Complete convergence for negatively orthant dependent random variables
© Qiu et al.; licensee Springer. 2014
Received: 14 November 2013
Accepted: 26 March 2014
Published: 9 April 2014
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.
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  proved the following result.
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. ), 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  in the following way.
where and are coordinatewise nondecreasing such that the covariance exists. An infinite sequence of is NA if every finite subfamily is NA.
- (a)negatively upper orthant dependent (NUOD) if
- (b)negatively lower orthant dependent (NLOD) if
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  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. )
If are Borel functions all of which are monotone increasing (or all monotone decreasing), then are NOD random variables.
Lemma 1.2 (Asadian et al. )
Proof By Lemma 1.2, the proof is similar to that of Theorem 2.3.1 in Stout , so it is omitted here. □
Lemma 1.4 (Kuczmaszewska )
if , then ;
We refer to Seneta  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 ).
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
Here , , .
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.
(2.5) ⇒ (2.6). The proof of (2.5) ⇒ (2.6) is similar to that of (2.2) ⇒ (2.4), so it is omitted. □
then (2.1)-(2.6) are equivalent.
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.
Thus, by (2.5) of Theorem 2.1, we have (2.15). □
Thus, by (2.2) of Theorem 2.1, we have (2.16). □
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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