Some mean convergence and complete convergence theorems for sequences of \(m\)-linearly negative quadrant dependent random variables. (English) Zbl 1299.60047

Appl. Math., Praha 58, No. 5, 511-529 (2013); correction ibid. 62, No. 2, 209-211 (2017).
In the paper, a new type of dependence in a sequence of random variables \(\{X_n:n\geq 1\}\), called \(m\)-linear negative quadrant dependence, is introduced. For such variables, the convergence of \(n^{-1/p}\sum _{k=1}^n (X_k-\operatorname{E} X_k)\) to zero is proved in \(L_p\) and in the sense of complete convergence if \(1 \leq p < 2.\) A Kolmogorov-type exponential inequality is also established as a by product.


60F15 Strong limit theorems
60F25 \(L^p\)-limit theorems
Full Text: DOI


[1] Chandra, T.K., Uniform integrability in the Cesàro sense and the weak law of large numbers, Sankhyā, Ser. A, 51, 309-317, (1989) · Zbl 0721.60024
[2] Fuk, D.H.; Nagaev, S.V., Probability inequalities for sums of independent random variables, Theory Probab. Appl., 16, 643-660, (1971) · Zbl 0259.60024
[3] Hsu, P. L.; Robbins, H., Complete convergence and the law of large numbers, Proc. Natl. Acad. Sci. USA, 33, 25-31, (1947) · Zbl 0030.20101
[4] Joag-Dev, K.; Proschan, F., Negative association of random variables, with applications, Ann. Stat., 11, 286-295, (1983) · 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, 137-143, (2007) · 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, 511-522, (2007) · Zbl 1126.60026
[7] Lehmann, E. L., Some concepts of dependence, Ann. Math. Stat., 37, 1137-1153, (1966) · Zbl 0146.40601
[8] Newman, C.M.; Tong, Y.L. (ed.), Asymptotic independence and limit theorems for positively and negatively dependent random variables, No. 5, 127-140, (1984), Hayward
[9] Ordóñez Cabrera, M.; 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, 644-658, (2005) · Zbl 1065.60022
[10] Pyke, R.; Root, D., On convergence in r-Mean of normalized partial sums, Ann. Math. Stat., 39, 379-381, (1968) · 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, 289-300, (2008) · Zbl 1136.60319
[12] Wan, C.G., Law of large numbers and complete convergence for pairwise NQD random sequences, Acta Math. Appl. Sin., 28, 253-261, (2005)
[13] Wang, J.F.; Zhang, L.X., A Berry-Esseen theorem for weakly negatively dependent random variables and its applications, Acta Math. Hung., 110, 293-308, (2006) · 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, 555-564, (2010) · 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, 56-62, (2006) · Zbl 1167.60315
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.