^{\gamma /p-1}P\bigl( \vert Y \vert >l^{1/p}\bigr)\bigr] \\ \leq& C\sum_{l=1}^{\infty} \bigl[l^{(\gamma/p-v/p-1)}E \vert Y \vert ^{v}I\bigl( \vert Y \vert \leq (l+1)^{1/p}\bigr)+l^{\gamma/p-1}P\bigl( \vert Y \vert >l^{1/p}\bigr)\bigr]\sum_{n=1}^{l} n^{\alpha -1-\gamma/p} \\ \leq& \textstyle\begin{cases} C\sum_{l=1}^{\infty}[l^{\alpha-v/p-1}E|Y|^{v}I(|Y|\leq (l+1)^{1/p})+l^{\alpha-1}P(|Y|>l^{1/p})], \\ \quad \text{if } \gamma< \alpha p, \\ C\sum_{l=1}^{\infty}[l^{\alpha-v/p-1}E|Y|^{v}I(|Y|\leq (l+1)^{1/p})+l^{\alpha-1}P(|Y|>l^{1/p})]\log(l+1), \\ \quad \text{if } \gamma =\alpha p, \\ C\sum_{l=1}^{\infty}[l^{\gamma/p-v/p-1}E|Y|^{v}I(|Y|\leq (l+1)^{1/p})+l^{\gamma/p-1}P(|Y|>l^{1/p})], \\ \quad \text{if } \gamma>\alpha p \end{cases}\displaystyle \\ \leq& \textstyle\begin{cases} C\sum_{k=1}^{\infty}k^{\alpha-v/p}E|Y|^{v}I((k-1)^{1/p}< |Y|\leq k^{1/p})+CE|Y|^{\alpha p}, \\ \quad \text{if } \gamma< \alpha p, \\ C\sum_{k=1}^{\infty}k^{\alpha-v/p}E|Y|^{v}I((k-1)^{1/p}< |Y|\leq k^{1/p})\log(k+1)+CE|Y|^{\alpha p}\log(|Y|+1), \\ \quad \text{if } \gamma =\alpha p, \\ C\sum_{k=1}^{\infty}k^{\gamma/p-v/p}E|Y|^{v}I((k-1)^{1/p}< |Y|\leq k^{1/p})+CE|Y| ^{\gamma}, \\ \quad \text{if } \gamma> \alpha p \end{cases}\displaystyle \\ \leq& \textstyle\begin{cases} CE|Y|^{\alpha p}< \infty, &\text{if } \gamma< \alpha p, \\ CE|Y|^{\alpha p}\log(|Y|+1)< \infty, & \text{if } \gamma=\alpha p, \\ CE|Y| ^{\gamma}< \infty, & \text{if } \gamma>\alpha p. \end{cases}\displaystyle \end{aligned}$$
(3.11)
For \(I_{322}\), we prove the claim by considering two cases.
If \(\alpha p<2\), setting \(v=2\), we have
$$I_{322}=C \sum_{n=1}^{\infty} n^{r-1-\gamma/p} \int_{n^{\gamma /p}}^{\infty}x^{-2/\gamma}\bigl\{ E \vert Y \vert ^{2}I\bigl( \vert Y \vert \leq x^{1/\gamma} \bigr)+x^{2/\gamma }P\bigl( \vert Y \vert >x^{1/\gamma}\bigr)\bigr\} \,dx. $$
Similarly to the proof of \(I_{321}\), we have
$$ I_{322}< \infty. $$
(3.12)
If \(\alpha p\geq2\), noting that \(E|Y|^{2}<\infty\), choosing \(v>\max\{ 2,\gamma,(\alpha-1)2p/(2-p)\}\), we obtain \(\alpha-v/p+v/2<1\). Therefore, we have
$$\begin{aligned} I_{322} \leq& C\sum_{n=1}^{\infty} n^{\alpha-2-\gamma/p+v/2} \int _{n^{\gamma/p}}^{\infty}x^{-v/\gamma}\bigl\{ E|Y|^{2}\bigr\} ^{v/2}\,dx \\ \leq& C\sum_{n=1}^{\infty} n^{\alpha-2+v/2-v/p} \bigl\{ E|Y|^{2}\bigr\} ^{v/2}< \infty. \end{aligned}$$
(3.13)
Then, by (3.6)–(3.13), we see that (2.4) is true.
Next, we prove (2.5).
$$\begin{aligned}& \sum_{n=1}^{\infty} n^{\alpha-2}E\Bigl\{ \sup_{ k\geq n} \bigl\vert k^{-1/p}S_{k} \bigr\vert -\varepsilon\Bigr\} _{+}^{\gamma} \\& \quad = \sum_{n=1}^{\infty} n^{\alpha-2} \int_{0}^{\infty}P\Bigl(\sup_{ k\geq n} \bigl\vert k^{-1/p}S_{k} \bigr\vert >\varepsilon+x^{1/\gamma} \Bigr)\,dx \\& \quad = \sum_{j=1}^{\infty}\sum _{n=2^{j-1}}^{2^{j}-1} n^{\alpha-2} \int _{0}^{\infty}P\Bigl(\sup_{ k\geq n} \bigl\vert k^{-1/p}S_{k} \bigr\vert >\varepsilon+x^{1/\gamma } \Bigr)\,dx \\& \quad \leq C \sum_{j=1}^{\infty} \int_{0}^{\infty}P\Bigl(\sup_{ k\geq 2^{j-1}} \bigl\vert k^{-1/p}S_{k} \bigr\vert >\varepsilon+x^{1/\gamma} \Bigr)\,dx\sum_{n=2^{j-1}}^{2^{j}-1} 2^{j(\alpha-2)} \\& \quad \leq C\sum_{j=1}^{\infty}2^{j(\alpha-1)} \int_{0}^{\infty}P\Bigl(\sup_{ k\geq2^{j-1}} \bigl\vert k^{-1/p}S_{k} \bigr\vert >\varepsilon+x^{1/\gamma} \Bigr)\,dx \\& \quad \leq C\sum_{j=1}^{\infty}2^{j(\alpha-1)} \sum_{l=j}^{\infty} \int _{0}^{\infty}P\Bigl(\max_{2^{l-1}\leq k\leq 2^{l}} \bigl\vert k^{-1/p}S_{k} \bigr\vert >\varepsilon+x^{1/\gamma} \Bigr)\,dx \\& \quad \leq C\sum_{l=1}^{\infty} \int_{0}^{\infty}P\Bigl(\max_{2^{l-1}\leq k\leq 2^{l}} \bigl\vert k^{-1/p}S_{k} \bigr\vert >\varepsilon+x^{1/\gamma} \Bigr)\,dx\sum_{j=1}^{l}2^{j(\alpha-1)} \\& \quad \leq C\sum_{l=1}^{\infty}2^{l(\alpha-1)} \int_{0}^{\infty}P\Bigl(\max_{2^{l-1}\leq k\leq2^{l}} \vert S_{k} \vert >\bigl(\varepsilon+x^{1/\gamma } \bigr)2^{(l-1)/p)}\Bigr)\,dx \\& \qquad \bigl(\mbox{Let }y=2^{(l-1)\gamma/p}x\bigr) \\& \quad \leq C\sum_{l=1}^{\infty}2^{l(\alpha-1-\gamma/p)} \int_{0}^{\infty }P\Bigl(\max_{1\leq k\leq2^{l}} \vert S_{k} \vert >\varepsilon2^{(l-1)/p}+y^{1/\gamma } \Bigr)\,dy \\& \quad \leq C\sum_{n=1}^{\infty}n^{(\alpha-2-\gamma/p)} \int_{0}^{\infty }P\Bigl(\max_{1\leq k\leq n} \vert S_{k} \vert >\varepsilon n^{1/p}2^{-1/p}+y^{1/\gamma } \Bigr)\,dy \\& \qquad \bigl(\mbox{Let }\varepsilon_{0}=2^{-1/p}\varepsilon \bigr) \\& \quad \leq C\sum_{n=1}^{\infty}n^{(\alpha-2-\gamma/p)}E \Bigl\{ \max_{1\leq k\leq n} \vert S_{k} \vert - \varepsilon_{0} n^{1/p}\Bigr\} _{+}^{\gamma}< \infty. \end{aligned}$$
Thus, (2.5) holds. □
Next, we prove Theorem 2.4.
Proof of Theorem 2.4
Since the proof of Theorem 2.4 is similar to that of Theorem 2.3, we only give the proof outline as follows.
By (3.1), we have
$$\begin{aligned}& \sum_{n=1}^{\infty} n^{-1-\gamma/p}E\Bigl\{ \max_{1\leq k\leq n} \vert S_{k} \vert -\varepsilon n^{1/p}\Bigr\} _{+}^{\gamma} \\& \quad \leq \sum_{n=1}^{\infty} n^{-1}P \Bigl(\max_{1\leq k\leq n} \vert S_{k} \vert >\varepsilon n^{1/p}\Bigr) +\sum_{n=1}^{\infty} n^{-1-\gamma/p} \int_{n^{\gamma/p}}^{\infty}P\Bigl(\max_{1\leq k\leq n} \vert S_{k} \vert >x^{1/\gamma}\Bigr)\,dx \\& \quad =: J_{1} +J_{2}. \end{aligned}$$
For \(J_{1}\), by Theorem 2.2, we have
$$ J_{1}=\sum_{n=1}^{\infty} n^{-1}P\Bigl(\max_{1\leq k\leq n} \vert S_{k} \vert >\varepsilon n^{1/p}\Bigr)< \infty. $$
(3.14)
For \(J_{2}\), similar to the proof of \(I_{3}\), we have
$$\begin{aligned} J_{2} \leq& C\sum_{n=1}^{\infty} n^{-1-\gamma/p} \int_{n^{\gamma /p}}^{\infty}P\Biggl\{ \max_{1\leq k\leq n} \Biggl\vert \sum_{j=-\infty}^{ +\infty}a_{j} \sum_{i=j+1}^{j+k}Y''_{i} \Biggr\vert \geq x^{1/\gamma}/2\Biggr\} \,dx \\ &{} +C\sum_{n=1}^{\infty} n^{ -1-\gamma/p} \int_{n^{\gamma/p}}^{\infty }P\Biggl\{ \max_{1\leq k\leq n} \Biggl\vert \sum_{j=-\infty}^{+\infty}a_{j} \sum_{i=j+1}^{j+k}\bigl(Y'_{i}-EY'_{i} \bigr) \Biggr\vert \geq x^{1/\gamma}/4\Biggr\} \,dx \\ =:&J_{21}+J_{22}. \end{aligned}$$
For \(J_{21}\), by Lemma 2.3, Markov and \(C_{r}\) inequalities, similarly to the proof of \(I_{31}\), we get
$$\begin{aligned} J_{21} \leq& C\sum_{n=1}^{\infty} n^{-1-\gamma/p} \int_{n^{\gamma /p}}^{\infty}x^{-\theta/\gamma}E\max _{1\leq k\leq n} \Biggl\vert \sum_{j=-\infty }^{+\infty}a_{j} \sum_{i=j+1}^{j+k}Y''_{i} \Biggr\vert ^{\theta} \,dx \\ \leq& C\sum_{n=1}^{\infty} n^{-\gamma/p} \sum_{l=n}^{\infty} \int _{l^{\gamma/p}}^{(l+1)^{\gamma/p}}x^{-\theta/\gamma}E \vert Y \vert ^{\theta }I\bigl( \vert Y \vert >x^{1/\gamma}\bigr)\,dx \\ \leq& C\sum_{n=1}^{\infty} n^{-\gamma/p} \sum_{l=n}^{\infty}l^{\gamma /p(1-\theta/\gamma)-1}E \vert Y \vert ^{\theta}I\bigl( \vert Y \vert >l^{/p}\bigr) \\ \leq& \textstyle\begin{cases} C\sum_{l=1}^{\infty}l^{-\theta/p}E|Y|^{\theta}I(|Y|>l^{1/p}),& \text{if } \gamma< p, \\ C\sum_{l=1}^{\infty}l^{-\theta/p}E|Y|^{\theta}I(|Y|>l^{1/p})\log(l+1), & \text{if } \gamma=p, \\ C\sum_{l=1}^{\infty}l^{\gamma/p-\theta/p-1}E|Y|^{\theta }I(|Y|>l^{1/p}), & \text{if } \gamma> p \end{cases}\displaystyle \\ \leq& \textstyle\begin{cases} CE|Y|^{p}< \infty, & \text{if } \gamma< p, \\ CE|Y|^{p}\log(|Y|+1)< \infty,& \text{if } \gamma=p, \\ CE|Y| ^{\gamma}< \infty,& \text{if } \gamma>p. \end{cases}\displaystyle \end{aligned}$$
(3.15)
For \(J_{22}\), similar to the proof of \(I_{32}\), taking \(v=2\), we obtain
$$\begin{aligned} J_{22} \leq& C\sum_{n=1}^{\infty} n^{-1-\gamma/p} \int_{n^{\gamma /p}}^{\infty}x^{-2/\gamma}E\Biggl\{ \max _{1\leq k\leq n}\Biggl|\sum_{j=-\infty }^{+\infty}a_{j} \sum_{i=j+1}^{j+k}\bigl(Y'_{i}-EY'_{i} \bigr)\Biggr|^{2}\Biggr\} \,dx \\ \leq& C\sum_{n=1}^{\infty} n^{-1-\gamma/p} \int_{n^{\gamma/p}}^{\infty }x^{-2/\gamma}E\Biggl\{ \sum _{j=-\infty}^{+\infty }|a_{j}|^{1/2} \Biggl(|a_{j}|^{1/2}\max_{1\leq k\leq n}\Biggl|\sum _{i=j+1}^{j+k}\bigl(Y'_{i}-EY'_{i} \bigr)\Biggr|\Biggr)\Biggr\} ^{2}\,dx \\ \leq& C\sum_{n=1}^{\infty} n^{-1-\gamma/p} \int_{n^{\gamma/p}}^{\infty }x^{-2/\gamma}\sum _{j=-\infty}^{+\infty}|a_{j}|\sum _{i=j+1}^{i+n}E\bigl|Y'_{i}\bigr|^{2} \,dx \\ \leq& C\sum_{n=1}^{\infty} n^{-\gamma/p} \int_{n^{\gamma/p}}^{\infty }x^{-2/\gamma}\bigl\{ E|Y|^{2}I\bigl(|Y|\leq x^{1/\gamma}\bigr)+x^{2/\gamma }P \bigl(|Y|>x^{1/\gamma}\bigr)\bigr\} \,dx \\ =& C\sum_{n=1}^{\infty} n^{-\gamma/p}\sum _{l=n}^{\infty} \int _{l^{\gamma/p}}^{(l+1)^{\gamma/p}}\bigl[x^{-2/\gamma}E|Y|^{2}I \bigl(|Y|\leq x^{1/\gamma}\bigr)+P\bigl(|Y|>x^{1/\gamma}\bigr)\bigr]\,dx \\ \leq& C\sum_{l=1}^{\infty} \bigl[l^{(\gamma/p-2/p-1)}E|Y|^{2}I\bigl(|Y|\leq (l+1)^{1/p} \bigr)+l^{\gamma/p-1}P\bigl(|Y|>l^{1/p}\bigr)\bigr]\sum _{n=1}^{l} n^{-\gamma /p} \\ \leq& \textstyle\begin{cases} C\sum_{k=1}^{\infty}[k^{1-2/p}E|Y|^{2}I((k-1)^{1/p}< |Y|\leq k^{1/p})]+CE|Y|^{p}, \\ \quad \text{if } \gamma< p, \\ C\sum_{k=1}^{\infty}[k^{1-2/p}E|Y|^{2}I((k-1)^{1/p}< |Y|\leq k^{1/p})]\log(l+1)+CE|Y|^{p}\log(|Y|+1), \\ \quad \text{if } \gamma= p, \\ C\sum_{k=1}^{\infty}[k^{\gamma/p-2/p}E|Y|^{2}I((k-1)^{1/p}< |Y|\leq k^{1/p})]+CE|Y|^{\gamma}, \\ \quad \text{if } \gamma> p \end{cases}\displaystyle \\ \leq& \textstyle\begin{cases} CE|Y|^{p}< \infty, & \text{if } \gamma< p, \\ CE|Y|^{p}\log(|Y|+1)< \infty, &\text{if } \gamma=p, \\ CE|Y| ^{\gamma}< \infty, & \text{if } \gamma>p. \end{cases}\displaystyle \end{aligned}$$
(3.16)
Therefore, from (3.15)–(3.16), we see that (2.6) holds. □