Abstract
Some exponential probability inequalities for widely negative orthant dependent (WNOD, in short) random variables are established, which can be treated as very important roles to prove the strong law of large numbers among others in probability theory and mathematical statistics. By using the exponential probability inequalities, we study the complete convergence for arrays of rowwise WNOD random variables. As an application, the Marcinkiewicz–Zygmund type strong law of large numbers is obtained. In addition, the complete moment convergence for arrays of rowwise WNOD random variables is studied by using the exponential probability inequality and complete convergence that we established.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
It is well known that the exponential probability inequality plays an important role to prove strong law of large numbers, strong convergence rate, complete convergence, consistency, asymptotic normality, and so on. There are much literature studying exponential probability inequalities for independent sequences and some dependent sequences, such as negatively associated (NA, in short) sequence, positively associated (PA, in short) sequence, negatively orthant dependent (NOD, in short) sequence, extended negatively dependent (END, in short) sequence, and so on. There are very little literatures studying exponential probability inequalities for widely negative orthant dependent (WNOD, short) sequence, which includes NA sequence, NOD sequence, END sequence and some positive dependent sequences as special cases. The main purpose of the paper is to provide some exponential probability inequalities for WNOD sequence and give applications in complete convergence and complete moment convergence.
The concept of widely negative dependence structure was introduced by Wang et al. [23] as follows.
Definition 1.1
For the random variables \(\{X_n, n\ge 1\}\), if there exists a finite positive sequence \(\{g_U (n), n \ge 1\}\) satisfying for each \(n \ge 1\) and for all \(x_i \in (-\infty ,\infty )\), \(1\le i\le n\),
then we say that the random variables \(\{X_n, n\ge 1\}\) are widely negative upper orthant dependent (WNUOD, in short); if there exists a finite positive sequence \(\{g_L (n), n \ge 1\}\) satisfying for each \(n \ge 1\) and for all \(x_i \in (-\infty ,\infty )\), \(1\le i\le n\),
then we say that the \(\{X_n, n\ge 1\}\) are widely negative lower orthant dependent (WNLOD, in short); if they are both WNUOD and WNLOD, then we say that the \(\{X_n, n\ge 1\}\) are widely negative orthant dependent (WNOD, in short), and \(g_U (n)\), \(g_L (n)\), \(n\ge 1\), are called dominating coefficients.
An array \(\{X_{ni}, i\ge 1, n\ge 1\}\) of random variables is called rowwise WNOD random variables if for every \(n\ge 1\), \(\{X_{ni}, i\ge 1\}\) is a sequence of WNOD random variables.
For examples of WNOD random variables with various dominating coefficients, we refer the reader to Wang et al. [23, 33]. These examples show that WNOD random variables contain some common negatively dependent random variables, some positively dependent random variables and some others.
Assume that \(g_U(n)\ge 1\), \(g_L(n)\ge 1\). It is easily seen that if both (1.1) and (1.2) hold for \(g_L(n) = g_U (n) = M\) for any \(n\ge 1\), where M is a positive constant, then the random variables \(\{X_n, n\ge 1\}\) are called extended negatively dependent (END, in short). This is the definition of END sequences. For the details about the concept and the probability limit theory of END sequence, one can refer to Liu [9], Chen et al. [3], Shen [17], Wang and Chen [34], Wang and Wang [24], Wang et al. [28, 29, 32], Qiu et al. [13], and so forth. If both (1.1) and (1.2) hold for \(g_L(n) = g_U (n)=1\) for any \(n\ge 1\), then the random variables \(\{X_n, n\ge 1\}\) are called negatively orthant dependent (NOD, in short). For more details about NOD sequence, one can refer to Fakoor and Azarnoosh [6], Asadian et al. [1], Wang et al. [26, 27], Wu [37, 38], Wu and Jiang [39], Sung [21], Nili Sani et al. [11], Li et al. [8], Shen [16, 18, 20], and so on. It is well known that NA random variables are NOD random variables. Hu [7] pointed out that negatively superadditive dependent (NSD, in short) random variables are NOD. Hence, the class of WNOD random variables includes independent sequence, NA sequence, NSD sequence, NOD sequence and END sequence as special cases. Studying the probability inequalities, moment inequalities and convergence theorem of WNOD random variables are of great interest.
There are only some literatures studying the probability limiting behavior of WNOD random variables, such as Wang et al. [23], Wang and Cheng [34], Wang et al. [35], Liu et al. [10], Chen et al. [4], Shen [19], Qiu and Chen [12], Qiu and Hu [14], Wang et al. [31], Yang et al. [43], and so on. In these literatures, Wang et al. [31] made great contribution to the probability limit theory and statistical large sample theory for WNOD random variables; they established some general results on complete convergence for weighted sums of arrays of rowwise WNOD random variables and presented some sufficient conditions to prove the complete consistency for the estimator of nonparametric regression model based on WNOD errors. In this work, we will provide some exponential probability inequalities for WNOD random variables. As applications, we will study the complete convergence and complete moment convergence for WNOD random variables by using the exponential probability inequalities that we established.
Throughout the paper, let \(\{X_n, n\ge 1\}\) be a sequence of WNOD random variables with dominating coefficients \(g_U (n)\), \(g_L (n)\), \(n\ge 1\). Let \(\{X_{ni}, i\ge 1, n \ge 1\}\) be an array of rowwise WOD random variables with dominating coefficients \(g_U (n)\), \(g_L (n)\), \(n\ge 1\) in each row. Denote \(g(n)=\max \{g_U (n), g_L (n)\}\), \(x\vee y=\max \{x, y\}\), \(S_n=\sum _{i=1}^nX_i\) and \(M_{t,n}=\sum _{i=1}^nE|X_i|^t\) for some \(t>0\) and each \(n\ge 1\). Let C denote a positive constant, which can be different in various places.
The organization of the paper is as follows: Some useful lemmas are presented in Sect. 2. The exponential probability inequalities for WNOD random variables are established in Sect. 3. The complete convergence and complete moment convergence for arrays of rowwise WNOD random variables are obtained in Sects. 4 and 5, respectively.
2 Preliminaries
In this section, we will present some important lemmas, which will be used to prove the main results of this work. The first one is a basic property for WNOD random variables, which can be found in Wang et al. [23].
Lemma 2.1
-
(i)
Let \(\{X_n, n\ge 1\}\) be WNLOD (WNUOD) with dominating coefficients \(g_L (n), n \ge 1\) (\(g_U (n), n \ge 1\)). If \(\{f_n(\cdot ), n\ge 1\}\) are nondecreasing, then \(\{f_n(X_n), n\ge 1\}\) are still WNLOD (WNUOD) with dominating coefficients \(g_L (n), n \ge 1\) (\(g_U (n), n \ge 1\)); if \(\{f_n(\cdot ), n\ge 1\}\) are nonincreasing, then \(\{f_n(X_n), n\ge 1\}\) are WNUOD (WNLOD) with dominating coefficients \(g_L (n), n \ge 1\) (\(g_U (n), n \ge 1\)).
-
(ii)
If \(\{X_n, n\ge 1\}\) are nonnegative and WNUOD with dominating coefficients \(g_U (n), n \ge 1\), then for each \(n\ge 1\),
$$\begin{aligned} E\prod _{i=1}^n X_i\le g_U (n)\prod _{i=1}^nEX_i. \end{aligned}$$(2.1)In particular, if \(\{X_n, n\ge 1\}\) are WNUOD with dominating coefficients \(g_U (n), n \ge 1\), then for each \(n\ge 1\) and any \(s>0\),
$$\begin{aligned} E\exp \left\{ s\sum _{i=1}^nX_i\right\} \le g_U (n)\prod _{i=1}^nE\exp \{sX_i\}. \end{aligned}$$(2.2)
By Lemma 2.1, we can get the following corollary immediately, which has been obtained by Shen [19].
Corollary 2.1
-
(i)
Let \(\{X_n, n\ge 1\}\) be WNOD. If \(\{f_n(\cdot ), n\ge 1\}\) are nondecreasing (or nonincreasing), then \(\{f_n(X_n), n\ge 1\}\) are still WNOD.
-
(ii)
If \(\{X_n, n\ge 1\}\) are WNOD, then for each \(n\ge 1\) and any \(s\in \mathbb {R}\),
$$\begin{aligned} E\exp \left\{ s\sum _{i=1}^nX_i\right\} \le g(n)\prod _{i=1}^nE\exp \{sX_i\}. \end{aligned}$$(2.3)
The following one is a very important property for stochastic domination. For the details of the proof, one can refer to Wu [36], or Wang et al. [30].
Lemma 2.2
Assume that the random variable Y is stochastically dominated by an nonnegative random variable X. That is to say, there exists a positive constant C such that
for all \(x\ge 0\). Then the following two statements
and
hold for all \(\alpha >0\) and \(b>0\).
Lemma 2.3
(cf. Shao [15]) For any \(x\ge 0\),
3 Exponential probability inequalities for WNOD random variables
In this section, we establish the exponential probability inequalities for WNOD random variables, which can be applied to establish the probability limit theorem for WNOD random variables, such as weak convergence, \(L_r\) convergence, strong convergence, complete convergence, complete moment convergence, consistency, asymptotic normality, and so on. The proofs of the exponential probability inequalities for WNOD random variables are mainly inspired by Fakoor and Azarnoosh [6] and Asadian et al. [1]. Our main results on exponential probability inequalities are as follows.
Theorem 3.1
Let \(0<t\le 2\) and \(\{X_n, n\ge 1\}\) be a sequence of WNOD random variables with \(E|X_n|^t<\infty \) for each \(n\ge 1\). Assume further that \(EX_n=0\) for each \(n\ge 1\) when \(1<t\le 2\). Then for all \(x>0\) and \(y>0\),
If \(xy^{t-1}>M_{t,n}\), then
Proof
For \(y>0\), denote \(Y_i=\min (X_i, y)\), \(i=1,2,\ldots ,n\) and \(S_n'=\sum _{i=1}^nY_i\), \(n\ge 1\). It is easily seen that for any \(h>0\),
It follows by (3.3) and Corollary 2.1 that
For \(0<t\le 1\), the function \((e^{hu}-1)/u^t\) is increasing on \(u>0\). Thus,
Combining the inequality above and (3.4), we can get that
Replacing \(X_i\) by \(-X_i\) in (3.5), we have
It follows by (3.5) and (3.6) that
Taking \(h=\frac{1}{y}\ln \left( 1+\frac{xy^{t-1}}{M_{t,n}}\right) \) in the right-hand side of (3.7), we can get (3.1) immediately.
For \(1<t\le 2\), we can still get (3.1) by the similar process of Theorem 3 in Fakoor and Azarnoosh [6] and Theorem 2.2 in Asadian et al. [1]. The details are omitted.
If \(xy^{t-1}>M_{t,n}\), then the right-hand side of (3.7) attains a minimum value when \(h=\frac{1}{y}\ln \left( \frac{xy^{t-1}}{M_{t,n}}\right) \). Substitute this value of h to the right-hand side of (3.7), we can get (3.2) immediately. This completes the proof of the theorem. \(\square \)
For \(t=2\), we have the following more precise exponential probability inequality than (3.1).
Theorem 3.2
Let \(\{X_n, n\ge 1\}\) be a sequence of WNOD random variables with \(EX_n=0\) and \(EX_n^2<\infty \) for each \(n\ge 1\). Then for any \(h,x,y>0\),
If we take \(h=\frac{1}{y}\ln \left( 1+\frac{xy}{M_{2,n}}\right) \), then
and
Proof
We use the same notations as those in Theorem 3.1. Similar to the proofs of Theorem 3.1 in the paper and Lemma 2.4 in Shen [20], we have
Replacing \(X_i\) by \(-X_i\) in the inequality above, we have
Therefore, the desired result (3.8) follows from (3.11) and (3.12) immediately.
Equation (3.9) can be easily obtained by taking \(h=\frac{1}{y}\ln \left( 1+\frac{xy}{M_{2,n}}\right) \) in (3.8).
By Lemma 2.3, we have that
Hence, the desired result (3.10) follows from (3.9) and (3.13) immediately. This completes the proof of the theorem. \(\square \)
Remark 3.1
Under the conditions of Theorem 3.1, Wang et al. [31] obtained the following inequality:
Noting that
we can see that (3.1) is more precise than (3.14). Additionally, we also get (3.2), which is not obtained in Wang et al. [31].
4 Complete convergence for arrays of rowwise WNOD random variables
In the previous section, we established some exponential probability inequalities for WNOD random variables. In this section, we will study the complete convergence for weighted sums of arrays of rowwise WNOD random variables by using the exponential probability inequalities that we established. The main ideas are inspired by Chen et al. [2] and Wu et al. [40].
Our main results on complete convergence for WNOD random variables are as follows. The first one (Theorem 4.1) is a very general result on complete convergence for weighted sums of WNOD random variables, which can be applied to establish other results on complete convergence and strong convergence. The proof is similar to Shen [20].
Theorem 4.1
Let \(\{X_{ni}, i\ge 1, n\ge 1\}\) be an array of rowwise WNOD random variables with finite second moments and \(\{a_{ni}, i\ge 1, n\ge 1\}\) be an array of constants. Let \(\{c_{n}, n\ge 1\}\) be a sequence of positive constants. Suppose that the following two conditions hold:
-
(i)
for every \(\epsilon >0\),
$$\begin{aligned} \sum \limits _{n=1}^\infty c_n g(n)\sum \limits _{i=1}^n P\left( \left| a_{ni}X_{ni}\right| >\epsilon \right) <\infty ; \end{aligned}$$ -
(ii)
for some \(\delta >0\) and \(J\ge 1\),
$$\begin{aligned} \sum \limits _{n=1}^\infty c_n g(n)\left[ \sum _{i=1}^n Ea_{ni}^2X_{ni}^2I\left( \left| a_{ni}X_{ni}\right| \le \delta \right) \right] ^J<\infty . \end{aligned}$$Then
$$\begin{aligned} \sum \limits _{n=1}^\infty c_nP\left( \left| \sum _{i=1}^n\left[ a_{ni}X_{ni}-Ea_{ni}X_{ni}I\left( \left| a_{ni}X_{ni}\right| \le \delta \right) \right] \right| >\varepsilon \right) <\infty \quad { for}~{any}\quad \varepsilon >0. \end{aligned}$$(4.1)
Proof
Without loss of generality, we assume that \(a_{ni}\ge 0\) for all \(i\ge 1\) and \(n\ge 1\) (Otherwise, we use \(a_{ni}^+\) and \(a_{ni}^-\) instead of \(a_{ni}\), respectively and note that \(a_{ni}=a_{ni}^+-a_{ni}^-\)). Denote
It is easy to check that \(\{Y_{ni}-EY_{ni}, i\ge 1, n\ge 1\}\) is an array of rowwise WNOD random variables by Corollary 2.1 (i).
For fixed \(n\ge 1\), denote
Thus,
By condition (i) and noting that \(g(n)\ge 1\), to prove (4.1), it suffices to show that for any \(\varepsilon >0\),
Let \(B_n^2=\sum _{i=1}^nVar(Y_{ni})\). For any \(\varepsilon >0\) and \(a>0\), denote \(d=\min \left\{ 1, \frac{a\varepsilon }{6\delta ^2}, \frac{a}{12\delta }\right\} \),
where \(\mathbf {N}\) is the set of positive integers.
It is easily seen that
and
Therefore,
which implies that \(\mathbf {N_1}\subset \mathbf {N_2}\bigcup \mathbf {N_3}\).
Note that
Therefore, in order to prove (4.2), we need only to show that
By Theorem 3.2, we can see that
Firstly, we will show that for any \(n\in \mathbf {N_4}\), \(\max \nolimits _{1\le i\le n}|EY_{ni}|\le \frac{a}{2}\). Actually, for any \(n\in \mathbf {N_4}\),
which implies that
It is easy to check that \(B_n^2\le a\varepsilon \) and \(\sum \nolimits _{i=1}^n P\left( \left| a_{ni}X_{ni}\right| > \delta \right) \le d\) when \(n\in \mathbf {N_4}\). Set \(a=\frac{\varepsilon }{6J}\). We have by conditions (i) and (ii) that
This completes the proof of the theorem. \(\square \)
By Theorem 4.1, we can get the following corollaries.
Corollary 4.1
Let \(\{X_{ni}, i\ge 1, n\ge 1\}\) be an array of rowwise WNOD random variables with finite second moments and \(\{a_{ni}, i\ge 1, n\ge 1\}\) be an array of constants. Let \(\{c_{n}, n\ge 1\}\) be a sequence of positive constants such that conditions (i) and (ii) of Theorem 4.1 hold for every \(\epsilon >0\) and some \(\delta >0\), \(J\ge 1\). Assume further that
Then
Corollary 4.2
Let \(\{X_{ni}, i\ge 1, n\ge 1\}\) be an array of rowwise WNOD random variables with mean zero and finite second moments, and \(\{a_{ni}, i\ge 1, n\ge 1\}\) be an array of constants. Let \(\{c_{n}, n\ge 1\}\) be a sequence of positive constants such that conditions (i) and (ii) of Theorem 4.1 hold for every \(\epsilon >0\) and some \(\delta >0\), \(J\ge 1\). Assume further that
Then (4.5) holds for any \(\varepsilon >0\).
If we take \(c_n\equiv 1\) and \(a_{ni}\equiv n^{-1/p}\) for \( i\ge 1\) and \(n\ge 1\) in Corollary 4.2, then we can get the following corollary by Corollary 4.2.
Corollary 4.3
Let \(p>0\) and \(\{X_{ni}, i\ge 1, n\ge 1\}\) be an array of rowwise WNOD random variables with mean zero and finite second moments. Suppose that the following conditions hold:
-
(i)
for every \(\epsilon >0\)
$$\begin{aligned} \sum \limits _{n=1}^\infty g(n)\sum \limits _{i=1}^n P\left( \left| X_{ni}\right| >n^{1/p}\epsilon \right) <\infty , \end{aligned}$$(4.7) -
(ii)
for some \(\delta >0\) and \(J\ge 1\),
$$\begin{aligned} \sum \limits _{n=1}^\infty g(n) \left[ n^{-2/p}\sum _{i=1}^n EX_{ni}^2I\left( \left| X_{ni}\right| \le n^{1/p}\delta \right) \right] ^J<\infty , \end{aligned}$$(4.8) -
(iii)
$$\begin{aligned} n^{-1/p}\sum \limits _{i=1}^nEX_{ni}I\left( \left| X_{ni}\right| > n^{1/p}\delta \right) \rightarrow 0\quad as\quad n\rightarrow \infty . \end{aligned}$$(4.9)
Then
$$\begin{aligned} \sum \limits _{n=1}^\infty P\left( \left| \sum _{i=1}^nX_{ni}\right| >n^{1/p}\varepsilon \right) <\infty \quad for~any\quad \varepsilon >0, \end{aligned}$$(4.10)and \(n^{-1/p}\sum _{i=1}^nX_{ni}\rightarrow 0\quad a.s.\)
As an application of Corollary 4.3, we can get the following complete convergence result and Marcinkiewicz–Zygmund type strong law of large numbers for WNOD random variables.
Corollary 4.4
Let \(\{X_{ni}, i\ge 1, n\ge 1\}\) be an array of rowwise WNOD random variables satisfying
for all \(x\ge 0\), \(i\ge 1\) and \(n\ge 1\), where X is some nonnegative random variable. Assume that \(EX_{ni}=0\) and \(g (n)=O(n^\lambda )\) for some \(\lambda >0\). If \(EX^{(2+\lambda )p}<\infty \) for some \(1\le p<2\), then (4.10) holds for any \(\varepsilon >0\) and \(n^{-1/p}\sum _{i=1}^nX_{ni}\rightarrow 0\) a.s.
Proof
We only need to prove that conditions (i)–(iii) of Corollary 4.3 hold true.
For any \(\epsilon >0\),
which implies (4.7).
Taking J large enough such that \(J>\frac{p(1+\lambda )}{2-p}\), which implies that \((1-2/p)J+\lambda <-1\). It follows by \(E|X|^{(2+\lambda )p}<\infty \) that \(EX^2<\infty \). Note that \(EX_{ni}^2\le CEX^2\) by Lemma 2.2, we can get that
which implies (4.8).
Noting that \(E|X|^{2p}<\infty \), we have by Lemma 2.2 that
which implies (4.9).
From the statements above and Corollary 4.3, we can get (4.10) immediately. This completes the proof of the corollary. \(\square \)
If we take \(c_n\equiv 1\) and \(a_{ni}=a_n^{-1}\), in Corollary 4.2, where \(\{a_{n}, n\ge 1\}\) is a sequence of positive constants, we can get another corollary by using Corollary 4.2 as follows.
Corollary 4.5
Let \(\{X_{ni}, i\ge 1, n\ge 1\}\) be an array of rowwise mean zero WNOD random variables with finite second moments and \(g(n)=O(1)\), and \(\{a_{n}, n\ge 1\}\) be a sequence of positive constants. Let \(\{g_i(t), i\ge 1\}\) be a sequence of nonnegative, even functions. Assume that there exist some \(\beta \in (1,2]\) and \(\gamma >0\) such that \(g_i(t)\ge \gamma t^\beta \) for \(0<t\le 1\) and there exists a \(\gamma >0\) such that \(g_i(t)\ge \gamma t\) for \(t>1\). If
then
Proof
We will prove that the conditions of Corollary 4.2 hold, where \(c_n\equiv 1\) and \(a_{ni}=a_n^{-1}\). Without loss of generality, we assume that \(0<\epsilon <1\). Note that \(g_i(t)\ge \gamma t^\beta \) for \(0<t\le 1\) and \(g_i(t)\ge \gamma t\) for \(t>1\), we have by (4.12) that
which implies that condition (i) in Theorem 4.1 holds.
For \(0<\delta <1\) and \(J\ge 1\), we have by the assumptions on \(g_i(t)\) and condition (4.12) that
and
By (4.14) and (4.15), we can see that condition (ii) in Theorem 4.1 and (4.4) in Corollary 4.2 are satisfied. Hence, the desired result (4.13) follows by Corollary 4.2 immediately. This completes the proof of the corollary. \(\square \)
Remark 4.1
It is easily seen that the conditions in Corollary 4.2 are weaker than those in Corollary 4.5. In addition, the results of Corollary 4.5 for NOD random variables have been obtained by Shen [16]. So our results of Theorem 4.1 and Corollary 4.2 generalize and improve the corresponding ones of Shen [16].
5 Complete moment convergence for arrays of rowwise WNOD random variables
In this section, we will study the complete moment convergence for arrays of rowwise WNOD random variables by using the exponential probability inequalities and complete convergence result that we obtained in Sects. 3 and 4. The main ideas are inspired by Wu et al. [40].
The concept of complete moment convergence was introduced by Chow [5] as follows: Let \(\{X_n, n\ge 1\}\) be a sequence of random variables and \(a_n>0\), \(b_n>0\), \(q>0\). If
for all \(\varepsilon >0\), then the above result was called the complete moment convergence. For more details about complete moment convergence, one can refer to Wu and Zhu [41], Wang and Hu [25], Yang et al. [42], Sung [22], and so on.
Our main results on complete moment convergence for arrays of rowwise WNOD random variables are as follows.
Theorem 5.1
Let \(\{X_{ni}, i\ge 1, n\ge 1\}\) be an array of rowwise WNOD random variables with mean zero and finite second moments, and \(\{a_{ni}, i\ge 1, n\ge 1\}\) be an array of constants. Let \(\{c_{n}, n\ge 1\}\) be a sequence of positive constants such that conditions (i) and (ii) of Theorem 4.1 hold for every \(\epsilon >0\) and some \(\delta >0\), \(J>1\). Assume further that
Then
Proof
Note that (5.1) implies
and thus (4.4) holds true. It follows by Corollary 4.2 that (4.5) holds for any \(\varepsilon >0\). Furthermore, by (5.2) and (5.5), we have
Therefore,
In order to prove (5.3), we only need to prove that
For fixed \(t>0\), denote for \(i\ge 1\) and \(n\ge 1\) that
Note that this decomposition does depend on t.
It is easily seen that
which implies that
By (5.4), we can get that
To prove (5.7), it suffices to show that \(H_2<\infty \). Firstly, we will show that
Note that \(|Z_{ni}|\le |a_{ni}X_{ni}|I(|a_{ni}X_{ni}|>t)\). It follows by \(EX_{ni}=0\) and (5.5) that
which implies (5.8) and \(\left| \sum _{i=1}^nEY_{ni}\right| \le t/2\) holds for any \(t\ge \delta \) and all n large enough. Hence,
It follows by Corollary 2.1 that \(\{Y_{ni}, i\ge 1, n\ge 1\}\) is an array of rowwise WNOD random variables. Denote \(B_n^2=\sum _{i=1}^n E(Y_{ni}-EY_{ni})^2\). Applying Theorem 3.1 with \(x=t/2\) and \(y=t/(2J)\), we can get by (5.9) and Theorem 3.1 that
Similar to the proof of (5.8), we can see that \(|EY_{ni}|\le t/(4J)\) holds for any \(t\ge \delta \) and all n large enough. Noting that \(|Y_{ni}|\le |a_{ni}X_{ni}|\), we have by (5.1) that
In the following, we will show that \(H_{22}<\infty \).
By the notation of \(Y_{ni}\), we can see that
which together with \(C_r\)-inequality yield that
Since \(J>1\), it follows by condition (ii) of Theorem 4.1 and (5.6) that
and
For any \(t\ge \delta \), it follows by (5.5) that
which implies that \(\sum _{i=1}^nP\left( \left| a_{ni}X_{ni}\right| > t\right) <1\) holds for any \(t\ge \delta \) and all n large enough. Hence, we have by (5.2) that
The desired result (5.3) follows from the statements above immediately. This completes the proof of the theorem. \(\square \)
References
Asadian, N., Fakoor, V., Bozorgnia, A.: Rosenthal’s type inequalities for negatively orthant dependent random variables. JIRSS 5(1–2), 69–75 (2006)
Chen, P.Y., Hu, T.C., Liu, X., Volodin, A.: On complete convergence for arrays of row-wise negatively associated random variables. Theory Probab. Appl. 52(2), 323–328 (2008)
Chen, Y., Chen, A., Ng, K.W.: The strong law of large numbers for extend negatively dependent random variables. J. Appl. Probab. 47, 908–922 (2010)
Chen, Y., Wang, L., Wang, Y.B.: Uniform asymptotics for the finite-time ruin probabilities of two kinds of nonstandard bidimensional risk models. J. Math. Anal. Appl. 401, 114–129 (2013)
Chow, Y.S.: On the rate of moment convergence of sample sums and extremes. Bull. Inst. Math. Acad. Sin. 16, 177–201 (1988)
Fakoor, V., Azarnoosh, H.A.: Probability inequalities for sums of negatively dependent random variables. Pak. J. Stat. 21, 257–264 (2005)
Hu, T.Z.: Negatively superadditive dependence of random variables with applications. Chin. J. Appl. Probab. Stat. 16, 133–144 (2000)
Li, X.Q., Yang, W.Z., Hu, S.H., Wang, X.J.: The Bahadur representation for sample quantile under NOD sequence. J. Nonparametr. Stat. 23(1), 59–65 (2011)
Liu, L.: Precise large deviations for dependent random variables with heavy tails. Stat. Probab. Lett. 79(9), 1290–1298 (2009)
Liu, X.J., Gao, Q.W., Wang, Y.B.: A note on a dependent risk model with constant interest rate. Stat. Probab. Lett. 82, 707–712 (2012)
Nili Sani, H.R., Amini, M., Bozorgnia, A.: Strong laws for weighted sums of negative dependent random variables. Islam. Repub. Iran J. Sci. 16(3), 261–266 (2005)
Qiu, D.H., Chen, P.Y.: Complete and complete moment convergence for weighted sums of widely orthant dependent random variables. Acta Math. Sin. Engl. Ser. 30(9), 1539–1548 (2014)
Qiu, D.H., Chen, P.Y., Antonini, R.G., Volodin, A.: On the complete convergence for arrays of rowwise extended negatively dependent random variables. J. Korean Math. Soc. 50(2), 379–392 (2013)
Qiu, D.H., Hu, T.C.: Strong limit theorems for weighted sums of widely orthant dependent random variables. J. Math. Res. Appl. 34(1), 105–113 (2014)
Shao, Q.M.: A comparison theorem on moment inequalities between negatively associated and independent random variables. J. Theor. Probab. 13(2), 343–356 (2000)
Shen, A.T.: Some strong limit theorems for arrays of rowwise negatively orthant-dependent random variables. J. Inequal. Appl. 2011, Article ID 93, 10 (2011)
Shen, A.T.: Probability inequalities for END sequence and their applications. J. Inequal. Appl. 2011, Article ID 98, 12 (2011)
Shen, A.T.: On strong convergence for weighted sums of a class of random variables. Abstr. Appl. Anal. 2013, Article ID 216236, 7 (2013)
Shen, A.T.: Bernstein-type inequality for widely dependent sequence and its application to nonparametric regression models. Abstr. Appl. Anal. 2013, Article ID 862602, 9 (2013)
Shen, A.T.: On the strong convergence rate for weighted sums of arrays of rowwise negatively orthant dependent random variables. Revista de la Real Academia de Ciencias Exactas Fisicas y Naturales. Serie A. Matematicas 107, 257–271 (2013)
Sung, S.H.: A note on the complete convergence for arrays of dependent random variables. J. Inequal. Appl. 2011, Article ID 76, 8 (2011)
Sung, S.H.: Complete \(q\) moment convergence for arrays of random variables. J. Inequal. Appl. 2013, Article ID 24, 11 (2013)
Wang, K.Y., Wang, Y.B., Gao, Q.W.: Uniform asymptotics for the finite-time ruin probability of a new dependent risk model with a constant interest rate. Methodol. Comput. Appl. Probab. 15(1), 109–124 (2013)
Wang, S.J., Wang, X.J.: Precise large deviations for random sums of END real-valued random variables with consistent variation. J. Math. Anal. Appl. 402, 660–667 (2013)
Wang, X.J., Hu, S.H.: Complete convergence and complete moment convergence for martingale difference sequence. Acta Math. Sin. Engl. Ser. 30(1), 119–132 (2014)
Wang, X.J., Hu, S.H., Yang, W.Z., Ling, N.X.: Exponential inequalities and inverse moment for NOD sequence. Stat. Probab. Lett. 80(5–6), 452–461 (2010)
Wang, X.J., Hu, S.H., Yang, W.Z.: Complete convergence for arrays of rowwise negatively orthant dependent random variables. RACSAM 106(2), 235–245 (2012)
Wang, X.J., Wang, S.J., Hu, S.H., Ling, J.M., Wei, Y.F.: On complete convergence of weighted sums for arrays of rowwise extended negatively dependent random variables. Stochastics: Int. J. Probab. Stoch. Process. 85(6), 1060–1072 (2013)
Wang, X.J., Hu, T.C., Volodin, A., Hu, S.H.: Complete convergence for weighted sums and arrays of rowwise extended negatively dependent random variables. Commun. Stat. Theory Methods 42, 2391–2401 (2013)
Wang, X.J., Deng, X., Zheng, L.L., Hu, S.H.: Complete convergence for arrays of rowwise negatively superadditive-dependent random variables and its applications. Statistics: J. Theor. Appl. Stat. 48(4), 834–850 (2014)
Wang, X.J., Xu, C., Hu, T.C., Volodin, A., Hu, S.H.: On complete convergence for widely orthant-dependent random variables and its applications in nonparametric regression models. TEST 23, 607–629 (2014)
Wang, X.J., Zheng, L.L., Xu, C., Hu, S.H.: Complete consistency for the estimator of nonparametric regression models based on extended negatively dependent errors. Statistics: J. Theor. Appl. Stat. 49(2), 396–407 (2015)
Wang, Y.B., Li, Y.W., Gao, Q.W.: On the exponential inequality for acceptable random variables. J. Inequal. Appl. 2011, Article ID 40, 1–10 (2011)
Wang, Y.B., Cheng, D.Y.: Basic renewal theorems for a random walk with widely dependent increments and their applications. J. Math. Anal. Appl. 384, 597–606 (2011)
Wang, Y.B., Cui, Z., Wang, K.Y., Ma, X.: Uniform asymptotics of the finite-time ruin probability for all times. J. Math. Anal. Appl. 390, 208–223 (2012)
Wu, Q.Y.: Probability Limit Theory for Mixing Sequences. Science Press of China, Beijing (2006)
Wu, Q.Y.: A strong limit theorem for weighted sums of sequences of negatively dependent random variables. J. Inequal. Appl. 2010, Article ID 383805, 1–8 (2010)
Wu, Q.Y.: A complete convergence theorem for weighted sums of arrays of rowwise negatively dependent random variables. J. Inequal. Appl. 2012, Article ID 50, 1–10 (2012)
Wu, Q.Y., Jiang, Y.Y.: The strong consistency of \(M\) estimator in a linear model for negatively dependent random samples. Commun. Stat. Theory Methods 40, 467–491 (2011)
Wu, Y.F., Cabrera, M.O., Volodin, A.: On limiting behavior for arrays of rowwise negatively orthant dependent random variables. J. Korean Stat. Soc. 42, 61–70 (2013)
Wu, Y.F., Zhu, D.J.: Convergence properties of partial sums for arrays of rowwise negatively orthant dependent random variables. J. Korean Stat. Soc. 39(2), 189–197 (2010)
Yang, W.Z., Hu, S.H., Wang, X.J.: Complete convergence for moving average process of martingale differences. Discret. Dyn. Nat. Soc. 2012, Article ID 128492, 1–16 (2012)
Yang, W.Z., Liu, T.T., Wang, X.J., Hu, S.H.: On the Bahadur representation of sample quantiles for widely orthant dependent sequences. Filomat 28(7), 1333–1343 (2014)
Acknowledgments
The authors are most grateful to the Editor-in-Chief Manuel Lopez Pellicer and anonymous referee for careful reading of the manuscript and valuable suggestions which helped in improving an earlier version of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by the National Natural Science Foundation of China (11201001, 11171001, 11426032), the Natural Science Foundation of Anhui Province (1308085QA03, 1508085J06), Open Project of School of Mathematical Sciences, Anhui University (ADSY201503), the Quality Improvement Projects for Undergraduate Education of Anhui University (ZLTS2015035) and the Research Teaching Model Curriculum of Anhui University (xjyjkc1407).
Rights and permissions
About this article
Cite this article
Shen, A., Yao, M., Wang, W. et al. Exponential probability inequalities for WNOD random variables and their applications. RACSAM 110, 251–268 (2016). https://doi.org/10.1007/s13398-015-0233-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13398-015-0233-7
Keywords
- Widely negative orthant dependent random variables
- Complete convergence
- Complete moment convergence
- Marcinkiewicz–Zygmund type strong law of large numbers