Sung, Soo Hak Complete convergence for weighted sums of \(\rho \ast \)-mixing random variables. (English) Zbl 1193.60045 Discrete Dyn. Nat. Soc. 2010, Article ID 630608, 13 p. (2010). Summary: We obtain the complete convergence for weighted sums of \(\rho ^{\ast }\)-mixing random variables. Our result extends the result of M. Peligrad and A. Gut [J. Theor. Probab. 12, No. 1, 87–104 (1999; Zbl 0928.60025)] on unweighted average to a weighted average under a mild condition of weights. Our result also generalizes and sharpens the result of J. An and D. Yuan [Stat. Probab. Lett. 78, No. 12, 1466–1472 (2008; Zbl 1155.60316)]. Cited in 6 ReviewsCited in 39 Documents MSC: 60F15 Strong limit theorems Citations:Zbl 0928.60025; Zbl 1155.60316 × Cite Format Result Cite Review PDF Full Text: DOI EuDML References: [1] R. C. Bradley, “On the spectral density and asymptotic normality of weakly dependent random fields,” Journal of Theoretical Probability, vol. 5, no. 2, pp. 355-373, 1992. · Zbl 0787.60059 · doi:10.1007/BF01046741 [2] W. Bryc and W. Smoleński, “Moment conditions for almost sure convergence of weakly correlated random variables,” Proceedings of the American Mathematical Society, vol. 119, no. 2, pp. 629-635, 1993. · Zbl 0785.60018 · doi:10.2307/2159950 [3] M. Peligrad and A. Gut, “Almost-sure results for a class of dependent random variables,” Journal of Theoretical Probability, vol. 12, no. 1, pp. 87-104, 1999. · Zbl 0928.60025 · doi:10.1023/A:1021744626773 [4] S. Utev and M. Peligrad, “Maximal inequalities and an invariance principle for a class of weakly dependent random variables,” Journal of Theoretical Probability, vol. 16, no. 1, pp. 101-115, 2003. · Zbl 1012.60022 · doi:10.1023/A:1022278404634 [5] S. Gan, “Almost sure convergence for \rho \~-mixing random variable sequences,” Statistics & Probability Letters, vol. 67, no. 4, pp. 289-298, 2004. · Zbl 1043.60023 · doi:10.1016/j.spl.2003.12.011 [6] A. Kuczmaszewska, “On Chung-Teicher type strong law of large numbers for \rho \ast -mixing random variables,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 140548, 10 pages, 2008. · Zbl 1145.60308 · doi:10.1155/2008/140548 [7] Q. Wu and Y. Jiang, “Some strong limit theorems for \rho \~-mixing sequences of random variables,” Statistics & Probability Letters, vol. 78, no. 8, pp. 1017-1023, 2008. · Zbl 1148.60020 · doi:10.1016/j.spl.2007.09.061 [8] J. An and D. Yuan, “Complete convergence of weighted sums for \rho \ast -mixing sequence of random variables,” Statistics & Probability Letters, vol. 78, no. 12, pp. 1466-1472, 2008. · Zbl 1155.60316 · doi:10.1016/j.spl.2007.12.020 [9] G.-H. Cai, “Strong law of large numbers for \rho \ast -mixing sequences with different distributions,” Discrete Dynamics in Nature and Society, vol. 2006, Article ID 27648, 7 pages, 2006. · Zbl 1110.60020 · doi:10.1155/DDNS/2006/27648 [10] A. Kuczmaszewska, “On complete convergence for arrays of rowwise dependent random variables,” Statistics & Probability Letters, vol. 77, no. 11, pp. 1050-1060, 2007. · Zbl 1120.60025 · doi:10.1016/j.spl.2006.12.007 [11] M.-H. Zhu, “Strong laws of large numbers for arrays of rowwise \rho \ast -mixing random variables,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 74296, 6 pages, 2007. · Zbl 1181.60044 · doi:10.1155/2007/74296 [12] P. L. Hsu and H. Robbins, “Complete convergence and the law of large numbers,” Proceedings of the National Academy of Sciences of the United States of America, vol. 33, pp. 25-31, 1947. · Zbl 0030.20101 · doi:10.1073/pnas.33.2.25 [13] P. Erdös, “On a theorem of Hsu and Robbins,” Annals of Mathematical Statistics, vol. 20, pp. 286-291, 1949. · Zbl 0033.29001 · doi:10.1214/aoms/1177730037 [14] L. E. Baum and M. Katz, “Convergence rates in the law of large numbers,” Transactions of the American Mathematical Society, vol. 120, pp. 108-123, 1965. · Zbl 0142.14802 · doi:10.2307/1994170 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.