1 Introduction
The concept of complete convergence of a sequence of random variables was introduced by Hsu and Robbins [1] as follows. A sequence \(\{U_{n}, n\ge1\}\) of random variables converges completely to the constant θ if
Moreover, they proved that the sequence of arithmetic means of independent and identically distributed (i.i.d.) random variables converges completely to the expected value if the variance of the summands is finite. This result has been generalized and extended in several directions, one can refer to [2‐13] and so forth.
$$\sum_{n=1}^{\infty}P\bigl(|U_{n}-\theta|> \varepsilon \bigr)< \infty \quad\mbox{for all } \varepsilon > 0. $$
When \(\{X_{n},n\ge1\}\) is a sequence of i.i.d. random variables with mean zero, Chow [14] first investigated the complete moment convergence, which is more exact than complete convergence. He obtained the following result.
Anzeige
Theorem A
Let
\(\{X_{n},n\ge1\}\)
be a sequence of i.i.d. random variables with
\(EX_{1}=0\). For
\(1\le p <2\)
and
\(r>1\), if
\(E\{|X_{1}|^{rp}+|X_{1}|\log(1+|X_{1}|)\} <\infty\), then
where (as in the following) \(x_{+}=\max\{0,x\}\).
$$\sum_{n=1}^{\infty}n^{r-2-1/p}E \Biggl\{ \Biggl\vert \sum_{k=1}^{n} X_{k} \Biggr\vert -\varepsilon n^{1/p} \Biggr\} _{+} < \infty\quad\textit{for all } \varepsilon >0, $$
Theorem A has been generalized and extended in several directions. One can refer to Wang and Su [15] and Chen [16] for random elements taking values in a Banach space, Wang and Zhao [17] for NA random variables, Chen et al. [5], Li and Zhang [18] for moving-average processes based on NA random variables, Chen and Wang [19] for φ-mixing random variables, Qiu and Chen [20] for weighted sums of arrays of rowwise NA random variables.
The aim of this paper is to extend and improve Theorem A to negatively orthant dependent (NOD) random variables. The sufficient and necessary conditions are obtained. In fact, the paper is the continued work of Qiu et al. [11] in which the complete convergence is obtained for NOD sequence. It is worth to point that Sung [21] has discussed the complete moment convergence for NOD, but the main result in our paper is more exact and the method is completely different.
The concepts of negatively associated (NA) and negatively orthant dependent (NOD) were introduced by Joag-Dev and Proschan [22] in the following way.
Anzeige
Definition 1.1
A finite family of random variables \(\{X_{i}, 1\le i\le n\}\) is said to be negatively associated (NA) if for every pair of disjoint nonempty subset \(A_{1}\), \(A_{2}\) of \(\{1,2,\ldots,n\}\),
where \(f_{1}\) and \(f_{2}\) are coordinatewise nondecreasing such that the covariance exists. An infinite sequence of \(\{X_{n},n\ge1\} \) is NA if every finite subfamily is NA.
$$\mathrm{Cov} \bigl(f_{1}(X_{i},i\in A_{1}), f_{2}(X_{j},j\in A_{2}) \bigr)\le0, $$
Definition 1.2
A finite family of random variables \(\{X_{i}, 1\le i\le n\}\) is said to be
(a)
negatively upper orthant dependent (NUOD) if
\(\forall x_{1},x_{2},\ldots,x_{n}\in R\),
$$P(X_{i}>x_{i},i=1,2,\ldots,n)\le\prod _{i=1}^{n}P(X_{i}>x_{i}) $$
(b)
negatively lower orthant dependent (NLOD) if
\(\forall x_{1},x_{2},\ldots,x_{n}\in R\),
$$P(X_{i}\le x_{i},i=1,2,\ldots,n)\le\prod _{i=1}^{n}P(X_{i}\le x_{i}) $$
(c)
negatively orthant dependent (NOD) if they are both NUOD and NLOD.
A sequence of random variables \(\{X_{n},n\ge1\}\) is said to be NOD if for each n, \(X_{1},X_{2},\ldots,X_{n}\) are NOD.
Obviously, every sequence of independent random variables is NOD. Joag-Dev and Proschan [22] pointed out NA implies NOD, neither NUOD nor NLOD implies being NA. They gave an example which possesses NOD, but does not possess NA. So we can see that NOD is strictly wider than NA. For more convergence properties about NOD random variables, one can refer to [2, 11, 20, 23‐26], and so forth.
In order to prove our main result, we need the following lemmas.
Lemma 1.1
(Bozorgnia et al. [23])
Let
\(X_{1},X_{2},\ldots,X_{n}\)
be NOD random variables.
(i)
If
\(f_{1},f_{2},\ldots,f_{n}\)
are Borel functions all of which are monotone increasing (or all monotone decreasing), then
\(f_{1}(X_{1}),f_{2}(X_{2}),\ldots,f_{n}(X_{n})\)
are NOD random variables.
(ii)
\(E\prod_{i=1}^{n} (X_{i})_{+}\le\prod_{i=1}^{n} E(X_{i})_{+}\), \(\forall n\ge2\).
Lemma 1.2
(Asadian et al. [27])
For any
\(v\ge2\), there is a positive constant
\(C(v)\)
depending only on
v
such that if
\(\{X_{n}, n\ge1\}\)
is a sequence of NOD random variables with
\(EX_{n}=0\)
for every
\(n\ge1\), then for all
\(n\ge1\),
$$E\Biggl\vert \sum_{i=1}^{n} X_{i}\Biggr\vert ^{v}\le C(v) \Biggl\{ \sum _{i=1}^{n} E|X_{i}|^{v} + \Biggl( \sum_{i=1}^{n} EX_{i}^{2} \Biggr)^{v/2} \Biggr\} . $$
Lemma 1.3
For any
\(v\ge2\), there is a positive constant
\(C(v)\)
depending only on
v
such that if
\(\{X_{n}, n\ge1\}\)
is a sequence of NOD random variables with
\(EX_{n}=0\)
for every
\(n\ge1\), then for all
\(n\ge1\),
where
\(\log x=\max\{1, \ln x\}\), and lnx
denotes the natural logarithm of
x.
$$E\max_{1\le j\le n}\Biggl\vert \sum_{i=1}^{j} X_{i}\Biggr\vert ^{v} \le C(v) \bigl(\log(4n) \bigr)^{v} \Biggl\{ \sum_{i=1}^{n} E|X_{i}|^{v} + \Biggl(\sum_{i=1}^{n} EX_{i}^{2} \Biggr)^{v/2} \Biggr\} , $$
Lemma 1.4
(Kuczmaszewska [8])
Let
β
be positive constant, \(\{X_{n},n\ge1\}\)
be a sequence of random variables and
X
be a random variable. Suppose that
holds for some
\(D>0\), then there exists a constant
\(C>0\)
depending only on
D
and
β
such that
$$ \sum_{i=1}^{n} P\bigl(|X_{i}|>x\bigr)\le Dn P\bigl(|X|>x\bigr),\quad \forall x>0, \forall n\ge1, $$
(1.1)
(i)
If
\(E|X|^{\beta}<\infty\), then
\(\frac{1}{n}\sum_{j=1}^{n} E|X_{j}|^{\beta}\le CE|X|^{\beta}\);
(ii)
\(\frac{1}{n}\sum_{j=1}^{n} E|X_{j}|^{\beta}I(|X_{j}|\le x)\le C \{E|X|^{\beta}I(|X|\le x)+x^{\beta}P(|X|>x) \}\);
(iii)
\(\frac{1}{n}\sum_{j=1}^{n} E|X_{j}|^{\beta}I(|X_{j}|>x) \le C E|X|^{\beta}I(|X|>x) \).
Throughout this paper, C will represent positive constants; their value may change from one place to another.
2 Main results and proofs
Theorem 2.1
Let
\(\gamma>0\), \(\alpha>1/2\), \(p>0\), \(\alpha p>1\). Let
\(\{X_{n},n\ge1\}\)
be a sequence of NOD random variables and
X
be a random variables possibly defined on a different space satisfying the condition (1.1). Moreover, assume that
\(EX_{n}=0\)
for all
\(n\ge1\)
in the case
\(\alpha\leq1\). Suppose that
Then the following statements hold:
where
\(S_{n}=\sum_{i=1}^{n} X_{i}\), \(S_{n}^{(k)}=S_{n}-X_{k}\), \(k=1,2,\ldots,n\).
$$ \begin{cases} E|X|^{p}< \infty,& \gamma<p,\\ E|X|^{p}\log(1+|X|)<\infty,& \gamma=p,\\ E|X|^{\gamma}<\infty,& \gamma>p. \end{cases} $$
(2.1)
$$\begin{aligned}& \sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-2 } E \Bigl\{ \max_{1\le k \le n} |S_{k}|-\varepsilon n^{\alpha}\Bigr\} _{+}^{\gamma}< \infty, \quad\forall \varepsilon >0, \end{aligned}$$
(2.2)
$$\begin{aligned}& \sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-2 } E \Bigl\{ \max_{1\le k \le n} \bigl|S_{n}^{(k)}\bigr|-\varepsilon n^{\alpha}\Bigr\} _{+}^{\gamma}< \infty, \quad\forall \varepsilon >0, \end{aligned}$$
(2.3)
$$\begin{aligned}& \sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-2 } E \Bigl\{ \max_{1\le k \le n} |X_{k}|-\varepsilon n^{\alpha}\Bigr\} _{+}^{\gamma}< \infty,\quad \forall \varepsilon >0, \end{aligned}$$
(2.4)
$$\begin{aligned}& \sum_{n=1}^{\infty}n^{\alpha p-2 } E \Bigl\{ \sup_{ k \ge n}k^{-\alpha} |S_{k}|-\varepsilon \Bigr\} _{+}^{\gamma}< \infty,\quad \forall \varepsilon >0, \end{aligned}$$
(2.5)
$$\begin{aligned}& \sum_{n=1}^{\infty}n^{\alpha p-2 } E \Bigl\{ \sup_{ k \ge n}k^{-\alpha} |X_{k}|-\varepsilon \Bigr\} _{+}^{\gamma}< \infty, \quad\forall \varepsilon >0, \end{aligned}$$
(2.6)
Proof
Firstly, we prove (2.2). Note that for all \(\varepsilon>0\)
Hence by Theorem 2.1 of Qiu et al. [11], in order to prove (2.2), it is enough to show that
Choose q such that \(1/(\alpha p)< q<1\). \(\forall j\ge1\), \(t>0\), let
then \(X_{j}=\sum_{l=1}^{5} X_{j}^{(t, l)}\). Note that
Therefore to prove (2.2), it suffices to show that \(I_{l}<\infty\) for \(l=1, 2, 3, 4, 5\).
$$\begin{aligned}& \sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-2 } E \Bigl\{ \max_{1\le k \le n} |S_{k}|-\varepsilon n^{\alpha}\Bigr\} _{+}^{\gamma}\\& \quad= \sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-2 }\int _{0}^{\infty}P \Bigl(\max_{1\le k\le n}|S_{k}|- \varepsilon n^{\alpha}>t^{1/\gamma} \Bigr)\,dt \\& \quad= \sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-2 }\int _{0}^{n^{\gamma\alpha }} P \Bigl(\max_{1\le k\le n}|S_{k}|- \varepsilon n^{\alpha}>t^{1/\gamma } \Bigr)\,dt \\& \quad\quad{} + \sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-2 }\int _{n^{\gamma\alpha }}^{\infty}P \Bigl(\max_{1\le k\le n}|S_{k}|- \varepsilon n^{\alpha}>t^{1/\gamma} \Bigr)\,dt \\& \quad\le \sum_{n=1}^{\infty}n^{\alpha p-2}P \Bigl(\max_{1\le k\le n}|S_{k}|>\varepsilon n^{\alpha}\Bigr) \\& \quad\quad{}+\sum_{n=1}^{\infty}n^{\alpha(p-\gamma)-2}\int _{n^{\gamma\alpha }}^{\infty}P\Bigl(\max_{1\le k\le n}|S_{k}|>t^{1/\gamma} \Bigr)\,dt. \end{aligned}$$
$$\sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-2 }\int _{n^{\gamma\alpha }}^{\infty}P \Bigl(\max_{1\le k\le n}|S_{k}|>t^{1/\gamma} \Bigr)\,dt< \infty. $$
$$\begin{aligned}& X_{j}^{(t,1)}=-t^{q/\gamma}I\bigl(X_{j}< -t^{q/\gamma} \bigr)+X_{j} I\bigl(|X_{j}|\le t^{q/\gamma} \bigr)+t^{q/\gamma}I\bigl(X_{j}>t^{q/\gamma}\bigr), \\& X_{j}^{(t,2)}=\bigl(X_{j}-t^{q/\gamma}\bigr)I \bigl(t^{q/\gamma}<X_{j}\le t^{q/\gamma }+t^{1/\gamma} \bigr)+t^{1/\gamma}I\bigl(X_{j}>t^{q/\gamma}+t^{1/\gamma} \bigr), \\& X_{j}^{(t,3)}=\bigl(X_{j}-t^{q/\gamma}-t^{1/\gamma} \bigr)I\bigl(X_{j}>t^{q/\gamma }+t^{1/\gamma}\bigr), \\& X_{j}^{(t,4)}=\bigl(X_{j}+t^{q/\gamma}\bigr)I \bigl(-t^{q/\gamma}-t^{1\gamma}\le X_{j}<-t^{q/\gamma} \bigr)-t^{1/\gamma}I\bigl(X_{j}<-t^{q/\gamma}-t^{1\gamma} \bigr), \\& X_{j}^{(t,5)}=\bigl(X_{j}+t^{q/\gamma}+t^{1/\gamma} \bigr)I\bigl(X_{j}<-t^{q/\gamma }-t^{1/\gamma}\bigr), \end{aligned}$$
$$\begin{aligned}& \sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-2 } \int_{n^{\gamma \alpha}}^{\infty}P \Bigl(\max_{1\le k\le n}|S_{k}|>t^{1/\gamma} \Bigr)\,dt \\& \quad\le \sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-2 }\int _{n^{\gamma\alpha }}^{\infty}P \Biggl(\max_{1\le k\le n} \Biggl\vert \sum_{j=1}^{k}X_{j}^{(t,1)} \Biggr\vert >t^{1/\gamma}/5 \Biggr)\,dt \\& \quad\quad{} +\sum_{l=2}^{3} \sum _{n=1}^{\infty}n^{\alpha( p-\gamma)-2 }\int_{n^{\gamma\alpha}}^{\infty}P \Biggl( \sum_{j=1}^{n}X_{j}^{(t,l)}>t^{1/\gamma}/5 \Biggr)\,dt \\& \quad\quad{} +\sum_{l=4}^{5} \sum _{n=1}^{\infty}n^{\alpha( p-\gamma)-2 }\int_{n^{\gamma\alpha}}^{\infty}P \Biggl(-\sum_{j=1}^{n}X_{j}^{(t,l)}>t^{1/\gamma}/5 \Biggr)\,dt \\& \quad\stackrel{\mathrm{def}}{=} \sum_{l=1}^{5} I_{l}. \end{aligned}$$
For \(I_{1}\), we first prove that
When \(\alpha\le1\). Since \(\alpha p>1\) implies \(p>1\), by Lemma 1.4 and \(EX_{j}=0\), \(j\ge1\), we have
When \(\alpha>1\) and \(p\ge1\).
When \(\alpha>1\) and \(p< 1\),
Therefore (2.7) holds. By (2.7), in order to prove \(I_{1}<\infty\), it is enough to show that
Fix any \(v\ge2\) and \(v>\max\{p/(1-q), \gamma/(1-q), 2\gamma /[2-(2-p)q], 2(\alpha p-1)/[2\alpha(1-q)+(\alpha p q-1)], (\alpha p-1)/(\alpha-1/2)\}\), by Markov’s inequality, Lemma 1.1, Lemma 1.3, and \(C_{r}\)-inequality, we have
Note that
If \(\max\{p,\gamma\}< 2\), by Lemma 1.4, we have
If \(\max\{p,\gamma\}\ge 2\), note that \(E|X|^{2}<\infty\), by Lemma 1.4, we have
Therefore, \(I_{1}^{*}<\infty\), so \(I_{1}<\infty\).
$$ \sup_{t\ge n^{\gamma\alpha}}t^{-1/\gamma}\max_{1\le k\le n}\Biggl\vert E\sum_{j=1}^{k}X_{j}^{(t,1)} \Biggr\vert \to0,\quad n\to\infty. $$
(2.7)
$$\begin{aligned}& \sup_{t\ge n^{\gamma\alpha}}t^{-1/\gamma}\max_{1\le k\le n} \Biggl\vert E\sum_{j=1}^{k}X_{j}^{(t,1)} \Biggr\vert \\& \quad\le\sup_{t\ge n^{\gamma\alpha}}t^{-1/\gamma}\sum _{j=1}^{n}E \bigl\{ |X_{j}|I \bigl(|X_{j}|>t^{q/\gamma}\bigr)+t^{q/\gamma}I \bigl(|X_{j}|>t^{q/\gamma}\bigr) \bigr\} \\& \quad\le2\sup_{t\ge n^{\gamma\alpha}}t^{-1/\gamma}\sum _{j=1}^{n}E|X_{j}|I\bigl(|X_{j}|>t^{q/\gamma} \bigr) \\& \quad\le Cn\sup_{t\ge n^{\gamma\alpha}}t^{-1/\gamma}E|X|I\bigl(|X|> t^{q/\gamma}\bigr) \\& \quad\le Cn^{1-\alpha}E|X| I\bigl(|X|> n^{\alpha q}\bigr) \\& \quad \le Cn^{1-\alpha p q -\alpha(1-q)}E|X|^{p}\to0,\quad n\to\infty. \end{aligned}$$
$$\begin{aligned}& \sup_{t\ge n^{\gamma\alpha}}t^{-1/\gamma}\max_{1\le k\le n} \Biggl\vert E\sum_{j=1}^{k}X_{j}^{(t,1)} \Biggr\vert \\& \quad\le \sup_{t\ge n^{\gamma\alpha}}t^{-1/\gamma}\sum _{j=1}^{n} E|X_{j}| \le Cn \sup _{t\ge n^{\gamma\alpha}}t^{-1/\gamma}E|X| \le Cn^{1-\alpha }\to0,\quad n\to \infty. \end{aligned}$$
$$\begin{aligned}& \sup_{t\ge n^{\gamma\alpha}}t^{-1/\gamma}\max_{1\le k\le n} \Biggl\vert E\sum_{j=1}^{k}X_{j}^{(t,1)} \Biggr\vert \\& \quad\le \sup_{t\ge n^{\gamma\alpha}}t^{-1/\gamma}\sum _{j=1}^{n}E \bigl\{ |X_{j}|I \bigl(|X_{j}|\le t^{q/\gamma}\bigr)+t^{q/\gamma}I \bigl(|X_{j}|>t^{q/\gamma}\bigr) \bigr\} \\& \quad\le \sup_{t\ge n^{\gamma\alpha}}t^{-1/\gamma}\sum _{j=1}^{n} t^{q(1-p)/\gamma} E|X_{j}|^{p} \le Cn \sup_{t\ge n^{\gamma\alpha}}t^{\{ q(1-p)-1\}/\gamma} E|X|^{p} \\& \quad\le Cn ^{1-\alpha p q-(1-q)\alpha}\to0,\quad n\to\infty. \end{aligned}$$
$$I_{1}^{*}:=\sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-2 }\int_{n^{\gamma\alpha }}^{\infty}P \Biggl(\max _{1\le k\le n}\Biggl\vert \sum_{j=1}^{k} \bigl(X_{j}^{(t,1)}-EX_{j}^{(t,1)} \bigr)\Biggr\vert >t^{1/\gamma}/10 \Biggr)\,dt< \infty. $$
$$\begin{aligned} I_{1}^{*} \le& C\sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-2 } \\ &{}\times\int_{n^{\gamma\alpha}}^{\infty}t^{-v/\gamma} \bigl(\log(4n)\bigr)^{v} \Biggl\{ \sum_{j=1}^{n}E \bigl\vert X_{j}^{(t,1)}\bigr\vert ^{v}+ \Biggl( \sum_{j=1}^{n}E \bigl(X_{j}^{(t,1)} \bigr)^{2} \Biggr)^{v/2} \Biggr\} \,dt \\ \le & C\sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-2 } \bigl(\log(4n)\bigr)^{v}\\ &{}\times\int_{n^{\gamma\alpha}}^{\infty}t^{-v/\gamma} \sum_{j=1}^{n} \bigl\{ E|X_{j}|^{v}I\bigl(|X_{j}|\le t^{\frac{q}{\gamma}} \bigr)+t^{qv/\gamma} P\bigl(|X_{j}|>t^{\frac {q}{\gamma}}\bigr) \bigr\} \,dt \\ &{} + C\sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-2 }\bigl( \log(4n)\bigr)^{v} \\ &{}\times\int_{n^{\gamma\alpha}}^{\infty}t^{\frac{-v}{\gamma}} \Biggl\{ \sum_{j=1}^{n} \bigl(EX_{j}^{2}I\bigl(|X_{j}|\le t^{\frac{q}{\gamma}} \bigr)+t^{\frac {2q}{\gamma}} P\bigl(|X_{j}|>t^{\frac{q}{\gamma}}\bigr) \bigr) \Biggr\} ^{\frac {v}{2}}\,dt \\ \stackrel{\mathrm{def}}{=} &I_{11}+I_{12}. \end{aligned}$$
$$\begin{aligned} I_{11} \le & C\sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-1 }\bigl(\log (4n)\bigr)^{v}\int_{n^{\gamma\alpha}}^{\infty}t^{-(1-q)v/\gamma} \,dt \\ \le & C\sum_{n=1}^{\infty}n^{\alpha p-\alpha(1-q)v-1 } \bigl(\log(4n)\bigr)^{v} < \infty. \end{aligned}$$
$$\begin{aligned} I_{12} \le & C\sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-2 }\bigl(\log(4n)\bigr)^{v} \int_{n^{\gamma\alpha}}^{\infty}t^{\frac{-v}{\gamma}} \Biggl\{ t^{(2-p)q/\gamma}\sum_{j=1}^{n}E|X_{j}|^{p} \Biggr\} ^{\frac{v}{2}}\,dt \\ \le & C\sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-2+v/2 } \bigl(\log(4n)\bigr)^{v} \int_{n^{\gamma\alpha}}^{\infty}t^{\frac{-[2-(2-p)q]v}{2\gamma}} \bigl(E|X|^{p} \bigr)^{\frac{v}{2}}\,dt \\ \le & C\sum_{n=1}^{\infty}n^{\alpha p-2-[\alpha(1-q)+(\alpha p q-1)/2]v } \bigl(\log(4n)\bigr)^{v} < \infty. \end{aligned}$$
$$\begin{aligned} I_{12} \le & C\sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-2+v/2 }\bigl(\log (4n)\bigr)^{v}\int_{n^{\gamma\alpha}}^{\infty}t^{-v/\gamma} \,dt \\ \le & C\sum_{n=1}^{\infty}n^{\alpha p-2-(\alpha-1/2)v } \bigl(\log (4n)\bigr)^{v}< \infty. \end{aligned}$$
For \(I_{2}\), we first prove
When \(p>1\), we have by Lemma 1.4 that
When \(0< p\le1\), we have by Lemma 1.4
Therefore (2.8) holds. By (2.8), in order to prove \(I_{2}<\infty\), it is enough to show that
Fix any \(v\ge2\) (to be specified later), by Markov’s inequality, Lemma 1.1, Lemma 1.2, \(C_{r}\)-inequality, Jensen’s inequality, and Lemma 1.4, we have
We get by the mean-value theorem and a standard computation
When \(\max\{p,\gamma\}< 2\), let \(v=2\). We have \(I_{24}=I_{22}<\infty\) and
When \(\max\{p,\gamma\}\ge 2\), let \(v>\max\{\gamma, (\alpha p -1)/(\alpha-1/2)\}\). Note that \(E|X|^{2}<\infty\)
and by the Markov inequality, we have
The proof of \(I_{21}<\infty\) is similar to that (2.9), so it is omitted. Therefore, \(I_{2}^{*}<\infty\), so \(I_{2}<\infty\).
$$ \sup_{t\ge n^{\gamma\alpha}} \Biggl\{ t^{-1/\gamma}\sum _{j=1}^{n}EX_{j}^{(t,2)} \Biggr\} \to0,\quad n \to\infty. $$
(2.8)
$$\begin{aligned}& \sup_{t\ge n^{\gamma\alpha}} \Biggl\{ t^{-1/\gamma}\sum _{j=1}^{n}EX_{j}^{(t,2)} \Biggr\} \\& \quad\le \sup_{t\ge n^{\gamma\alpha}}t^{-1/\gamma}\sum _{j=1}^{n} \bigl\{ EX_{j}I \bigl(X_{j}>t^{q/\gamma}\bigr)+t^{1/\gamma}P \bigl(X_{j}>t^{q/\gamma}+t^{1/\gamma }\bigr) \bigr\} \\& \quad\le \sup_{t\ge n^{\gamma\alpha}}t^{-1/\gamma}\sum _{j=1}^{n} \bigl\{ EX_{j}I \bigl(X_{j}>t^{q/\gamma}\bigr)+EX_{j}I \bigl(X_{j}>t^{q/\gamma}+t^{1/\gamma}\bigr) \bigr\} \\& \quad\le Cn\sup_{t\ge n^{\gamma\alpha}}t^{-1/\gamma }E|X|I\bigl(|X|>t^{q/\gamma} \bigr) \le Cn^{1-\alpha}E |X| I\bigl(|X|>n^{q\alpha}\bigr) \\& \quad\le Cn^{1-q \alpha p-\alpha(1-q)} E|X|^{p} \to0,\quad n\to\infty. \end{aligned}$$
$$\begin{aligned}& \sup_{t\ge n^{\gamma\alpha}} \Biggl\{ t^{-1/\gamma}\sum _{j=1}^{n}EX_{j}^{(t,2)} \Biggr\} \\& \quad\le \sup_{t\ge n^{\gamma\alpha}}t^{-1/\gamma}\sum _{j=1}^{n} \bigl\{ E|X_{j}|I \bigl(|X_{j}|\le2 t^{1/\gamma}\bigr)+t^{1/\gamma}P \bigl(|X_{j}|>2t^{q/\gamma }\bigr) \bigr\} \\& \quad\le Cn\sup_{t\ge n^{\gamma\alpha}}t^{-1/\gamma} \bigl\{ E|X|I\bigl(|X|\le 2 t^{1/\gamma}\bigr)+2 t^{1/\gamma} P\bigl(|X|> 2 t^{1/\gamma}\bigr)+ t^{1/\gamma }P\bigl(|X|>2t^{q/\gamma}\bigr) \bigr\} \\& \quad\le Cn\sup_{t\ge n^{\gamma\alpha}} \bigl\{ t^{-p/\gamma }E|X|^{p}+t^{-pq/\gamma} E|X|^{p} \bigr\} \\& \quad\le Cn^{1-\alpha p q} \to0,\quad n\to\infty. \end{aligned}$$
$$I_{2}^{*} := \sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-2 }\int_{n^{\gamma \alpha}}^{\infty}P \Biggl( \sum _{j=1}^{n} \bigl(X_{j}^{(t,2)}-EX_{j}^{(t,2)} \bigr)>t^{1/\gamma}/10 \Biggr)\,dt < \infty. $$
$$\begin{aligned} I_{2}^{*} \le & C\sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-2 }\int_{n^{\gamma\alpha}}^{\infty}t^{-v/\gamma} \Biggl\{ \sum_{j=1}^{n}E\bigl\vert X_{j}^{(t,2)}\bigr\vert ^{v}+ \Biggl(\sum _{j=1}^{n}E \bigl(X_{j}^{(t,2)} \bigr)^{2} \Biggr)^{v/2}\,dt \Biggr\} \\ \le & C\sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-2 }\int _{n^{\gamma \alpha}}^{\infty}t^{-v/\gamma} \sum _{j=1}^{n} \bigl\{ E|X_{j}|^{v}I \bigl(|X_{j}|\le 2t^{1/\gamma}\bigr)+t^{v/\gamma} P \bigl(X_{j}>t^{1/\gamma}\bigr) \bigr\} \,dt \\ &{} + C\sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-2 }\int _{n^{\gamma\alpha }}^{\infty}t^{-v/\gamma} \Biggl\{ \sum _{j=1}^{n}E \bigl(X_{j}^{2}I \bigl(|X_{j}|\le 2t^{1/\gamma}\bigr)+t^{2/\gamma} P \bigl(X_{j}>t^{1/\gamma}\bigr) \bigr) \Biggr\} ^{v/2}\,dt \\ \le & C\sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-1 }\int _{n^{\gamma \alpha}}^{\infty}t^{-v/\gamma} E|X|^{v}I \bigl(|X|\le2t^{1/\gamma}\bigr)\,dt \\ &{} +C\sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-1 }\int _{n^{\gamma\alpha }}^{\infty}P\bigl(|X|>t^{1/\gamma}\bigr)\,dt \\ &{} + C\sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-2+v/2 }\int _{n^{\gamma \alpha}}^{\infty}t^{-v/\gamma} \bigl\{ E|X|^{2}I\bigl(|X|\le2t^{1/\gamma }\bigr) \bigr\} ^{v/2}\,dt \\ &{} +C\sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-2+v/2 }\int _{n^{\gamma \alpha}}^{\infty}\bigl( P\bigl(|X|>t^{1/\gamma}\bigr) \bigr)^{v/2}\,dt \\ \stackrel{\mathrm{def}}{=} &I_{21}+I_{22}+I_{23}+I_{24}. \end{aligned}$$
$$\begin{aligned} I_{22} = & C\sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-1 }\sum_{j=n}^{\infty}\int _{j^{\gamma\alpha}}^{(j+1)^{\gamma\alpha}} P\bigl(|X|>t^{1/\gamma}\bigr)\,dt \\ \le & C\sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-1 }\sum _{j=n}^{\infty}j^{\gamma\alpha-1} P \bigl(|X|>j^{\alpha}\bigr) \\ = & C\sum_{j=1}^{\infty}j^{\gamma\alpha-1} P \bigl(|X|>j^{\alpha}\bigr) \sum_{n=1}^{j} n^{\alpha( p-\gamma)-1 } \\ \le & \begin{cases} C\sum_{j=1}^{\infty}j^{\alpha p -1} P(|X|>j^{\alpha}), & \gamma< p,\\ C\sum_{j=1}^{\infty}j^{\alpha p-1} \log j P(|X|>j^{\alpha}) , & \gamma =p,\\ \sum_{j=1}^{\infty}j^{\gamma\alpha-1} P(|X|>j^{\alpha}) , & \gamma>p \end{cases} \\ \le & \begin{cases} CE|X|^{p}, & \gamma<p,\\ CE|X|^{p}\log(1+|X|) , & \gamma=p,\\ CE|X|^{\gamma},& \gamma>p \end{cases} \\ < & \infty. \end{aligned}$$
$$\begin{aligned} I_{21} =& I_{23} = C\sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-1 }\sum_{j=n}^{\infty}\int _{j^{\gamma\alpha}}^{(j+1)^{\gamma\alpha}} t^{-2/\gamma} E|X|^{2}I \bigl(|X|\le2t^{1/\gamma}\bigr)\,dt \\ \le & C\sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-1 } \sum_{j=n}^{\infty}j^{(-2+\gamma)\alpha-1} E|X|^{2}I\bigl(|X|\le2(j+1)^{\alpha}\bigr) \\ = & C\sum_{j=1}^{\infty}j^{(-2+\gamma)\alpha-1} E|X|^{2}I\bigl(|X|\le 2(j+1)^{\alpha}\bigr) \sum _{n=1}^{j} n^{\alpha( p-\gamma)-1 } \\ \le & \begin{cases} C\sum_{j=1}^{\infty}j^{\alpha(p-2)-1} E|X|^{2}I(|X|\le2(j+1)^{\alpha}), & \gamma< p,\\ C\sum_{j=1}^{\infty}j^{\alpha(p-2)-1} \log j E|X|^{2}I(|X|\le 2(j+1)^{\alpha}) , & \gamma=p,\\ C\sum_{j=1}^{\infty}j^{\alpha(\gamma-2) -1} E|X|^{2}I(|X|\le 2(j+1)^{\alpha}) , & \gamma>p \end{cases} \\ \le & \begin{cases} CE|X|^{p}, & \gamma<p,\\ CE|X|^{p}\log(1+|X|) , & \gamma=p,\\ CE|X|^{\gamma},& \gamma>p \end{cases} \\ < & \infty. \end{aligned}$$
(2.9)
$$I_{23} \le C\sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-2+v/2 }\int_{n^{\gamma\alpha}}^{\infty}t^{-v/\gamma}\, dt = C\sum_{n=1}^{\infty}n^{\alpha p-2-(\alpha-1/2)v}< \infty, $$
$$I_{24} \le C\sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-2+v/2 }\int_{n^{\gamma\alpha}}^{\infty}t^{-v/\gamma} \,dt< \infty. $$
For \(I_{3}\), we get
By similar proofs to \(I_{2}<\infty\) and \(I_{3}<\infty\), we have \(I_{4}<\infty \) and \(I_{5}<\infty\), respectively. Therefore, (2.2) holds.
$$\begin{aligned} I_{3} \le& \sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-2 }\int_{n^{\gamma \alpha}}^{\infty}P \Biggl\{ \bigcup_{j=1}^{n}\bigl(X_{j}^{(t,3)}>0\bigr) \Biggr\} \,dt \\ = & \sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-2 }\int _{n^{\gamma\alpha }}^{\infty}\sum_{j=1}^{n} P \bigl( X_{j}>t^{1/\gamma}+t^{q/\gamma} \bigr)\,dt \\ \le & C \sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-1 } \int_{n^{\gamma \alpha}}^{\infty}P\bigl(|X|>t^{1/\gamma}\bigr)\,dt =CI_{22}< \infty. \end{aligned}$$
Equation (2.2) ⇒ (2.3). Note that \(|S_{n}^{(k)}|=|S_{n}-X_{k}|\le |S_{n}|+|X_{k}|=|S_{n}|+|S_{k}-S_{k-1}|\le|S_{n}|+|S_{k}|+|S_{k-1}|\le 3\max_{1\le j\le n}|S_{j}|\), \(\forall 1\le k\le n\), hence
Equation (2.3) holds.
$$\begin{aligned}& \sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-2 } E \Bigl\{ \max_{1\le k \le n} \bigl|S_{n}^{(k)}\bigr|-\varepsilon n^{\alpha}\Bigr\} _{+}^{\gamma} \\& \quad= \sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-2 }\int _{0}^{\infty}P \Bigl(\max_{1\le k\le n}\bigl|S_{n}^{(k)}\bigr|- \varepsilon n^{\alpha}>t^{1/\gamma} \Bigr)\,dt \\& \quad\le \sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-2 }\int _{0}^{\infty}P \Bigl(\max_{1\le k\le n}|S_{n}|> \varepsilon n^{\alpha}/3+t^{1/\gamma}/3 \Bigr)\,dt \\& \quad= 3^{\gamma}\sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-2 }\int_{0}^{\infty}P \Bigl(\max _{1\le k\le n}|S_{n}|>\varepsilon n^{\alpha}/3+t^{1/\gamma} \Bigr)\,dt \\& \quad= 3^{\gamma}\sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-2 }E \Bigl\{ \max_{1\le k\le n}|S_{n}|- \varepsilon n^{\alpha}/3 \Bigr\} ^{\gamma}< \infty. \end{aligned}$$
(2.10)
Equation (2.3) ⇒ (2.4). Since \(\frac{1}{2} \vert S_{n} \vert \le\frac{n-1}{n}\vert S_{n} \vert =\vert \frac{1}{n}\sum_{k=1}^{n} S_{n}^{(k)} \vert \le\max_{1\le k\le n} \vert S_{n}^{(k)} \vert \), \(\forall n\ge2\), and \(|X_{k}|=\vert S_{n}-S_{n}^{(k)}\vert \le|S_{n}|+|S_{n}^{(k)}|\le3\max_{1\le k\le n}|S_{n}^{(k)}|\), we have (2.4) by a similar argument to (2.10).
Theorem 2.2
Let
\(\gamma>0\), \(\alpha>1/2\), \(p>0\), \(\alpha p>1\). Let
\(\{X_{n},n\ge1\}\)
be a sequence of NOD random variables and
X
be a random variables possibly defined on a different space. Moreover, assume that
\(EX_{n}=0\)
for all
\(n\ge1\)
when
\(\alpha\le1\). If there exist constants
\(D_{1}>0\)
and
\(D_{2}>0\)
such that
Then (2.1)-(2.6) are equivalent.
$$\frac{D_{1}}{n} \sum_{i=n}^{2n-1} P\bigl(|X_{i}|>x\bigr) \le P\bigl(|X|>x\bigr)\le\frac {D_{2}}{n} \sum _{i=n}^{2n-1} P\bigl(|X_{i}|>x\bigr),\quad \forall x>0, n \ge1. $$
Proof
By Theorem 2.1, in order to prove Theorem 2.2, it is enough to show that (2.4) ⇒ (2.1) and (2.6) ⇒ (2.1). We only prove (2.4) ⇒ (2.1), the proof of (2.6) ⇒ (2.1) is similar and omitted. Note that
by Theorem 2.2 of Qiu et al. [11], the proof of (2.4) ⇒ (2.1) is completed. □
$$\begin{aligned} \infty > & \sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-2 } E \Bigl\{ \max_{1\le k \le n} |X_{k}|-\varepsilon n^{\alpha}\Bigr\} _{+}^{\gamma}\\ = & \sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-2 } \int _{0} ^{\infty}P \Bigl(\max_{1\le k \le n} |X_{k}|-\varepsilon n^{\alpha}>t^{1/\gamma}\Bigr)\,dt \\ \ge & \sum_{n=1}^{\infty}n^{\alpha( p-\gamma)-2 } \int _{0} ^{\varepsilon ^{\gamma}n^{\alpha\gamma}} P \Bigl(\max_{1\le k \le n} |X_{k}|>\varepsilon n^{\alpha}+t^{1/\gamma}\Bigr)\,dt \\ \ge & \gamma^{\alpha}\sum_{n=1}^{\infty}n^{\alpha p-2 } P \Bigl(\max_{1\le k \le n} |X_{k}|>2 \varepsilon n^{\alpha}\Bigr) \end{aligned}$$
Acknowledgements
The authors would like to thank the referees and the editors for the helpful comments and suggestions. The work of Qiu was supported by the National Natural Science Foundation of China (Grant No. 61300204), the work of Liu was supported by the National Natural Science Foundation of China (Grant No. 71471075), the work of Chen was supported by the National Natural Science Foundation of China (Grant No. 11271161).
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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.