Some inequalities for a LNQD sequence with applications
© Li et al.; licensee Springer 2012
Received: 3 February 2012
Accepted: 18 September 2012
Published: 2 October 2012
In this paper, some inequalities for a linearly negative quadrant dependent (LNQD) sequence are obtained. As their application, the asymptotic normality of the weight function estimate for a regression function is established, which extends the results of Roussas et al. (J. Multivar. Anal. 40:162-291, 1992) and Yang (Acta. Math. Sin. Engl. Ser. 23(6):1013-1024, 2007) for the strong mixing case to the LNQD case.
MSC:60E15, 62G08, 62E20.
We first recall the definitions of some dependent sequences.
Definition 1.1 (Lehmann )
A sequence of random variables is said to be pairwise negatively quadrant dependent (PNQD) if every pair of random variables in the sequence is NQD.
Definition 1.2 (Newman )
A sequence of random variables is said to be linearly negative quadrant dependent (LNQD) if for any disjoint subsets and positive ’s, and are NQD.
Definition 1.3 (Joag-Dev and Proschan )
where and are increasing for every variable (or decreasing for every variable) so that this covariance exists. An infinite sequence of random variables is said to be NA if every finite subfamily is NA.
Remark 1.1 (i) If is a sequence of LNQD random variables, then is still a sequence of LNQD random variables, where a and b are real numbers. (ii) NA implies LNQD from the definitions, but LNQD does not imply NA.
Because of wide applications of LNQD random variables, the concept of LNQD random variables has received more and more attention recently. For example, Newman  established the central limit theorem for a strictly stationary LNQD process; Wang and Zhang  provided uniform rates of convergence in the central limit theorem for LNQD sequence; Ko et al. obtained the Hoeffding-type inequality for LNQD sequence; Ko et al. studied the strong convergence for weighted sums of LNQD arrays; Wang et al. obtained some exponential inequalities for a linearly negative quadrant dependent sequence; Wu and Guan  obtained the mean convergence theorems for weighted sums of dependent random variables. In addition, from Remark 1.1, it is shown that LNQD is much weaker than NA and independent random variables. So, it is interesting to study some inequalities and their applications to a regression function for LNQD sequence.
The main results of this paper depend on the following lemmas.
Lemma 1.1 (Lehmann )
If f and g are both nondecreasing (or both nonincreasing) functions, then and are NQD.
Lemma 1.2 (Zhang )
2 Main results
Now, we state our main results with their proofs.
Proof The proof follows easily from the brief outline of the main points of the proof of Theorem 4.1 in Roussas , p.773]. □
By Theorem 2.1, we establish an inequality for characteristic function (c.f.) as follows:
This result, along with (2.8), completes the proof of the theorem. □
Remark 2.1 Let , in Theorem 2.3, we can get Lemma 3.1 of Ko et al.; let , , we also get Lemma 1.4 of Wang et al.. Thus, our Theorem 2.3 improves and extends Lemma 3.1 in Ko et al. and Lemma 1.4 in Wang et al..
Therefore, the proof is complete by (2.9) and (2.10). □
Proof We obtain the result from the proving process of Theorem 2.3 in Wang et al.. □
Theorem 2.5 Letbe a LNQD random variable sequence with zero mean and finite second moment, . Assume thatis a real constant sequence satisfying. Then for any, .
Combining (2.11)-(2.13), we get the result of the theorem. □
where a weight function , , depends on the fixed design points and on the number of observations n.
Here, our purpose is to use the inequalities in Section 2 to establish asymptotic normality for the estimate (3.2) under LNQD condition. The results obtained generalize the results of Roussas et al. and Yang  based on strong mixing sequence to LNQD sequence. Adopting the basic assumptions of Yang , we assume the following:
Assumption (A1) (i) is a bounded function defined on the compact subset A of ; (ii) is a strictly stationary and LNQD time series with , ; (iii) For each n, the joint distribution of is the same as that of .
Assumption (A2) (i) for all ; (ii) ; (iii) .
Assumption (A3) for and .
Here, we will prove the following result.
Thus (3.4) holds.
We now proceed with the proof of (3.5). Let and , then . Apply relation (3.4) to obtain . This would also imply that , provided we show that .
So, (3.9) holds. Thus, the proof is complete. □
This research is supported by the National Natural Science Foundation of China (11061029) and the Science Foundation of Jiangxi Education Department (GJJ12604).
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