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Complete convergence for negatively orthant dependent random variables
Journal of Inequalities and Applications volume 2014, Article number: 145 (2014)
Abstract
In this paper, necessary and sufficient conditions of the complete convergence are obtained for the maximum partial sums of negatively orthant dependent (NOD) random variables. The results extend and improve those in Kuczmaszewska (Acta Math. Hung. 128(1-2):116-130, 2010) for negatively associated (NA) random variables.
MSC:60F15, 60G50.
1 Introduction
The concept of complete convergence for a sequence of random variables was introduced by Hsu and Robbins [1] as follows. A sequence of random variables converges completely to the constant θ if
Moreover, they proved that the sequence of arithmetic means of independent identically distribution (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 by many authors. One can refer to [2–16], and so forth. Kuczmaszewska [8] proved the following result.
Theorem A Let be a sequence of negatively associated (NA) random variables and X be a random variables possibly defined on a different space satisfying the condition
for all , all and some positive constant D. Let and . Moreover, additionally assume that for all if . Then the following statements are equivalent:
-
(i)
,
-
(ii)
, .
The aim of this paper is to extend and improve Theorem A to negatively orthant dependent (NOD) random variables. The tool in the proof of Theorem A is the Rosenthal maximal inequality for NA sequence (cf. [17]), but no one established the kind of maximal inequality for NOD sequence. So the truncated method is different and the proofs of our main results are more complicated and difficult.
The concept of negatively associated (NA) and negatively orthant dependent (NOD) was introduced by Joag-Dev and Proschan [18] in the following way.
Definition 1.1 A finite family of random variables is said to be negatively associated (NA) if for every pair of disjoint nonempty subset , of ,
where and are coordinatewise nondecreasing such that the covariance exists. An infinite sequence of is NA if every finite subfamily is NA.
Definition 1.2 A finite family of random variables is said to be
-
(a)
negatively upper orthant dependent (NUOD) if
for ,
-
(b)
negatively lower orthant dependent (NLOD) if
for ,
-
(c)
negatively orthant dependent (NOD) if they are both NUOD and NLOD.
A sequence of random variables is said to be NOD if for each n, are NOD.
Obviously, every sequence of independent random variables is NOD. Joag-Dev and Proschan [18] pointed out that NA implies NOD, neither being NUOD nor being NLOD implies being NA. They gave an example that possesses NOD, but does not possess NA, which shows that NOD is strictly wider than NA. For more details of NOD random variables, one can refer to [3, 6, 11, 14, 19–21], and so forth.
In order to prove our main results, we need the following lemmas.
Lemma 1.1 (Bozorgnia et al. [19])
Let be NOD random variables.
-
(i)
If are Borel functions all of which are monotone increasing (or all monotone decreasing), then are NOD random variables.
-
(ii)
, .
Lemma 1.2 (Asadian et al. [22])
For any , there is a positive constant depending only on q such that if is a sequence of NOD random variables with for every , then for all ,
Lemma 1.3 For any , there is a positive constant depending only on q such that if is a sequence of NOD random variables with for every , then for all ,
Proof By Lemma 1.2, the proof is similar to that of Theorem 2.3.1 in Stout [23], so it is omitted here. □
Lemma 1.4 (Kuczmaszewska [8])
Let β, γ be positive constants. Suppose that is a sequence of random variables and X is a random variable. There exists constant such that
-
(i)
if , then ;
-
(ii)
;
-
(iii)
.
Recall that a function is said to be slowly varying at infinity if it is real valued, positive, and measurable on , and if for each
We refer to Seneta [24] for other equivalent definitions and for a detailed and comprehensive study of properties of slowly varying functions.
We frequently use the following properties of slowly varying functions (cf. Seneta [24]).
Lemma 1.5 If is a function slowly varying at infinity, then for any
and
where depend only on s.
Throughout this paper, C will represent positive constants of which the value may change from one place to another.
2 Main results and proofs
Theorem 2.1 Let , , and be a slowly varying function at infinity. Let 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, additionally assume that for , for all . If
then the following statements hold:
Here , , .
Proof First, we prove (2.2). Choose q such that . Let , , , , . Note that
and
In order to prove (2.2), it suffices to show that for . Obviously, for , the condition (2.1) implies . Therefore, we choose , and if . In order to prove , we first prove that
This holds when . Since , . By , , and Lemma 1.4, we have
When , ,
When , ,
Therefore, (2.8) holds. So, in order to prove , it is enough to prove that
By Lemma 1.1 for , is a sequence of NOD random variables. When , by and , we have . Taking v such that , we get by the Markov inequality, the inequality, the Hölder inequality, and Lemma 1.3,
By the inequality, Lemma 1.4, and Lemma 1.5, we have
By the inequality and Lemma 1.4,
When ,
When ,
Therefore, (2.9) holds for . Define , , , since , we have
By Lemma 1.5, (2.1), and a standard computation, we have
Now we prove . By (2.1) and Lemma 1.4, we have
Therefore, in order to prove , it is enough to prove that
Taking v such that , we get by Lemma 1.1, the Markov inequality, the inequality, the Hölder inequality, and Lemma 1.2,
By the inequality, Lemma 1.4, Lemma 1.5, (2.1), and a standard computation, we have
and
Therefore, (2.12) holds. By (2.10)-(2.12) we get . In a similar way of we can obtain . Thus, (2.2) holds.
(2.2) ⇒ (2.3). Note that , we have , hence, from (2.2), (2.3) holds.
(2.3) ⇒ (2.4). Since , , and , we have , , hence, from (2.3), (2.4) holds.
(2.2) ⇒ (2.5). By Lemma 1.5 and (2.3), we have
(2.5) ⇒ (2.6). The proof of (2.5) ⇒ (2.6) is similar to that of (2.2) ⇒ (2.4), so it is omitted. □
Theorem 2.2 Let , , and be a slowly varying function at infinity. Let be a sequence of NOD random variables and X be a random variables possibly defined on a different space. Moreover, additionally assume that for , for all . If there exist constant and such that
then (2.1)-(2.6) are equivalent.
Proof From the proof of Theorem 2.1, in order to prove Theorem 2.2, it is enough to show that (2.4) ⇒ (2.6) and (2.6) ⇒ (2.1). The proof of (2.4) ⇒ (2.6) is similar to that of (2.2) ⇒ (2.5). Now, we prove (2.6) ⇒ (2.1). Firstly we prove that
Otherwise, there are , , and a sequence of positive integers , such that , . Without loss of generality, we can assume that , . Therefore, we have
By we have
which is in contradiction with (2.6), thus, (2.13) holds. By Lemma 1.1, we get
By (2.13), we have , . Therefore, when n is large enough, we have
In a similar way, when n is large enough,
Thus, when n is large enough, we have
Taking , by (2.6), (2.14), Lemma 1.5, and a standard computation, we have
Thus, (2.1) holds. □
In the following, let be a sequence of non-negative, integer valued random variables and τ a positive random variable. All random variables are defined on the same probability space.
Theorem 2.3 Let , , and be a slowly varying function as . Let be a sequence of NOD random variables and X be a random variables possibly defined on a different space satisfying the condition (1.1) and (2.1). Moreover, additionally assume that for , for all . If there exists such that , then
Proof Note that
Thus, by (2.5) of Theorem 2.1, we have (2.15). □
Theorem 2.4 Let , , and be a slowly varying function at infinity. Let be a sequence of NOD random variables and X be a random variables possibly defined on a different space satisfying the condition (1.1) and (2.1). Moreover, additionally assume that for , for all . If there exists such that with for some , then
Proof Note that
Thus, by (2.2) of Theorem 2.1, we have (2.16). □
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The authors would like to thank the referees and the editors for the helpful comments and suggestions. This work was supported by the National Natural Science Foundation of China (Grant No. 11271161).
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Qiu, D., Wu, Q. & Chen, P. Complete convergence for negatively orthant dependent random variables. J Inequal Appl 2014, 145 (2014). https://doi.org/10.1186/1029-242X-2014-145
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DOI: https://doi.org/10.1186/1029-242X-2014-145