The following three lemmas play important roles in the proof of our theorems. Lemma 3.1 can be obtained directly from the definition of END sequences.
Lemma 3.1
Let
\(\{X_{n}; n\geq1\}\)
be a sequence of END random variables and
\(\{f_{n}; n\geq1\}\)
be a sequence of Borel functions, all of which are monotone increasing (or all are monotone decreasing). Then
\(\{f_{n}(X_{n}); n\geq1\}\)
is a sequence of END r.v.’s.
Lemma 3.2
Liu et al. 2015 [13]
Let
\(\{ X_{n};n\geq1\}\)
be a sequence of END random variables with
\(\mathbb{E}X_{n}=0\)
and
\(\mathbb {E}|X_{n}|^{p}<\infty\), \(p\geq 2\). Then there exists a positive constant
c
depending only on
p
such that
$$\mathbb{E}\bigl(|S_{n}|^{p}\bigr)\leq c \Biggl\{ \sum ^{n} _{i=1}\mathbb{E}|X_{i}|^{p} + \Biggl(\sum^{n} _{i=1} \mathbb{E}X_{i}^{2} \Biggr)^{p/2} \Biggr\} $$
and
$$\mathbb{E}\Bigl(\max_{1\leq i \leq n}|S_{i}|^{p} \Bigr)\leq c\ln^{p}n \Biggl\{ \sum^{n} _{i=1}\mathbb{E}|X_{i}|^{p} + \Biggl(\sum ^{n} _{i=1}\mathbb{E}X_{i}^{2} \Biggr)^{p/2} \Biggr\} . $$
Lemma 3.3
Let
\(\{X_{n};n\geq1\}\)
be a sequence of END random variables. Then, for any
\(x\geq0\), there exists a positive constant
c
such that for all
\(n\geq1\),
$$\Bigl(1-P\Bigl(\max_{1\leq k\leq n} \vert X_{k} \vert >x \Bigr) \Bigr)^{2}\sum_{k=1} ^{n}P \bigl( \vert X_{k} \vert >x\bigr)\leq cP \Bigl(\max _{1\leq k\leq n} \vert X_{k} \vert >x \Bigr). $$
Further, if \(P(\max_{1\leq k\leq n}|X_{k}|>x)\rightarrow0\) as \(n\rightarrow\infty\), then there exists a positive constant c such that for all \(n\geq1\),
$$\sum_{k=1} ^{n}P\bigl( \vert X_{k} \vert >x\bigr)\leq cP \Bigl(\max_{1\leq k\leq n} \vert X_{k} \vert >x \Bigr). $$
Proof
From the proof of Lemma 1.4 in Wu (2012 [27]) and by Lemma 3.2, we can prove Lemma 3.3. □
Proof of Theorem 2.1
We first prove that (2.1) ⇒ (2.2). Note that
$$\begin{aligned}& \sum_{n=1}^{\infty}n^{r-2-q/p}\mathbb{E} \Bigl\{ \max_{1\leq k\leq n}|S_{k}|-\varepsilon n^{1/p} \Bigr\} _{+}^{q} \\& \quad = \sum_{n=1}^{\infty}n^{r-2-q/p} \int_{0}^{n^{1/p}} q x^{q-1} P \Bigl(\max _{1\leq k\leq n}|S_{k}|-\varepsilon n^{1/p}>x \Bigr)\, \mathrm{d}x \\& \qquad {}+\sum_{n=1}^{\infty}n^{r-2-q/p} \int_{n^{1/p}}^{\infty} q x^{q-1}P \Bigl(\max _{1\leq k\leq n}|S_{k}|-\varepsilon n^{1/p}>x \Bigr)\, \mathrm{d}x \\& \quad \ll \sum_{n=1}^{\infty}n^{r-2}P \Bigl(\max_{1\leq k\leq n}|S_{k}|>\varepsilon n^{1/p} \Bigr) \\& \qquad {}+\sum_{n=1}^{\infty}n^{r-2-q/p} \int_{n^{1/p}}^{\infty} x^{q-1}P \Bigl(\max _{1\leq k\leq n}|S_{k}|>x \Bigr)\,\mathrm{d}x \\& \quad \mathop{\hat{=}} I+J. \end{aligned}$$
By Corollary 2.1 in Liu et al. (2015 [13]), \(I<\infty\). Hence, in order to establish (2.2), it is enough to prove that
$$ J\mathop{\hat{=}}\sum_{n=1}^{\infty}n^{r-2-q/p} \int_{n^{1/p}}^{\infty} x^{q-1}P \Bigl(\max _{1\leq k\leq n}|S_{k}|>x \Bigr)\,\mathrm{d}x< \infty. $$
(3.1)
Let \(x\geq n^{1/p}\), \(r^{-1}<\alpha<1\) and an integer \(N>\max ((r-1)(\alpha r-1)^{-1}, q(\alpha rp)^{-1})\). Define, for \(1\leq k\leq n\), \(n\geq1\),
$$\begin{aligned}& X_{k}^{(1)}=-x^{\alpha}I\bigl(X_{k}< -x^{\alpha}\bigr)+X_{k}I\bigl(|X_{k}|\leq x^{\alpha}\bigr)+x^{\alpha}I\bigl(X_{k}>x^{\alpha}\bigr), \\& X_{k}^{(2)}=\bigl(X_{k}-x^{\alpha}\bigr) I \bigl(x^{\alpha}< X_{k}< x/(4N)\bigr), \\& X_{k}^{(3)}=\bigl(X_{k}+x^{\alpha}\bigr) I \bigl(-x/(4N)< X_{k}< -x^{\alpha}\bigr), \\& X_{k}^{(4)}=\bigl(X_{k}+x^{\alpha}\bigr) I \bigl(X_{k}\leq-x/(4N)\bigr)+\bigl(X_{k}-x^{\alpha}\bigr) I\bigl(X_{k}\geq x/(4N)\bigr), \\& S^{(j)}_{k}=\sum_{i=1}^{k}X^{(j)}_{i}, \quad j=1, 2, 3, 4. \end{aligned}$$
It is obvious that \(S_{k}=\sum_{j=1}^{4}S^{(j)}_{k}\). Hence, in order to establish (3.1), it suffices to prove that
$$ J^{(j)}\mathop{\hat{=}}\sum_{n=1}^{\infty}n^{r-2-q/p} \int_{n^{1/p}}^{\infty} x^{q-1}P \Bigl(\max _{1\leq k\leq n} \bigl\vert S^{(j)}_{k} \bigr\vert >x/4 \Bigr)\,\mathrm{d}x< \infty, \quad j=1, 2, 3, 4. $$
(3.2)
Note that
$$ \sum_{n=1}^{j}n^{r-1-q/p}\ll \textstyle\begin{cases} j^{r-q/p},&\mbox{if }q< rp, \\ \ln j,&\mbox{if }q=rp, \\ 1,&\mbox{if }q>rp. \end{cases} $$
By combining this with (2.1), Markov’s inequality and
$$\begin{aligned} P\Bigl(\max_{1\leq k\leq n} \bigl\vert S^{(4)}_{k} \bigr\vert >x/4\Bigr) \leq& P\bigl(\mbox{there is } k, k\in[1, n] \mbox{ such that } X^{(4)}_{k}\neq0\bigr) \\ \leq& nP\bigl( \vert X \vert \geq x/(4N)\bigr), \end{aligned}$$
we get
$$\begin{aligned} J^{(4)} \leq& \sum_{n=1}^{\infty}n^{r-1-q/p} \int_{n^{1/p}}^{\infty} x^{q-1}P\bigl( \vert X \vert >x/(4N)\bigr)\,\mathrm{d}x \\ \leq&\sum_{n=1}^{\infty}n^{r-1-q/p}\sum _{j=n}^{\infty}\int _{j^{1/p}}^{(j+1)^{1/p}} x^{q-1}P\bigl(4N \vert X \vert >j^{1/p}\bigr)\,\mathrm{d}x \\ \ll&\sum_{n=1}^{\infty}n^{r-1-q/p}\sum _{j=n}^{\infty}j^{q/p-1}P\bigl(4N \vert X \vert >j^{1/p}\bigr) \\ =&\sum_{j=1}^{\infty}\sum _{n=1}^{j} n^{r-1-q/p} j^{q/p-1}P\bigl(4N \vert X \vert >j^{1/p}\bigr) \\ \ll& \textstyle\begin{cases} \sum_{j=1}^{\infty}j^{r-1}P(4N \vert X \vert >j^{1/p}),&\mbox{if }q< rp, \\ \sum_{j=1}^{\infty}j^{r-1}\ln j P(4N \vert X \vert >j^{1/p}),&\mbox{if }q=rp, \\ \sum_{j=1}^{\infty}j^{q/p-1}P(4N \vert X \vert >j^{1/p}),&\mbox{if }q>r p \end{cases}\displaystyle \\ \ll& \textstyle\begin{cases} \mathbb{E} \vert X \vert ^{rp} < \infty,&\mbox{if }q< rp, \\ \mathbb{E} \vert X \vert ^{rp} \ln \vert X \vert < \infty,&\mbox{if }q=rp, \\ \mathbb{E} \vert X \vert ^{q} < \infty,&\mbox{if }q>rp. \end{cases}\displaystyle \end{aligned}$$
(3.3)
From the definition of \(X^{(2)}_{k}\), it is clear that \(X^{(2)}_{k}>0\). Thus, by Definition 1.1 and (2.1),
$$\begin{aligned} P \Bigl(\max_{1\leq k\leq n} \bigl\vert S^{(2)}_{k} \bigr\vert >x/4 \Bigr) =&P \Biggl(\sum_{k=1}^{n}X^{(2)}_{k}>x/4 \Biggr) \\ \leq& P\bigl(\mbox{there are at least } N \mbox{ indices } i\in[1, n] \mbox{ such that } X_{i}>x^{\alpha}\bigr) \\ \leq&\sum_{1\leq i_{1}< i_{2}< \cdots< i_{N}\leq n}\prod_{j=1}^{N}P \bigl( \vert X_{i_{j}} \vert >x^{\alpha}\bigr) \\ \leq& n^{N}\bigl(P\bigl( \vert X \vert >x^{\alpha}\bigr) \bigr)^{N}\leq n^{N}x^{-\alpha rpN}\bigl(\mathbb{E} \vert X \vert ^{rp}\bigr)^{N} \\ \ll& n^{N}x^{-\alpha rpN}. \end{aligned}$$
Hence, by the definition of N, \(-\alpha rpN+q-1<-1\) and \(-(\alpha r-1)N+r-2<-1\),
$$\begin{aligned} J^{(2)} \ll&\sum_{n=1}^{\infty}n^{r-2-q/p+N} \int_{n^{1/p}}^{\infty} x^{-\alpha rpN +q-1}\,\mathrm{d}x \\ \ll& \sum_{n=1}^{\infty}n^{r-2-q/p+N}n^{(-\alpha rpN +q)/p} \\ =&\sum_{n=1}^{\infty}n^{-(\alpha r-1)N-2+r} \\ < &\infty. \end{aligned}$$
(3.4)
Similarly, we can show
$$ J^{(3)}< \infty. $$
(3.5)
In order to estimate \(J^{(1)}\), we first verify that
$$ \sup_{x\geq n^{1/p}}\max_{1\leq k\leq n}x^{-1}\bigl| \mathbb{E}S^{(1)}_{k}\bigr|\rightarrow0\quad \mbox{as } n \rightarrow\infty. $$
(3.6)
When \(0< p<1\), by Markov’s inequality and \(\mathbb{E}|X|^{p}<\infty\),
$$\begin{aligned} \sup_{x\geq n^{1/p}}\max_{1\leq k\leq n}x^{-1} \bigl\vert \mathbb{E}S^{(1)}_{k} \bigr\vert \leq&\sup _{x\geq n^{1/p}}x^{-1}n \bigl(x^{\alpha}P\bigl( \vert X \vert >x^{\alpha}\bigr)+\mathbb{E} \vert X \vert I\bigl( \vert X \vert \leq x^{\alpha}\bigr) \bigr) \\ \leq&\sup_{x\geq n^{1/p}} \bigl(nx^{-1+\alpha-\alpha p} \mathbb{E} \vert X \vert ^{p}+x^{-1}n\mathbb{E} \vert X \vert ^{p}x^{\alpha(1-p)} \bigr) \\ \ll& n^{-(1-\alpha)(p^{-1}-1)} \\ \rightarrow&0\quad \mbox{as } n\rightarrow\infty. \end{aligned}$$
When \(1\leq p<2\), by \(\mathbb{E}X=0\) and \(\mathbb{E}|X|^{rp}<\infty\), we get
$$\begin{aligned} \sup_{x\geq n^{1/p}}\max_{1\leq k\leq n}x^{-1} \bigl\vert \mathbb{E}S^{(1)}_{k} \bigr\vert \leq&\sup _{x\geq n^{1/p}} \bigl(n x^{-1+\alpha}P\bigl( \vert X \vert >x^{\alpha}\bigr)+x^{-1}n\mathbb {E} \vert X \vert I\bigl( \vert X \vert > x^{\alpha}\bigr) \bigr) \\ \ll&\sup_{x\geq n^{1/p}}nx^{-1-\alpha rp+\alpha}\mathbb{E} \vert X \vert ^{rp}\ll n^{-(\alpha r-1)-(1-\alpha)/p} \\ \rightarrow&0 \quad \mbox{as } n\rightarrow\infty. \end{aligned}$$
That is, (3.6) holds. Hence, in order to prove \(J^{(1)}<\infty\), it suffices to prove that
$$ \tilde{J}^{(1)} \mathop{\hat{=}} \sum_{n=1}^{\infty}n^{r-2-q/p} \int_{n^{1/p}}^{\infty} x^{q-1}P \Bigl(\max _{1\leq k\leq n} \bigl\vert S^{(1)}_{k}- \mathbb{E}S^{(1)}_{k} \bigr\vert >x/8 \Bigr)\,\mathrm{d}x< \infty. $$
(3.7)
Obviously, \(X^{(1)}_{k}\) is increasing on \(X_{k}\), thus by Lemma 3.1, \(\{X^{(1)}_{k}; k\geq1\}\) is also a sequence of END random variables. In view of Lemma 3.2, taking \(u=\max(rp,q)\) and
$$s>\max \biggl(2, u, \frac{r-1}{\frac{1}{2} (\frac{\alpha u}{p}-1 )+\frac{1-\alpha}{p}}, \frac{r-1}{\frac{1}{p}-\frac{1}{2}}, \frac{q}{1-\alpha+\frac{\alpha u}{2}}, \frac{q-\alpha rp}{1-\alpha} \biggr), $$
we obtain
$$\begin{aligned} \tilde{J}^{(1)} \ll& \sum_{n=1}^{\infty}n^{r-2-q/p} \int_{n^{1/p}}^{\infty} x^{q-1-s}\mathbb{E} \Bigl(\max _{1\leq k\leq n} \bigl\vert S^{(1)}_{k}- \mathbb{E}S^{(1)}_{k} \bigr\vert ^{s} \Bigr)\, \mathrm{d}x \\ \ll& \sum_{n=1}^{\infty}n^{r-1-q/p} \ln^{s} n \int_{n^{1/p}}^{\infty} x^{q-1-s}\mathbb{E} \bigl\vert X_{1}^{(1)} \bigr\vert ^{s}\,\mathrm{d}x \\ &{}+\sum_{n=1}^{\infty}n^{r-2-q/p+s/2} \ln^{s} n \int_{n^{1/p}}^{\infty} x^{q-1-s}\bigl(\mathbb{E} \bigl(X_{1}^{(1)}\bigr)^{2}\bigr)^{s/2}\, \mathrm{d}x \\ \mathop{\hat{=}}& \tilde{J}^{(1)}_{1}+ \tilde{J}^{(1)}_{2}. \end{aligned}$$
(3.8)
If \(u<2\), then by \(\mathbb{E} |X|^{u}<\infty\) and \(|X_{1}^{(1)}|\leq x^{\alpha}\),
$$\begin{aligned} \tilde{J}^{(1)}_{2} \leq&\sum_{n=1}^{\infty}n^{r-2-q/p+s/2}\ln^{s} n \int_{n^{1/p}}^{\infty} x^{q-1-s}\bigl(\mathbb{E} |X|^{u}\bigr)^{s/2}x^{\alpha(2-u)s/2}\,\mathrm{d}x \\ \ll&\sum_{n=1}^{\infty}n^{r-2-((\alpha u/p-1)/2+(1-\alpha)/p)s} \ln^{s} n< \infty \end{aligned}$$
(3.9)
from \(q-1-(1-\alpha+\alpha u/2)s<-1\) and \(r-2-((\alpha u/p-1)/2+(1-\alpha)/p)s<-1\).
If \(u\geq2\), then \(\mathbb{E}(X_{1}^{(1)})^{2}\leq\mathbb{E}X^{2}<\infty\). Hence,
$$ \tilde{J}^{(1)}_{2}\ll\sum_{n=1}^{\infty}n^{r-2-q/p+s/2}\ln^{s} n n^{(q-s)/p} =\sum _{n=1}^{\infty}n^{r-2-(1/p-1/2)s}\ln^{s} n< \infty $$
(3.10)
from \(s>q\) and \(r-2-(1/p-1/2)s<-1\).
For \(\tilde{J}^{(1)}_{1}\), by the \(C_{r}\) inequality and (2.1),
$$\begin{aligned} \tilde{J}^{(1)}_{1} \ll&\sum_{n=1}^{\infty}n^{r-1-q/p}\ln^{s} n \int_{n^{1/p}}^{\infty} x^{q-1-s} \bigl(x^{\alpha s}P \bigl(|X|>x^{\alpha}\bigr)+ \mathbb{E}|X|^{s}I\bigl(|X|\leq x^{\alpha}\bigr) \bigr)\,\mathrm{d}x \\ \leq&\sum_{n=1}^{\infty}n^{r-1-q/p} \ln^{s} n \int_{n^{1/p}}^{\infty} \bigl(x^{q-1-s+\alpha s-\alpha pr} \mathbb{E}|X|^{rp}+x^{q-1-s}\mathbb{E}|X|^{rp}x^{\alpha(s-rp)} \bigr)\,\mathrm{d}x \\ \ll& \sum_{n=1}^{\infty}n^{r-1-\alpha r-s(1-\alpha)/p} \ln^{s} n \\ < &\infty \end{aligned}$$
from \(q-1-\alpha rp-s(1-\alpha)<-1\) and \(r-1-\alpha r-s(1-\alpha)/p<-1\). By combining this with (3.3)-(3.5) and (3.7)-(3.10), we get that (3.2) holds. This ends the proof of (2.1) ⇒ (2.2).
Secondly we prove that (2.2) ⇒ (2.3). By (2.2) holds for any \(\varepsilon>0\), we get
$$\begin{aligned}& \sum_{n=1}^{\infty}n^{r-2}\mathbb{E} \Bigl\{ \sup_{k\geq n}k^{-1/p}|S_{k}|-\varepsilon \Bigr\} _{+}^{q} \\& \quad =\sum_{j=1}^{\infty}\sum _{n=2^{j-1}}^{2^{j}-1} n^{r-2} \int_{0}^{\infty}P \Bigl(\sup_{k\geq n}k^{-1/p}|S_{k}|> \varepsilon+x^{1/q} \Bigr) \,\mathrm{d}x \\& \quad \ll\sum_{j=1}^{\infty}2^{(r-1)j} \int_{0}^{\infty}P \Bigl(\sup_{k\geq 2^{j-1} }k^{-1/p}|S_{k}|> \varepsilon+x^{1/q} \Bigr) \,\mathrm{d}x \\& \quad \ll\sum_{j=1}^{\infty}2^{(r-1)j} \sum_{i=j}^{\infty}\int_{0}^{\infty}P \Bigl(\max_{2^{i-1}\leq k< 2^{i} }k^{-1/p}|S_{k}|> \varepsilon+x^{1/q} \Bigr) \,\mathrm{d}x \\& \quad \ll\sum_{i=1}^{\infty}2^{(r-1)i} \int_{0}^{\infty}P \Bigl(\max_{2^{i-1}\leq k< 2^{i}}|S_{k}|> \bigl(\varepsilon+x^{1/q}\bigr)2^{(i-1)/p} \Bigr) \,\mathrm{d}x \\& \quad \ll\sum_{i=1}^{\infty}2^{(r-1-q/p)i} \int_{0}^{\infty}P \Bigl(\max_{1\leq k< 2^{i}}|S_{k}|> \varepsilon2^{(i-1)/p}+x^{1/q} \Bigr) \,\mathrm{d}x \\& \quad \ll\sum_{n=1}^{\infty}n^{r-2-q/p} \int_{0}^{\infty}P \Bigl(\max_{1\leq k\leq n}|S_{k}|> \varepsilon2^{-1/p}n^{1/p}+x^{1/q} \Bigr) \,\mathrm{d}x \\& \quad =\sum_{n=1}^{\infty}n^{r-2-q/p} \mathbb{E} \Bigl(\max_{1\leq k\leq n}|S_{k}|-\bigl( \varepsilon2^{-1/p}\bigr)n^{1/p} \Bigr)^{q}_{+} \\& \quad < \infty. \end{aligned}$$
That is, (2.3) holds.
Finally, we prove that (2.3) ⇒ (2.1). By (2.3) and \(\max_{1\leq k\leq n}|X_{k}|\leq2\max_{1\leq k\leq n}|S_{k}|\), it follows that
$$\sum_{n=1}^{\infty}n^{r-2}\mathbb{E} \Bigl\{ \sup_{k\geq n}k^{-1/p}|X_{k}|-\varepsilon \Bigr\} _{+}^{q}< \infty \quad \mbox{for any } \varepsilon>0. $$
Therefore,
$$\begin{aligned}& \infty >\sum_{j=1}^{\infty}\sum _{n=2^{j-1}}^{2^{j}-1} n^{r-2}\mathbb{E} \Bigl(\sup _{k\geq 2^{j} }k^{-1/p}|X_{k}|-\varepsilon \Bigr)^{q}_{+} \\& \quad \gg\sum_{j=1}^{\infty}2^{(r-1)j} \mathbb{E} \Bigl(\max_{2^{j}\leq k< 2^{j+1}}2^{-(j+1)/p}|X_{k}|- \varepsilon \Bigr)^{q}_{+} \\& \quad \geq\sum_{j=1}^{\infty}2^{(r-1)j} \int_{0}^{\varepsilon^{q}}P \Bigl(\max_{2^{j}\leq k< 2^{j+1}}2^{-(j+1)/p}|X_{k}|> \varepsilon+x^{1/q} \Bigr) \,\mathrm{d}x \\& \quad \gg\sum_{j=1}^{\infty}2^{(r-1)j}P \Bigl(\max_{2^{j}\leq k< 2^{j+1}}|X_{k}|>2^{j/p}2^{1/p+1} \varepsilon \Bigr), \end{aligned}$$
(3.11)
it implies that \(P(\max_{2^{j}\leq k< 2^{j+1}}|X_{k}|>2^{j/p}x)\rightarrow0\), \(j\rightarrow\infty\) for any \(x>0\). Thus, by Lemma 3.3, for any \(x>0\), there is \(c>0\) such that for sufficiently large j
$$2^{j}P\bigl(|X|>2^{j/p}x\bigr)\leq c P \Bigl(\max _{2^{j}\leq k< 2^{j+1}}|X_{k}|>2^{j/p}x \Bigr). $$
Consequently, taking \(\varepsilon=2^{-1/p}\) in (3.11),
$$\begin{aligned} \infty >&\sum_{j=1}^{\infty}2^{(r-1)j} \int_{0}^{\infty}P \Bigl(\max_{2^{j}\leq k< 2^{j+1}}2^{-j/p}|X_{k}|-1>x^{1/q} \Bigr) \,\mathrm{d}x \\ \geq&\sum_{j=1}^{\infty}2^{rj} \int_{0}^{\infty}P \bigl(|X|>2^{j/p} \bigl(1+x^{1/q}\bigr) \bigr) \,\mathrm{d}x \\ =&\sum_{j=1}^{\infty}2^{(r-q/p)j} \int_{0}^{\infty}P \bigl(|X|>2^{j/p}+x^{1/q} \bigr) \,\mathrm{d}x \\ \geq&\sum_{j=1}^{\infty}2^{(r-q/p)j}\sum _{i=j}^{\infty}\int_{2^{iq/p}}^{2^{(i+1)q/p}} P \bigl(|X|>2^{j/p}+x^{1/q} \bigr) \,\mathrm{d}x \\ \gg&\sum_{j=1}^{\infty}2^{(r-q/p)j}\sum _{i=j}^{\infty}2^{iq/p} P \bigl(|X|>2^{i/p}\bigl(2^{1/p}+1\bigr) \bigr) \\ =&\sum_{i=1}^{\infty}\sum _{j=1}^{i}2^{(r-q/p)j} 2^{iq/p}P \bigl(|X|>c2^{i/p} \bigr) \\ \gg& \textstyle\begin{cases} \sum_{i=1}^{\infty}2^{ri}P (|X|>c2^{i/p} ),&\mbox{if }q< rp, \\ \sum_{i=1}^{\infty}2^{ri}\ln2^{i}P (|X|>c2^{i/p} ),&\mbox{if }q=rp, \\ \sum_{i=1}^{\infty}2^{iq/p}P (|X|>c2^{i/p} ),&\mbox{if }q>rp \end{cases}\displaystyle \\ \gg& \textstyle\begin{cases} \sum_{n=1}^{\infty}n^{r-1}P (|X|>cn^{1/p} )\gg\mathbb{E}|X|^{rp},&\mbox{if }q< rp, \\ \sum_{n=1}^{\infty}n^{r-1}\ln nP (|X|>cn^{1/p} )\gg\mathbb{E}|X|^{rp}\ln|X|,&\mbox{if }q=rp, \\ \sum_{n=1}^{\infty}n^{q/p-1}P (|X|>cn^{1/p} )\gg\mathbb{E}|X|^{q},&\mbox{if }q>rp. \end{cases}\displaystyle \end{aligned}$$
Hence, (2.1) holds. This completes the proof of Theorem 2.1. □