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Some inequalities for a LNQD sequence with applications
Journal of Inequalities and Applications volume 2012, Article number: 216 (2012)
Abstract
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.
1 Introduction
We first recall the definitions of some dependent sequences.
Definition 1.1 (Lehmann [1])
Two random variables X and Y are said to be negative quadrant dependent (NQD) if
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 [2])
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 [3])
Random variables are said to be negatively associated (NA) if for every pair of disjoint subsets and of ,
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 [2] established the central limit theorem for a strictly stationary LNQD process; Wang and Zhang [4] provided uniform rates of convergence in the central limit theorem for LNQD sequence; Ko et al.[5] obtained the Hoeffding-type inequality for LNQD sequence; Ko et al.[6] studied the strong convergence for weighted sums of LNQD arrays; Wang et al.[7] obtained some exponential inequalities for a linearly negative quadrant dependent sequence; Wu and Guan [8] 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 [1])
Let random variables X and Y be NQD, then
-
(i)
;
-
(ii)
If f and g are both nondecreasing (or both nonincreasing) functions, then and are NQD.
Lemma 1.2 (Zhang [4])
Suppose thatis a sequence of LNQD random variables with. Then for any, there exists a positive constant D such that
2 Main results
Now, we state our main results with their proofs.
Theorem 2.1 Let X and Y be NQD random variables with finite second moments. If f and g are complex-valued functions defined on R with bounded derivativesand, then
Proof The proof follows easily from the brief outline of the main points of the proof of Theorem 4.1 in Roussas [9], p.773]. □
By Theorem 2.1, we establish an inequality for characteristic function (c.f.) as follows:
Theorem 2.2 Ifare LNQD random variables with finite second moments, letandbe c.f.’s ofand, respectively, then for all nonnegative (or nonpositive) real numbers,
Proof Write
Further notice that . Thus,
By the definition of LNQD, it is easy to see that and are NQD for . Then by Theorem 2.1, we can obtain that
Similarly as above, we have
From (2.2) to (2.4), we obtain
Therefore, in view of (2.1) and (2.5), we obtain that
For , using the same decomposition as in (2.1) above, we obtain
Similarly to the calculation of , we get
Thus, from (2.6) and (2.7), constantly repeating the above procedure, we get
Note that for , and are NQD by the definition of LNQD. Similarly as above, we obtain that
This result, along with (2.8), completes the proof of the theorem. □
Theorem 2.3 Letbe a sequence of LNQD random variables, and letbe all nonnegative (or nonpositive) real numbers. Then
Remark 2.1 Let , in Theorem 2.3, we can get Lemma 3.1 of Ko et al.[5]; let , , we also get Lemma 1.4 of Wang et al.[7]. Thus, our Theorem 2.3 improves and extends Lemma 3.1 in Ko et al.[5] and Lemma 1.4 in Wang et al.[7].
Proof For , it is easy to see that and are NQD by the definition of LNQD, which implies that and are also NQD for by Lemma 1.1(ii). Then by Lemma 1.1(i) and induction,
For , it is easy to see that and and are NQD by the definition of LNQD, which implies that and are also NQD for by Lemma 1.1(ii). Similar to the proof of (2.9), we obtain
Therefore, the proof is complete by (2.9) and (2.10). □
Theorem 2.4 Suppose thatis a LNQD random variable sequence with zero mean anda.s. (). Letand. Then for any,
Proof We obtain the result from the proving process of Theorem 2.3 in Wang et al.[7]. □
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, .
Proof Let , . Notice that
Let . Then is still a sequence of LNQD random variables with by Remark 1.1. Note that . By Lemma 1.2, we obtain
Similarly as above, we have
Combining (2.11)-(2.13), we get the result of the theorem. □
3 Application
To show the application of the inequalities in Section 2, in this section we discuss the asymptotic normality of the general linear estimator for the following regression model:
where the design points , which is a compact set of , g is a bounded real valued function on A, and the are regression errors with zero mean and finite variance . As an estimate of , we consider the following general linear smoother:
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.[10] and Yang [11] based on strong mixing sequence to LNQD sequence. Adopting the basic assumptions of Yang [11], 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 .
Denote
Assumption (A2) (i) for all ; (ii) ; (iii) .
Assumption (A3) for and .
Assumption (A4) There exist positive integers and such that for sufficiently large n and as ,
Here, we will prove the following result.
Theorem 3.1 Let Assumptions (A1)∼(A4) be satisfied. Then
Proof We first give some denotations. For convenience of writing, omit everywhere the argument x and set , for , so that . Let . Then may be split as , where
Thus, to prove the theorem, it suffices to show that
By Theorem 2.5, Assumptions (A2)(ii)∼(iii) and (A4)(i)∼(iii), we have
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 .
Indeed, by Assumption (A3) and , we obtain . Then by stationarity and Assumption (A2), it can be shown that
Next, in order to establish asymptotic normality, we assume that are independent random variables, and the distribution of is the same as that for . Then and . Let , , then are independent random variables with and . Let be the characteristic function of X, then
By Theorem 2.2, relation (3.6) and Assumption (A2), we obtain that
Thus, it suffices to show that which, on account of , will follow from the convergence . By the Lyapunov condition, it suffices to show that for some ,
Using Theorem 2.5 and Assumptions (A2) and (A4)(iv), we have
So, (3.9) holds. Thus, the proof is complete. □
References
Lehmann EL: Some concepts of dependence. Ann. Math. Stat. 1966, 37: 1137–1153. 10.1214/aoms/1177699260
Newman CM: Asymptotic independence and limit theorems for positively and negatively dependent random variables. 5. In Statistics and Probability. Edited by: Tong YL. Inst. Math. Statist., Hayward; 1984:127–140.
Joag-Dev K, Proschan F: Negative association of random variables with applications. Ann. Stat. 1983, 11(1):286–295. 10.1214/aos/1176346079
Zhang LX: A functional central limit theorem for asymptotically negatively dependent random fields. Acta Math. Hung. 2000, 86: 237–259. 10.1023/A:1006720512467
Ko MH, Choi YK, Choi YS: Exponential probability inequality for linearly negative quadrant dependent random variables. Commun. Korean Math. Soc. 2007, 22: 137–143. 10.4134/CKMS.2007.22.1.137
Ko MH, Ryu DH, Kim TS: Limiting behaviors of weighted sums for linearly negative quadrant dependent random variables. Taiwan. J. Math. 2007, 11(2):511–522.
Wang XJ, Hu SH, Yang WZ, Li XQ: Exponential inequalities and complete convergence for a LNQD sequence. J. Korean Stat. Soc. 2010, 39: 555–564. 10.1016/j.jkss.2010.01.002
Wu YF, Guan M: Mean convergence theorems and weak laws of large numbers for weighted sums of dependent random variables. J. Math. Anal. Appl. 2011, 377: 613–623. 10.1016/j.jmaa.2010.11.042
Roussas GG: Positive and negative dependence with some statistical applications. In Asymptotics, Nonparametrics, and Time Series. Edited by: Ghosh S, Puri ML. CRC Press, Boca Raton; 1999:757–787.
Roussas GG, Tran LT, Ioannides DA: Fixed design regression for time series: asymptiotic normality. J. Multivar. Anal. 1992, 40: 162–291.
Yang SC: Maximal moment inequality for partial sum of strong mixing sequences and application. Acta Math. Sin. Engl. Ser. 2007, 23(6):1013–1024. 10.1007/s10114-005-0841-9
Acknowledgements
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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Li, Y., Guo, J. & Li, N. Some inequalities for a LNQD sequence with applications. J Inequal Appl 2012, 216 (2012). https://doi.org/10.1186/1029-242X-2012-216
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DOI: https://doi.org/10.1186/1029-242X-2012-216