# Article

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Keywords:
$m$-linearly negative quadrant dependence; mean convergence; complete convergence
Summary:
The structure of linearly negative quadrant dependent random variables is extended by introducing the structure of $m$-linearly negative quadrant dependent random variables ($m=1,2,\dots$). For a sequence of $m$-linearly negative quadrant dependent random variables $\{X_n, n\ge 1\}$ and $1<p<2$ (resp. $1\le p <2$), conditions are provided under which $n^{-1/p} \sum _{k=1}^{n} (X_k - EX_k) \to 0$ in $L^1$ (resp. in $L^p$). Moreover, for $1\le p < 2$, conditions are provided under which $n^{-1/p} \sum _{k=1}^{n} (X_k - EX_k)$ converges completely to $0$. The current work extends some results of Pyke and Root (1968) and it extends and improves some results of Wu, Wang, and Wu (2006). An open problem is posed.
References:
[1] Chandra, T. K.: Uniform integrability in the Cesàro sense and the weak law of large numbers. Sankhyā, Ser. A 51 (1989), 309-317. MR 1175608 | Zbl 0721.60024
[2] Fuk, D. H., Nagaev, S. V.: Probability inequalities for sums of independent random variables. Theory Probab. Appl. 16 (1971), 643-660. MR 0293695 | Zbl 0259.60024
[3] Hsu, P. L., Robbins, H.: Complete convergence and the law of large numbers. Proc. Natl. Acad. Sci. USA 33 (1947), 25-31. DOI 10.1073/pnas.33.2.25 | MR 0019852 | Zbl 0030.20101
[4] Joag-Dev, K., Proschan, F.: Negative association of random variables, with applications. Ann. Stat. 11 (1983), 286-295. DOI 10.1214/aos/1176346079 | MR 0684886 | Zbl 0508.62041
[5] Ko, M.-H., Choi, Y.-K., Choi, Y.-S.: Exponential probability inequality for linearly negative quadrant dependent random variables. Commun. Korean Math. Soc. 22 (2007), 137-143. DOI 10.4134/CKMS.2007.22.1.137 | MR 2286902 | Zbl 1168.60336
[6] Ko, M.-H., Ryu, D.-H., Kim, T.-S.: Limiting behaviors of weighted sums for linearly negative quadrant dependent random variables. Taiwanese J. Math. 11 (2007), 511-522. DOI 10.11650/twjm/1500404705 | MR 2333362 | Zbl 1126.60026
[7] Lehmann, E. L.: Some concepts of dependence. Ann. Math. Stat. 37 (1966), 1137-1153. DOI 10.1214/aoms/1177699260 | MR 0202228 | Zbl 0146.40601
[8] Newman, C. M.: Asymptotic independence and limit theorems for positively and negatively dependent random variables. Inequalities in Statistics and Probability. IMS Lecture Notes Monogr. Ser. 5 Y. L. Tong Inst. Math. Statist. Hayward (1984), 127-140. MR 0789244
[9] Cabrera, M. Ordóñez, Volodin, A. I.: Mean convergence theorems and weak laws of large numbers for weighted sums of random variables under a condition of weighted integrability. J. Math. Anal. Appl. 305 (2005), 644-658. DOI 10.1016/j.jmaa.2004.12.025 | MR 2131528
[10] Pyke, R., Root, D.: On convergence in $r$-mean of normalized partial sums. Ann. Math. Stat. 39 (1968), 379-381. DOI 10.1214/aoms/1177698400 | MR 0224137 | Zbl 0164.47303
[11] Sung, S. H., Lisawadi, S., Volodin, A.: Weak laws of large numbers for arrays under a condition of uniform integrability. J. Korean Math. Soc. 45 (2008), 289-300. DOI 10.4134/JKMS.2008.45.1.289 | MR 2375136 | Zbl 1136.60319
[12] Wan, C. G.: Law of large numbers and complete convergence for pairwise NQD random sequences. Acta Math. Appl. Sin. 28 (2005), 253-261 Chinese. MR 2157985
[13] Wang, J. F., Zhang, L. X.: A Berry-Esseen theorem for weakly negatively dependent random variables and its applications. Acta Math. Hung. 110 (2006), 293-308. DOI 10.1007/s10474-006-0024-x | MR 2213231 | Zbl 1121.60024
[14] Wang, X., Hu, S., Yang, W., Li, X.: Exponential inequalities and complete convergence for a LNQD sequence. J. Korean Statist. Soc. 39 (2010), 555-564. DOI 10.1016/j.jkss.2010.01.002 | MR 2780225 | Zbl 1294.60037
[15] Wu, Q., Wang, Y., Wu, Y.: On some limit theorems for sums of NA random matrix sequences. Chin. J. Appl. Probab. Stat. 22 (2006), 56-62 Chinese. MR 2275261 | Zbl 1167.60315

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