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Some limit theorems for weighted negative quadrant dependent random variables with infinite mean
- Fuqiang Ma1Email authorView ORCID ID profile,
- Jianmin Li1 and
- Tiantian Hou1
Journal of Inequalities and Applications20182018:62
https://doi.org/10.1186/s13660-018-1655-5
© The Author(s) 2018
Received: 19 October 2017
Accepted: 7 March 2018
Published: 20 March 2018
Abstract
In the present paper, we will investigate weak laws of large numbers for weighted pairwise NQD random variables with infinite mean. The almost sure upper and lower bounds for a particular normalized weighted sum of pairwise NQD nonnegative random variables are established also.
Keywords
Strong lawWeak law of large numbersPairwise negative quadrant dependent sequence
MSC
60F0560F15
1 Introduction
With Markov’s truncation method, Kolmogorov got a weak law of large numbers for independent identically random variables with a necessary and sufficient condition, which is called the Kolmogorov–Feller weak law of large numbers.
Theorem 1.1
([1])
Let
\(\{X_{i}, i\ge 1\}\)
be a sequence of i.i.d. random variables with partial sums
\(S_{n}=X_{1}+\cdots +X_{n}\). Then
if and only if
$$ \frac{S_{n}-n\mathbb {E}X_{1}1\{\vert X_{1} \vert \le n\}}{n} \xrightarrow{\mathbb {P}} 0,\quad \textit{as } n\to \infty, $$
$$ x\mathbb {P}\bigl(\vert X_{1} \vert >x\bigr)\to 0, \quad \textit{as } x\to \infty . $$
(1.1)
The theorem states the condition of the mean’s existence is not necessary, and St. Petersburg game (see [2]) and Feller game (see [3]), which are well known as the typical examples, are formulated by a nonnegative random variable X with the tail probability
for each fixed \(0 <\alpha \le 1\), where \(a_{n}\asymp b_{n}\) denotes
Nakata [4] considered truncated random variables and studied strong laws of large numbers and central limit theorems in this situation. In [5], Nakata studied the weak laws of large numbers for weighted independent random variables with the tail probability (1.2) and explored the case that the decay order of the tail probability is −1. In this paper, we shall consider the more general random sequence with infinite mean.
$$ \mathbb {P}(X > x)\asymp x^{-\alpha } $$
(1.2)
$$ 0< \liminf_{n\to \infty }\frac{a_{n}}{b_{n}}\le \limsup _{n\to \infty }\frac{a _{n}}{b_{n}}< \infty . $$
Let us recall the concept of negative quadrant dependent (NQD) random variables, which was introduced by Lehmann [6].
Definition 1.1
Two random variables X and Y are said to be NQD if for any \(x,y\in \mathbb {R}\)
A sequence of random variables \(\{X_{n}, n\ge 1\}\) is said to be pairwise NQD if, for all \(i,j\in N\), \(i\ne j\), \(X_{i}\) and \(X_{j}\) are NQD.
$$ \mathbb {P}(X\le x, Y\le y)\le \mathbb {P}(X\le x)\mathbb {P}(Y\le y). $$
Because pairwise NQD includes the independent, NA (negatively associated), NOD (negatively orthant dependent) and LNQD (linearly negative quadrant dependent), it is a more general dependence structure. It is necessary to study its probabilistic properties.
Definition 1.2
A finite sequence \(\{X_{1}, \ldots , X_{n}\}\) of random variables is said to be NA if for any disjoint subsets A, B of \(\{1, \ldots , n \}\) and any real coordinatewise nondecreasing functions f on \(\mathbb {R}^{\vert A \vert }\) and g on \(\mathbb {R}^{\vert B \vert }\),
whenever the covariance exists, where \(\vert A \vert \) denotes the cardinality of A. An infinite family of random variables is NA if every finite subfamily is NA.
$$ \operatorname{Cov}\bigl(f(X_{k}, k\in A), g(X_{k}, k\in B) \bigr) \le 0 $$
(1.3)
The concept of a NA sequence was introduced by Joag-Dev and Proschan [7], and it is easy to see that a sequence of NA random variables is a pairwise NQD sequence.
In the present paper, we suppose that all random variables satisfy the condition
In Sect. 2, we will investigate weak laws of large numbers for weighted pairwise NQD random variables with the common distribution (1.4). The almost sure upper and lower bounds for a particular normalized weighted sum of pairwise NQD nonnegative random variables will be established in Sect. 3. Throughout this paper, the symbol C represents positive constants whose values may change from one place to another.
$$ \mathbb {P}\bigl(\vert X \vert >x\bigr)\asymp x^{-\alpha } \quad \text{for a fixed } 0< \alpha \le 1. $$
(1.4)
2 Weak law of large numbers
In this section, we extend the corresponding results in Nakata [5] from the case of i.i.d. random variables to pairwise NQD random variables.
2.1 Main results
We state our weak law of large numbers for different weighted sums of pairwise NQD random variables.
Theorem 2.1
Let
\(\{X_{i}, i\ge 1\}\)
be a sequence of pairwise NQD random variables whose distributions satisfy
and
If there exist two positive sequences
\(\{a_{j}\}\)
and
\(\{b_{j}\}\)
satisfying
then it follows that
In particular, if there exists a constant
A
such that
then we have
$$ \mathbb {P}\bigl(\vert X_{j} \vert >x\bigr)\asymp x^{-\alpha } \quad \textit{for } j\ge 1 $$
$$ \limsup_{x\to \infty } \sup_{j\ge 1}x^{\alpha }\mathbb {P}\bigl(\vert X_{j} \vert >x\bigr)< \infty . $$
$$ \sum_{j=1}^{n} a_{j}^{\alpha }=o\bigl(b_{n}^{\alpha }\bigr), $$
(2.1)
$$ \lim_{n\to \infty }\frac{1}{b_{n}}\sum _{j=1}^{n} a_{j} \biggl( X_{j}- \mathbb {E}X _{j}1 \biggl\{ \vert X_{j} \vert \le \frac{b_{n}}{a_{j}} \biggr\} \biggr) =0 \quad \textit{in probability}. $$
(2.2)
$$ \lim_{n\to \infty }\frac{1}{b_{n}}\sum_{j=1}^{n} a_{j}\mathbb {E}X_{j}1 \biggl\{ \vert X_{j} \vert \le \frac{b_{n}}{a_{j}} \biggr\} =A, $$
$$ \lim_{n\to \infty }\frac{1}{b_{n}}\sum_{j=1}^{n} a_{j} X_{j}=A \quad \textit{in probability}. $$
From Theorem 2.1 and by using the same methods as in Nakata [5] to calculate the constant A, we can obtain the following four corollaries for the pairwise NQD random variables.
Corollary 2.1
Under the assumptions of Theorem
2.1, if
\(0<\alpha <1\), then we have
$$ \lim_{n\to \infty }\frac{1}{b_{n}}\sum_{j=1}^{n} a_{j} X_{j}=0 \quad \textit{in probability}. $$
Corollary 2.2
Let
\(\{X_{i}, i\ge 1\}\)
be a sequence of nonnegative pairwise NQD random variables whose distributions satisfy
If
\(q_{j}\ge 1\)
for any positive integer
j,and
Then we have
$$ \mathbb {P}\bigl(\vert X_{j} \vert >x\bigr)= (x+q_{j})^{-1} \quad \textit{for } x\ge 0. $$
$$ Q_{n}:=\sum_{j=1}^{n}q_{j}^{-1} \to \infty , \quad \textit{as } n\to \infty . $$
(2.3)
$$ \lim_{n\to \infty }\frac{\sum_{j=1}^{n} q_{j}^{-1}X_{j}}{Q_{n}\log Q _{n}}=1 \quad \textit{in probability}. $$
Corollary 2.3
Let us suppose the assumptions of Corollary
2.2
and
\(q_{j}=j\)
in Eq. (2.3). Then, for any
\(\gamma >-1\)
and real
δ, we have
$$ \lim_{n\to \infty }\frac{\sum_{j=1}^{n} j^{-1}(\log j)^{\gamma }( \log \log j)^{\delta }X_{j}}{(\log n)^{\gamma +1}(\log \log n)^{ \delta +1}}=\frac{1}{\gamma +1} \quad \textit{in probability}. $$
Corollary 2.4
Let
\(\{X_{i}, i\ge 1\}\)
be a sequence of pairwise NQD random variables whose common distribution satisfies (1.4) with
\(\alpha =1\). If there exists a real
Q
such that
Then, for each real
\(\beta >-1\)
and slowly varying sequence
\(l(n)\), it follows that
$$ \lim_{x\to \infty }\frac{\mathbb {E}X1(\vert X \vert \le x)}{\log x}=Q. $$
$$ \lim_{n\to \infty }\frac{\sum_{j=1}^{n} j^{\beta }l(j)X_{j}}{n^{ \beta +1}l(n)\log n}=\frac{Q}{1+\beta } \quad \textit{in probability}. $$
Theorem 2.2
Let
\(\{X_{i}, i\ge 1\}\)
be a sequence of pairwise NQD random variables whose distributions satisfy
and
If there exist two positive sequences
\(\{a_{j}\}\)
and
\(\{b_{j}\}\)
satisfying
Then it follows that
$$ \mathbb {P}\bigl(\vert X_{j} \vert >x\bigr)\asymp x^{-\alpha } \quad \textit{for } j\ge 1 $$
$$ \limsup_{x\to \infty } \sup_{j\ge 1}x^{\alpha }\mathbb {P}\bigl(\vert X_{j} \vert >x\bigr)< \infty . $$
$$ \bigl( \log^{2} n \bigr) \sum _{j=1}^{n} a_{j}^{\alpha }=o \bigl(b_{n}^{\alpha }\bigr). $$
(2.4)
$$ \lim_{n\to \infty }\frac{1}{b_{n}}\max _{1\le k\le n}\Biggl\vert \sum_{j=1} ^{k} a_{j} \biggl( X_{j}-\mathbb {E}X_{j}1 \biggl\{ \vert X_{j} \vert \le \frac{b_{n}}{a_{j}} \biggr\} \biggr) \Biggr\vert =0 \quad \textit{in probability}. $$
(2.5)
Corollary 2.5
Under the assumptions of Theorem
2.2, if
\(0<\alpha <1\), then we have
$$ \lim_{n\to \infty }\frac{1}{b_{n}}\max_{1\le k\le n}\Biggl\vert \sum_{j=1} ^{k} a_{j} X_{j}\Biggr\vert =0 \quad \textit{in probability}. $$
2.2 Proofs of Theorem 2.1 and Theorem 2.2
We first give some useful lemmas.
Lemma 2.1
([5])
If a random variable
X
satisfies (1.4), then it follows that
and
$$ \mathbb {E}\bigl(\vert X \vert 1\bigl(\vert X \vert \le x\bigr)\bigr)\asymp \textstyle\begin{cases} x^{1-\alpha },& \textit{if } 0< \alpha < 1, \\ \log x,& \textit{if } \alpha =1, \end{cases} $$
$$ \mathbb {E}\bigl(\vert X \vert ^{2}1\bigl(\vert X \vert \le x\bigr) \bigr)\asymp x^{2-\alpha } \quad \textit{for } 0< \alpha \le 1. $$
Lemma 2.2
([6])
Let \(\{X_{n},n\ge 1\}\) be a sequence of pairwise NQD random variables. Let \(\{f_{n}, n\ge 1\}\) be a sequence of increasing functions. Then \(\{f_{n}(X_{n}), n\ge 1\}\) is a sequence of pairwise NQD random variables.
Lemma 2.3
([9])
Let
\(\{X_{n},n\ge 1\}\)
be a sequence of pairwise NQD random variables with mean zero and
\(\mathbb {E}X_{n}^{2}<\infty \), and
\(T_{j}(k)= \sum_{i=j+1}^{j+k} X_{i}\), \(j\ge 0\). Then
$$ \mathbb {E}\bigl(T_{j}(k)\bigr)^{2}\le C\sum _{i=j+1}^{j+k} \mathbb {E}X_{i}^{2}, \qquad \mathbb {E}\max_{1\le k\le n} \bigl(T_{j}(k)\bigr)^{2}\le C\log^{2} n\sum_{i=j+1}^{j+n} \mathbb {E}X _{i}^{2}. $$
Proof of Theorem 2.1
For any \(1\le i\le n\), let us define
and
Then from Lemma 2.2 it follows that \(\{Y_{ni}, 1\le i\le n, n \ge 1\}\) and \(\{Z_{ni}, 1\le i\le n, n\ge 1\}\) are both pairwise NQD, and
Furthermore, let us define \(X_{ni}= X_{i}1(a_{i}\vert X_{i} \vert \le b_{n})\), then the limit (2.2) holds if we show
and
Using the proof of Lemma 2.2 in [4], we get
and
From Lemma 2.1 and Lemma 2.3, we have
which implies (2.6). Similarly, for any \(r>0\), we have
which yields (2.7). Finally, (2.8) holds since
Based on the above discussions, the desired results are obtained. □
$$ Y_{ni}=-b_{n}1(a_{i}X_{i}< -b_{n})+a_{i}X_{i}1 \bigl(a_{i}\vert X_{i} \vert \le b_{n}\bigr)+b _{n}1(a_{i}X_{i}>b_{n}) $$
$$ Z_{ni}=(a_{i}X_{i}+b_{n})1(a_{i}X_{i}< -b_{n})+(a_{i}X_{i}-b_{n})1(a _{i}X_{i}>b_{n}). $$
$$ \sum_{i=1}^{n} a_{i} X_{i}=\sum_{i=1}^{n} (Y_{ni}+Z_{ni}). $$
$$\begin{aligned}& \lim_{n\to \infty } \frac{1}{b_{n}}\sum _{i=1}^{n} (Y_{ni}-\mathbb {E}Y_{ni})=0 \quad \text{in probability}, \end{aligned}$$
(2.6)
$$\begin{aligned}& \lim_{n\to \infty } \frac{1}{b_{n}}\sum _{i=1}^{n} (a_{i}X_{i}- Y_{ni})=0 \quad \text{in probability}, \end{aligned}$$
(2.7)
$$ \lim_{n\to \infty } \frac{1}{b_{n}}\sum _{i=1}^{n} (a_{i}\mathbb {E}X_{ni}- \mathbb {E}Y _{ni})=0. $$
(2.8)
$$ \frac{1}{b^{2}_{n}}\sum_{i=1}^{n} a_{i}^{2}\mathbb {E}\bigl[ X_{i}^{2}1\bigl(a _{i}\vert X_{i} \vert \le b_{n}\bigr) \bigr] \to 0 $$
$$ \sum_{i=1}^{n} \mathbb {P}\bigl(a_{i} \vert X_{i} \vert \ge b_{n}\bigr)\to 0. $$
$$\begin{aligned} \frac{1}{b^{2}_{n}}\operatorname{Var} \Biggl( \sum_{i=1}^{n}(Y_{ni}-\mathbb {E}Y_{ni}) \Biggr) &\le \frac{C}{b^{2}_{n}}\sum _{i=1}^{n}\mathbb {E}Y_{ni}^{2} \\ & \le \frac{C}{b ^{2}_{n}}\sum_{i=1}^{n} a_{i}^{2}\mathbb {E}\bigl[ X_{i}^{2}1 \bigl(a_{i}\vert X_{i} \vert \le b_{n}\bigr) \bigr] \\ &\quad {}+C\sum_{i=1}^{n} \mathbb {P}\bigl(a_{i}\vert X_{i} \vert \ge b_{n}\bigr) \to 0, \end{aligned}$$
$$\begin{aligned} \mathbb {P}\Biggl( \frac{1}{b_{n}}\Biggl\vert \sum_{i=1}^{n} Z_{ni}\Biggr\vert >r \Biggr)&\le \mathbb {P}\Biggl( \bigcup_{i=1}^{n} \bigl\{ a_{i}\vert X_{i} \vert >b_{n}\bigr\} \Biggr) \\ &\le \sum_{i=1}^{n} \mathbb {P}\bigl( a_{i}\vert X_{i} \vert >b_{n} \bigr) \to 0, \end{aligned}$$
$$ \frac{1}{b_{n}}\Biggl\vert \sum_{i=1}^{n} (a_{i}\mathbb {E}X_{ni}- \mathbb {E}Y_{ni})\Biggr\vert \le C \sum_{i=1}^{n}\mathbb {P}\bigl(a_{i} \vert X_{i} \vert >b_{n}\bigr)\to 0. $$
Proof of Theorem 2.2
By a proof similar to that of Theorem 2.1, it is enough to show
and
From Lemma 2.1 and Lemma 2.3, we have
which implies (2.9). Similarly, for any \(r>0\), we have
which yields (2.10). At last
Based on the above discussions, the desired result is obtained. □
$$\begin{aligned}& \lim_{n\to \infty } \frac{1}{b_{n}}\max _{1\le k\le n}\Biggl\vert \sum_{i=1} ^{k} (Y_{ni}-\mathbb {E}Y_{ni})\Biggr\vert =0 \quad \text{in probability}, \end{aligned}$$
(2.9)
$$\begin{aligned}& \lim_{n\to \infty } \frac{1}{b_{n}}\max _{1\le k\le n}\Biggl\vert \sum_{i=1} ^{k} (a_{i}X_{i}- Y_{ni})\Biggr\vert =0 \quad \text{in probability}, \end{aligned}$$
(2.10)
$$ \lim_{n\to \infty } \frac{1}{b_{n}}\max _{1\le k\le n}\Biggl\vert \sum_{i=1} ^{k} (a_{i}\mathbb {E}X_{ni}- \mathbb {E}Y_{ni}) \Biggr\vert =0. $$
(2.11)
$$\begin{aligned} &\frac{1}{b^{2}_{n}}\mathbb {E}\Biggl( \max_{1\le k\le n}\Biggl\vert \sum_{i=1}^{k} (Y _{ni}-\mathbb {E}Y_{ni})\Biggr\vert ^{2} \Biggr) \\ &\quad \le \frac{C\log^{2} n}{b^{2}_{n}} \sum_{i=1}^{n}\mathbb {E}Y_{ni}^{2} \\ &\quad \le \frac{C\log^{2} n}{b^{2}_{n}} \sum_{i=1}^{n} a_{i}^{2}\mathbb {E}\bigl[ X_{i}^{2}1 \bigl(a_{i}\vert X_{i} \vert \le b_{n}\bigr) \bigr] +C \log^{2} n\sum_{i=1}^{n} \mathbb {P}\bigl(a_{i}\vert X_{i} \vert \ge b_{n} \bigr) \\ &\quad \le \frac{C \log^{2} n}{b^{2}_{n}}\sum_{i=1}^{n} a_{i}^{2}(b_{n}/a_{i})^{2-\alpha }+ \frac{C\log^{2} n}{b_{n}^{\alpha }}\sum_{i=1}^{n} a_{i}^{\alpha } \to 0, \end{aligned}$$
$$ \begin{aligned}\mathbb {P}\Biggl( \frac{1}{b_{n}}\max_{1\le k\le n}\Biggl\vert \sum_{i=1}^{k} Z_{ni}\Biggr\vert >r \Biggr) \le \mathbb {P}\Biggl( \bigcup_{i=1}^{n} \bigl\{ a_{i}\vert X_{i} \vert >b_{n}\bigr\} \Biggr) \le \sum_{i=1}^{n} \mathbb {P}\bigl( a_{i}\vert X_{i} \vert >b_{n} \bigr) \to 0,\end{aligned}$$
$$ \frac{1}{b_{n}}\max_{1\le k\le n}\Biggl\vert \sum _{i=1}^{k} (a_{i}\mathbb {E}X_{ni}- \mathbb {E}Y_{ni})\Biggr\vert \le C\sum_{i=1}^{n} \mathbb {P}\bigl(a_{i}\vert X_{i} \vert >b_{n}\bigr) \to 0. $$
3 One side strong law
Adler [10] considered the almost sure upper and lower bounds for a particular normalized weighted sum of independent nonnegative random variables (see Corollary 2.2). In this section, we extend his work from the independent case to pairwise NQD nonnegative random variables.
3.1 Main results
Theorem 3.1
Let
\(\{X_{i}, i\ge 1\}\)
be a sequence of nonnegative pairwise NQD random variables whose distributions satisfy
where
\(q_{j}\ge 1\)
for any positive integer
j, and
If
then we have
$$ \mathbb {P}\bigl(\vert X_{j} \vert >x\bigr)= (x+q_{j})^{-1} \quad \textit{for } x\ge 0 $$
$$ Q_{n}:=\sum_{j=1}^{n}q_{j}^{-1} \to \infty , \quad \textit{as } n\to \infty . $$
(3.1)
$$ \sum_{n=1}^{\infty } \frac{q_{n}^{-1}}{Q_{n}\log Q_{n}}=\infty , $$
(3.2)
$$ \limsup_{n\to \infty }\frac{\sum_{j=1}^{n} q_{j}^{-1}X_{j}}{Q_{n} \log Q_{n}}=\infty\quad\textit{almost surely}. $$
Theorem 3.2
Let
\(\{X_{i}, i\ge 1\}\)
be a sequence of nonnegative pairwise NQD random variables whose distributions satisfy
where
\(q_{j}\ge 1\)
for any positive integer
j, and
If there is a sequence
\(\{d_{n}, n\ge 1\}\)
such that
and
then we have
$$ \mathbb {P}(X_{j}>x)= (x+q_{j})^{-1} \quad \textit{for } x \ge 0 $$
$$ Q_{n}:=\sum_{j=1}^{n}q_{j}^{-1} \to \infty , \quad \textit{as } n\to \infty . $$
(3.3)
$$ \lim_{n\to \infty } \frac{1}{Q_{n}\log Q_{n}}\sum _{j=1}^{n} q^{-1} _{j}\log \bigl(q_{j}^{-1}d_{j}\bigr)=1 $$
(3.4)
$$ \sum_{n=1}^{\infty } \frac{q_{n}^{-2}d_{n}\log^{2}n}{Q_{n}^{2}\log^{2}Q _{n}}< \infty , $$
(3.5)
$$ \liminf_{n\to \infty }\frac{1}{Q_{n}\log Q_{n}}\sum _{j=1}^{n} q_{j} ^{-1}X_{j}=1\quad \textit{almost surely.} $$
Remark 3.1
For the independent case, the assumption (3.5) can be weakened by the following condition (see [10]):
If \(\{X_{i}, i\ge 1\}\) is a sequence of NA random variables, then from the maximal inequality of NA random variables (see [8, Theorem 2]), the condition (3.5) can be weakened by (3.6).
$$ \sum_{n=1}^{\infty } \frac{q_{n}^{-2}d_{n}}{Q_{n}^{2}\log^{2}Q_{n}}< \infty . $$
(3.6)
Remark 3.2
Let us give an example to show that the conditions (3.4) and (3.5) can be satisfied. If we choose \(q_{j}^{-1}=j^{\alpha}\), \(d_{j}=j/\log^{2} j\) where \(\alpha>-1\), then it is easy to show
and
Hence we have
and
$$\begin{aligned}& Q_{n}\sim \frac{n^{\alpha+1}}{\alpha+1},\qquad \ Q_{n}\log Q_{n}\sim n^{\alpha+1}\log n, \\& \sum_{i=1}^{n}q_{i}^{-1}\log q^{-1}_{i}=\alpha\sum_{i=1}^{n}i^{\alpha}\log i\sim\frac{\alpha}{\alpha+1}n^{\alpha+1}\log n, \end{aligned}$$
$$ \sum_{i=1}^{n}q_{i}^{-1}\log d_{i}=\sum_{i=1}^{n}i^{\alpha}(\log i-2\log\log i)\sim\frac{1}{\alpha+1}n^{\alpha+1}\log n. $$
$$ \frac{1}{Q_{n}\log Q_{n}}\sum_{j=1}^{n} q^{-1}_{j}\log\bigl(q_{j}^{-1}d_{j}\bigr)\to 1 $$
$$ \sum_{n=1}^{\infty}\frac{q_{n}^{-2}d_{n}\log^{2}n}{Q_{n}^{2}\log^{2}Q_{n}}=\sum_{n=1}^{\infty}\frac{n^{2\alpha}n}{n^{2(\alpha+1)}\log^{2} n}< \infty. $$
3.2 Proofs of Theorem 3.1 and Theorem 3.2
Before giving our proofs, we need the following useful lemmas.
Lemma 3.1
([11])
Let
\(\{A_{n}, n\ge 1\}\)
be a sequence of events, such that
\(\sum_{n=1}^{\infty }\mathbb {P}(A_{n})=\infty \). Then
$$ \mathbb {P}( A_{n}, i.o. ) \ge \limsup_{n\to \infty } \frac{ ( \sum_{k=1}^{n} \mathbb {P}(A_{k}) ) ^{2}}{\sum_{i,k=1}^{n}\mathbb {P}(A_{i}A_{k})}. $$
Lemma 3.2
([9])
Let
\(\{X_{n}, n\ge 1\}\)
be pairwise NQD random sequences. If
then we have
$$ \sum_{n=1}^{\infty }\log^{2} n \operatorname{Var}(X_{n})< \infty , $$
$$ \sum_{n=1}^{\infty }(X_{n}-\mathbb {E}X_{n}) \quad \textit{converges almost surely.} $$
Proof of Theorem 3.1
For any \(M>0\), we have
Let us define \(A_{n}= \{ X_{n}>\frac{MQ_{n}\log Q_{n}}{q_{n}^{-1}} \} \), then we have
where we used the fact that \(\{X_{n}, n\ge 1\}\) is a sequence of nonnegative pairwise NQD random variables. Hence from Lemma 3.1, we get
which yields
□
$$ \begin{aligned}\sum_{n=1}^{\infty }\mathbb {P}\biggl( \frac{ q_{n}^{-1}X_{n}}{Q_{n}\log Q_{n}}>M \biggr) = \sum_{n=1}^{\infty } \mathbb {P}\biggl( X_{n}> \frac{MQ_{n}\log Q_{n}}{q_{n}^{-1}} \biggr) =\infty .\end{aligned}$$
$$ \mathbb {P}(A_{i}A_{k})\le \mathbb {P}(A_{i})\mathbb {P}(A_{k}), $$
$$ \limsup_{n\to \infty }\frac{ q_{n}^{-1}X_{n}}{Q_{n}\log Q_{n}}=\infty\quad\text{almost surely}, $$
$$ \limsup_{n\to \infty }\frac{ \sum_{i=1}^{n}q_{i}^{-1}X_{i}}{Q_{n} \log Q_{n}}\ge \limsup _{n\to \infty }\frac{ q_{n}^{-1}X_{n}}{Q_{n} \log Q_{n}}=\infty \quad \text{almost surely}. $$
Proof Theorem 3.2
For \(n\ge 1\), let us define
and
Then from Lemma 2.2 it follows that \(\{Y_{n}, n\ge 1\}\) and \(\{Z_{n}, n\ge 1\}\) are both pairwise NQD, and
Since
then from the condition (3.5), we have
Thus by Lemma 3.2, we have
which, by the Kronecker lemma, implies that
Furthermore, since
then by the condition (3.4) we have
Hence, from (3.7), (3.8) and (3.9), we have
By Corollary 2.2, we have
So, from (3.10) and (3.11), we have
□
$$ Y_{n}=X_{n}1(X_{n}\le d_{n})+d_{n}1(X_{n}> d_{n}) $$
$$ Z_{n}=(X_{n}-d_{n})1(X_{n}> d_{n}). $$
$$\begin{aligned} \sum_{i=1}^{n} q_{i}^{-1} X_{i} =\sum _{i=1}^{n} q_{i}^{-1}(Y_{i}+Z_{i}) = \sum_{i=1}^{n} q_{i}^{-1}(Y_{i}- \mathbb {E}Y_{i})+\sum_{i=1}^{n} q_{i}^{-1}Z _{i}+\sum _{i=1}^{n} q_{i}^{-1}\mathbb {E}Y_{i}. \end{aligned}$$
(3.7)
$$\begin{aligned} \mathbb {E}Y_{n}^{2}&\le C \bigl( \mathbb {E}X_{n}^{2}1(X_{n}\le d_{n})+d_{n}^{2} \mathbb {P}(X _{n}>d_{n}) \bigr) \\ &= C \biggl( \int_{0}^{d_{n}} \frac{x^{2}}{(x+q_{n})^{2}}\,dx+d_{n}^{2} \int_{d_{n}}^{\infty }\frac{1}{(x+q _{n})^{2}}\,dx \biggr) \le C d_{n}, \end{aligned}$$
$$ \sum_{n=1}^{\infty }\frac{q_{n}^{-2}\log^{2} n\operatorname{Var}(Y_{n})}{Q_{n}^{2} \log^{2} Q_{n}}\le \sum _{n=1}^{\infty }\frac{q_{n}^{-2}\log^{2} nd _{n}}{Q_{n}^{2}\log^{2} Q_{n}}< \infty . $$
$$ \sum_{n=1}^{\infty }\frac{q_{n}^{-1} (Y_{n}-\mathbb {E}Y_{n})}{Q_{n}\log Q _{n}} \quad \text{converges almost surely}, $$
$$ \lim_{n\to \infty }\sum_{i=1}^{n} \frac{q_{i}^{-1} (Y_{i}-\mathbb {E}Y_{i})}{Q _{n}\log Q_{n}}=0 \quad\text{almost surely}. $$
(3.8)
$$\begin{aligned} \mathbb {E}Y_{n}&= \mathbb {E}X_{n}1(X_{n}\le d_{n})+d_{n}\mathbb {P}(X_{n}> d_{n}) \\ &= \int_{0}^{d_{n}}\frac{x}{(x+q_{n})^{2}}\,dx+d_{n} \int^{\infty }_{d_{n}}\frac{1}{(x+q _{n})^{2}}\,dx \\ &= \int_{q_{n}}^{d_{n}+q_{n}}\frac{1}{x}\,dx-q_{n} \int_{q_{n}}^{d_{n}+q_{n}}\frac{1}{x^{2}}\,dx+d_{n} \int^{\infty }_{d _{n}}\frac{1}{(x+q_{n})^{2}}\,dx \\ &= \log (1+d_{n}/q_{n})-1+\frac{q _{n}}{d_{n}+q_{n}}+ \frac{d_{n}}{d_{n}+q_{n}} \\ &=\log (1+d_{n}/q_{n}) \end{aligned}$$
$$ \lim_{n\to \infty }\sum_{i=1}^{n} \frac{q_{i}^{-1} \mathbb {E}Y_{i}}{Q_{n} \log Q_{n}}=1 \quad\text{almost surely}. $$
(3.9)
$$ \liminf_{n\to \infty }\sum_{i=1}^{n} \frac{q_{i}^{-1} X_{i}}{Q_{n} \log Q_{n}}\ge \lim_{n\to \infty }\sum _{i=1}^{n} \frac{q_{i}^{-1}(Y _{i}-\mathbb {E}Y_{i})}{Q_{n}\log Q_{n}}+\lim _{n\to \infty }\sum_{i=1}^{n} \frac{q _{i}^{-1}\mathbb {E}Y_{i}}{Q_{n}\log Q_{n}}=1. $$
(3.10)
$$ \liminf_{n\to \infty }\sum_{i=1}^{n} \frac{q_{i}^{-1} X_{i}}{Q_{n} \log Q_{n}}\leq 1 \quad\text{almost surely}. $$
(3.11)
$$ \liminf_{n\to \infty }\sum_{i=1}^{n} \frac{q_{i}^{-1} X_{i}}{Q_{n} \log Q_{n}}=1\quad\text{almost surely}. $$
Declarations
Acknowledgements
The authors are most grateful to the editor and anonymous referee for careful reading of the manuscript and valuable suggestions, which helped in improving an earlier version of this paper. This work is supported by IRTSTHN (14IRTSTHN023), NSFC (11471104).
Authors’ contributions
All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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.
Authors’ Affiliations
(1)
College of Mathematics and Statistics, Pingdingshan University, Pingdingshan, China
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