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Complete convergence of weighted sums for arrays of rowwise \(\varphi \)-mixing random variables. (English) Zbl 1340.60045

The authors consider arrays of rowwise \(\varphi\)-mixing random variables and study necessary and sufficient conditions under which weighted sums of such random variables converge completely and satisfy also the complete moment convergence. Using the standard technique, they establish a variant of the Baum-Katz-type theorem for these arrays of weakly dependent random variables and generalize some recent results in this field (see [X.-J. Wang and S.-H. Hu, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 106, No. 1, 321–331 (2012; Zbl 1266.60056)]).

MSC:

60F15 Strong limit theorems
60E15 Inequalities; stochastic orderings
60F99 Limit theorems in probability theory
60B10 Convergence of probability measures

Citations:

Zbl 1266.60056
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References:

[1] J.-I. Baek, I.-B. Choi, S.-L. Niu: On the complete convergence of weighted sums for arrays of negatively associated variables. J. Korean Stat. Soc. 37 (2008), 73-80. · Zbl 1298.60039 · doi:10.1016/j.jkss.2007.08.001
[2] J.-I. Baek, S.-T. Park: Convergence of weighted sums for arrays of negatively dependent random variables and its applications. RETRACTED. J. Theor. Probab. 23 (2010), 362-377; retraction ibid. 26 (2013), 899-900. · Zbl 1196.60045 · doi:10.1007/s10959-008-0198-y
[3] L. E. Baum, M. Katz: Convergence rates in the law of large numbers. Trans. Am. Math. Soc. 120 (1965), 108-123. · Zbl 0142.14802 · doi:10.1090/S0002-9947-1965-0198524-1
[4] P. Chen, T.-C. Hu, X. Liu, A. Volodin: On complete convergence for arrays of row-wise negatively associated random variables. Theory Probab. Appl. 52 (2008), 323-328; and Teor. Veroyatn. Primen. 52 (2007), 393-397. · Zbl 1146.60025 · doi:10.1137/S0040585X97983079
[5] P. Chen, T.-C. Hu, A. Volodin: Limiting behaviour of moving average processes under negative association assumption. Theory Probab. Math. Stat. 77 (2008), 165-176; and Teor. Jmovirn. Mat. Stat. 77 (2007), 149-160. · Zbl 1199.60074 · doi:10.1090/S0094-9000-09-00755-8
[6] R. L. Dobrushin: Central limit theorem for non-stationary Markov chains. I, II. Teor. Veroyatn. Primen. 1 (1956), 72-89; Berichtigung. Ibid. 3 (1958), 477. · Zbl 0093.15001
[7] P. Erdős: On a theorem of Hsu and Robbins. Ann. Math. Stat. 20 (1949), 286-291. · Zbl 0033.29001 · doi:10.1214/aoms/1177730037
[8] M. L. Guo: Complete moment convergence of weighted sums for arrays of rowwise φ-mixing random variables. Int. J. Math. Math. Sci. 2012 (2012), Article ID 730962, 13 pp. · Zbl 1253.60050
[9] P. L. Hsu, H. Robbins: Complete convergence and the law of large numbers. Proc. Natl. Acad. Sci. USA 33 (1947), 25-31. · Zbl 0030.20101 · doi:10.1073/pnas.33.2.25
[10] T.-C. Hu, M. Ordóñez Cabrera, S. H. Sung, A. Volodin: Complete convergence for arrays of rowwise independent random variables. Commun. Korean Math. Soc. 18 (2003), 375-383. · Zbl 1101.60324 · doi:10.4134/CKMS.2003.18.2.375
[11] A. Jun, Y. Demei: Complete convergence of weighted sums for ϱ*-mixing sequence of random variables. Stat. Probab. Lett. 78 (2008), 1466-1472. · Zbl 1155.60316 · doi:10.1016/j.spl.2007.12.020
[12] V. M. Kruglov, A. I. Volodin, T.-C. Hu: On complete convergence for arrays. Stat. Probab. Lett. 76 (2006), 1631-1640. · Zbl 1100.60014 · doi:10.1016/j.spl.2006.04.006
[13] A. Kuczmaszewska: On complete convergence for arrays of rowwise dependent random variables. Stat. Probab. Lett. 77 (2007), 1050-1060. · Zbl 1120.60025 · doi:10.1016/j.spl.2006.12.007
[14] A. Kuczmaszewska: On complete convergence for arrays of rowwise negatively associated random variables. Stat. Probab. Lett. 79 (2009), 116-124. · Zbl 1154.60319 · doi:10.1016/j.spl.2008.07.030
[15] M. Peligrad: Convergence rates of the strong law for stationary mixing sequences. Z. Wahrscheinlichkeitstheor. Verw. Geb. 70 (1985), 307-314. · Zbl 0554.60038 · doi:10.1007/BF02451434
[16] M. Peligrad, A. Gut: Almost-sure results for a class of dependent random variables. J. Theor. Probab. 12 (1999), 87-104. · Zbl 0928.60025 · doi:10.1023/A:1021744626773
[17] D. H. Qiu, T.-C. Hu, M. O. Cabrera, A. Volodin: Complete convergence for weighted sums of arrays of Banach-space-valued random elements. Lith. Math. J. 52 (2012), 316-325. · Zbl 1263.60024 · doi:10.1007/s10986-012-9175-3
[18] Q. M. Shao: A moment inequality and its applications. Acta Math. Sin. 31 (1988), 736-747. (In Chinese.) · Zbl 0698.60025
[19] A. T. Shen, X. H. Wang, J. M. Ling: On complete convergence for non-stationary φ-mixing random variables. Commun. Stat. Theory Methods, DOI:10.1080/03610926.2012.725501. · Zbl 0698.60025
[20] G. Stoica: Baum-Katz-Nagaev type results for martingales. J. Math. Anal. Appl. 336 (2007), 1489-1492. · Zbl 1130.60020 · doi:10.1016/j.jmaa.2007.03.012
[21] G. Stoica: A note on the rate of convergence in the strong law of large numbers for martingales. J. Math. Anal. Appl. 381 (2011), 910-913. · Zbl 1237.60025 · doi:10.1016/j.jmaa.2011.04.008
[22] S. H. Sung: Moment inequalities and complete moment convergence. J. Inequal. Appl. 2009 (2009), Article ID 271265, 14 pp. · Zbl 1146.60025
[23] S. H. Sung: Complete convergence for weighted sums ofϱ*-mixing random variables. Discrete Dyn. Nat. Soc. 2010 (2010), Article ID 630608, 13 pp. · Zbl 0142.14802
[24] S. H. Sung: On complete convergence for weighted sums of arrays of dependent random variables. Abstr. Appl. Anal. 2011 (2011), Article ID 630583, 11 pp. · Zbl 1231.60025
[25] S. H. Sung, A. I. Volodin, T.-C. Hu: More on complete convergence for arrays. Stat. Probab. Lett. 71 (2005), 303-311. · Zbl 1087.60030 · doi:10.1016/j.spl.2004.11.006
[26] X. J. Wang, S. H. Hu: Some Baum-Katz type results for φ-mixing random variables with different distributions. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 106 (2012), 321-331. · Zbl 1266.60056 · doi:10.1007/s13398-011-0056-0
[27] X. J. Wang, S. H. Hu, W. Z. Yang, Y. Shen: On complete convergence for weighted sums of φ-mixing random variables. J. Inequal. Appl. 2010 (2010), Article ID 372390, 13 pp. · Zbl 1208.60031
[28] X. J. Wang, S. H. Hu, W. Z. Yang, X. H. Wang: Convergence rates in the strong law of large numbers for martingale difference sequences. Abstr. Appl. Anal. 2012 (2012), Article ID 572493, 13 pp. · Zbl 1253.60045
[29] X. J. Wang, S. H. Hu, W. Z. Yang, X. H. Wang: On complete convergence of weighted sums for arrays of rowwise asymptotically almost negatively associated random variables. Abstr. Appl. Anal. 2012 (2012), Article ID 315138, 15 pp. · Zbl 1253.60044
[30] Q. Y. Wu: Probability Limit Theory for Mixed Sequence. China Science Press, Beijing, 2006. (In Chinese.)
[31] Q. Y. Wu: A complete convergence theorem for weighted sums of arrays of rowwise negatively dependent random variables. J. Inequal. Appl. 2012 (2012), Article ID 50, 10 pp. (electronic only). · Zbl 1293.62053
[32] L. X. Zhang, J. W. Wen: The strong law of large numbers for B-valued random fields. Chin. Ann. Math., Ser. A 22 (2001), 205-216. (In Chinese.) · Zbl 0983.60016
[33] X. C. Zhou, J. G. Lin: On complete convergence for arrays of rowwise ϱ-mixing random variables and its applications. J. Inequal. Appl. 2010 (2010), Article ID 769201, 12 pp. · Zbl 1208.60032
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