Complete moment convergence for randomly weighted sums of arrays of rowwise \(m_n\)-extended negatively dependent random variables and its applications. (English) Zbl 1507.60040
Authors’ abstract: In this paper, a new concept of rowwise \(m_n\)-extended negatively dependent (rowwise \(m_n\)-END) random variables, which is wider than some known dependent structures such as END, widely orthant dependence (WOD) and so on, is introduced, and complete moment convergence for randomly weighted sums of rowwise \(m_n\)-END arrays are investigated under some proper and sufficient conditions. In addition, a relationship between \(\{m_n,\,n\geq 1\}\), dominating sequence and moment conditions for convergence is in a sense revealed. The results obtained in the paper generalise some corresponding ones for independent and some dependent random variables. As applications, strong convergence for elements of a linear autoregression model and complete convergence for randomly indexed sums of sequences of WOD random variables are established.
Reviewer: Edward Omey (Brussel)
MSC:
| 60F15 | Strong limit theorems |
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
Keywords:
complete moment convergence; rowwise \(m_n\)-end arrays; strong law of large numbers; randomly indexed sumsReferences:
| [1] | Adler, A.; Rosalsky, A., Some general strong laws for weighted sums of stochastically dominated random variables, Stoch. Anal. Appl., 5, 1, 1-16 (1987) · Zbl 0617.60028 · doi:10.1080/07362998708809104 |
| [2] | Adler, A.; Rosalsky, A.; Taylor, RL, Strong laws of large numbers for weighted sums of random elements in normed linear spaces, Int. J. Math. Math. Sci., 12, 507-529 (1989) · Zbl 0679.60009 · doi:10.1155/S0161171289000657 |
| [3] | Gut, A., Complete convergence and convergence rates for randomly indexed partial sums with an application to some first passage times, Acta Math. Hung., 42, 3-4, 225-232 (1983) · Zbl 0536.60036 · doi:10.1007/BF01956770 |
| [4] | Kuczmaszewska, A.; Yan, JG, On complete convergence in Marcinkiewicz-Zygmund type SLLN for random variables, Appl. Math. J. Chin. Univ., 36, 3, 342-353 (2021) · Zbl 1488.60054 · doi:10.1007/s11766-021-3816-4 |
| [5] | Shen, A. T.: Probability inequalities for END sequence and their applications. J. inequal. Appl. 98(1) (2011). doi:10.1186/1029-242X-2011-98 · Zbl 1276.60019 |
| [6] | Stone, CJ, Consistent nonparametric regression, Ann. Stat., 5, 4, 595-620 (1977) · Zbl 0366.62051 · doi:10.1214/aos/1176343886 |
| [7] | Zhang, CY; Liang, Q., On complete convergence of randomly indexed partial sums and first passage times of NA sequences, J. Eng. Math., 15, 2, 85-90 (1998) · Zbl 0930.60014 |
| [8] | Buraczewski, D.; Damek, E.; Mikosch, T., Stochastic Models with Power Law Tails. The Equation \(X=AX+B (2016)\), Cham: Springer, Cham · Zbl 1357.60004 · doi:10.1007/978-3-319-29679-1 |
| [9] | Szynal, D., On almost complete convergence for the sum of a random number of independent random variables, Bull. Acad. Sci. Polon. Ser. Math. Astronom. Phys., 20, 571-574 (1972) · Zbl 0253.60057 |
| [10] | Lu, DW; Song, LX; Zhang, T., Large deviations for sum of UEND and \(\varphi \)-mixing random variables with heavy tails, Commun. Stat. Theory Methods, 45, 7, 2118-2129 (2013) · Zbl 1339.60022 · doi:10.1080/03610926.2013.873134 |
| [11] | Seneta, E., Regularly Varying Function (1976), Berlin: Springer, Berlin · Zbl 0324.26002 · doi:10.1007/BFb0079658 |
| [12] | Tang, GW, On complete convergence of randomly indexed partial Sum of strongly mixing random variables sequences, J. Guangxi Teach. Edu. Univ. (Natural Science Ed.), 21, 2, 9-12 (2004) |
| [13] | Fang, H. Y., Ding, S. S., Li, X. Q., Yang, W. Z.: Asymptotic approximations of ratio moments based on dependent sequences. Mathematics 8(361) (2020). doi:10.3390/math8030361 |
| [14] | Lita da Silva, J., Convergence in \(p\)-mean for arrays of row-wise extended negatively dependent random variables, Acta Math. Hungar., 150, 2, 346-362 (2016) · Zbl 1399.60067 · doi:10.1007/s10474-016-0645-7 |
| [15] | Lita da Silva, J., Limiting behaviour for arrays of row-wise upper extended negatively dependent random variables, Acta Math. Hungar., 148, 2, 481-492 (2016) · Zbl 1374.60049 · doi:10.1007/s10474-016-0585-2 |
| [16] | Lita da Silva, J., Almost sure convergence for weighted sums of extended negatively dependent random variables, Acta Math. Hungar., 146, 1, 56-70 (2015) · Zbl 1363.60036 · doi:10.1007/s10474-015-0502-0 |
| [17] | Lita da Silva, J., Strong laws of large numbers for arrays of row-wise extended negatively dependent random variables with applications, J. Nonparametr. Stat., 32, 1, 20-41 (2020) · Zbl 1435.62132 · doi:10.1080/10485252.2019.1688326 |
| [18] | Yan, JG, The uniform convergence and precise asymptotics of generalized stochastic order statistics, J. Math. Anal. Appl., 343, 644-653 (2008) · Zbl 1135.62037 · doi:10.1016/j.jmaa.2008.01.072 |
| [19] | Yan, JG, Strong stability of a type of Jamison weighted sums for END random variables, J. Korean Math. Soc., 54, 3, 897-907 (2017) · Zbl 1365.60026 · doi:10.4134/JKMS.j160300 |
| [20] | Yan, JG, Almost sure convergence for weighted sums of WNOD random variables and its applications to non parametric regression models, Commun. Stat. Theory Methods, 47, 16, 3893-3909 (2018) · Zbl 1508.60044 · doi:10.1080/03610926.2017.1364390 |
| [21] | Yan, JG, Complete convergence and complete moment convergence for maximal weighted sums of extended negatively dependent random variables, Acta Math. Sin. Engl. Ser., 34, 10, 1501-1516 (2018) · Zbl 1404.60042 · doi:10.1007/s10114-018-7133-7 |
| [22] | Yan, JG, Complete convergence in Marcinkiewicz-Zygmund type SLLN for END random variables and its applications, Commun. Stat. Theory Methods, 48, 20, 5074-5098 (2019) · Zbl 07529839 · doi:10.1080/03610926.2018.1508709 |
| [23] | Wang, KY; Wang, YB; Gao, QW, Uniform asymptotics for the finite-time ruin probability of a new dependent risk model with a constant interest rate, Methodol. Comput. Appl. Probab., 15, 1, 109-124 (2013) · Zbl 1263.91027 · doi:10.1007/s11009-011-9226-y |
| [24] | Joag-Dev, K.; Proschan, F., Negative association of random variables with applications, Ann. Stat., 11, 286-295 (1983) · Zbl 0508.62041 · doi:10.1214/aos/1176346079 |
| [25] | Baum, LE; Katz, M., Convergence rates in the law of large numbers, Trans. Am. Math. Soc., 120, 1, 108-123 (1965) · Zbl 0142.14802 · doi:10.1090/S0002-9947-1965-0198524-1 |
| [26] | Liu, L., Precise large deviations for dependent random variables with heavy tails, Statist. Probab. Lett., 79, 1290-1298 (2009) · Zbl 1163.60012 · doi:10.1016/j.spl.2009.02.001 |
| [27] | Csörgö, M.; Révész, P., Strong Approximations in Probability and Statistics (1981), Budapest: Akaddmiai Kiadó, Budapest · Zbl 0539.60029 |
| [28] | Ilienko, MK, A note on the Kolmogorov-Marcinkiewicz-Zygmund type strong law of large numbers for elements of autoregression sequences, Theory Stoch. Process., 22, 38, 22-29 (2017) · Zbl 1399.60047 |
| [29] | Ilienko, MK, A refinement of conditions for the almost sure convergence of series of multidimensional regression sequences, Theory Probab. Math. Stat., 6, 93, 71-78 (2016) · Zbl 1357.40001 |
| [30] | Runovska, MK, Convergence of series of Gaussian Markov sequences, Theory Probab. Math. Stat., 83, 149-162 (2011) · Zbl 1253.65003 · doi:10.1090/S0094-9000-2012-00848-X |
| [31] | Ilienko, MK, On the convergence of the Baum-Katz series for elements of a linear autoregression, Acta Math. Hungar., 164, 2, 413-427 (2021) · Zbl 1488.60051 · doi:10.1007/s10474-021-01157-3 |
| [32] | Katz, ML, The probability in the tail of a distribution, Ann. Math. Stat., 34, 1, 312-318 (1963) · Zbl 0209.49503 · doi:10.1214/aoms/1177704268 |
| [33] | Cheng, N., Li, X. Q., Wang, M. H., Wang, X. J., Xi, M. M.: Complete moment convergence for \(m\)-END random variables with application to non-parametric regression models,.Commun. Stat. Theory Methods (2020).doi:10.1080/03610926.2020.1800040 · Zbl 07533648 |
| [34] | Erdös, P., On a theorem of Hsu and Robbins, Ann. Math. Stat., 20, 286-291 (1949) · Zbl 0033.29001 · doi:10.1214/aoms/1177730037 |
| [35] | Li, P. H., Li, X. Q., Wu, K. H.: Complete convergence of randomly weighted END sequences and its applications. J. Inequal. Appl. 182(1) (2017). doi:10.1186/s13660-017-1457-1. · Zbl 1373.60061 |
| [36] | Hsu, PL; Robbins, H., Complete convergence and the law of large numbers, Proc. Nat. Acad. Sci. USA, 33, 2, 25-31 (1947) · Zbl 0030.20101 · doi:10.1073/pnas.33.2.25 |
| [37] | Hu, SH; Zhu, CH; Chen, YB; Wang, LC, Fixed-design regression for linear time series, Acta Math. Sci., 22B, 9-18 (2002) · Zbl 1010.62085 · doi:10.1016/S0252-9602(17)30450-2 |
| [38] | Buldygin, V.. V., Runovska (Ilienko), M.. K.: Almost sure convergence of the series of Gaussian Markov sequences. Commun. Statist. Theory Methods 40, 3407-3424 (2011) · Zbl 1315.60047 |
| [39] | Stout, WF, Almost Sure Convergence (1974), New York: Academic Press, New York · Zbl 0321.60022 |
| [40] | Wang, XJ; Hu, TC; Volodin, A.; Hu, SH, Complete convergence for weighted sums and arrays of rowwise extended negatively dependent random variables, Commun. Stat. Theory Methods, 42, 2391-2401 (2013) · Zbl 1276.60036 · doi:10.1080/03610926.2011.609321 |
| [41] | Wang, XJ; Wu, Y.; Hu, SH, Exponential probability inequality for \(m\)-END random variables and its applications, Metrika, 79, 127-147 (2016) · Zbl 1330.60031 · doi:10.1007/s00184-015-0547-7 |
| [42] | Xu, X.; Yan, JG, Complete moment convergence for randomly weighted sums of END sequences and its applications, Commun. Stat. Theory Methods, 50, 12, 2877-2899 (2021) · Zbl 07533702 · doi:10.1080/03610926.2019.1678637 |
| [43] | Chen, YQ; Chen, AY; Ng, KW, The Strong law of large numbers for extended negatively dependent random variables, J. Appl. Probab., 47, 908-922 (2010) · Zbl 1213.60058 · doi:10.1017/S0021900200007257 |
| [44] | Chow, YS, On the rate of moment convergence of sample sums and extremes, Bull. Inst. Math., 16, 3, 177-201 (1988) · Zbl 0655.60028 |
| [45] | Wu, Y., Wang, X. J.: Strong laws for weighted sums of \(m\)-extended negatively dependent random variables and its applications, J. Math. Anal. Appl. 494(2) (2021). doi:10.1016/j.jmaa.2020.124566 · Zbl 1476.60064 |
| [46] | Wu, Y.; Wang, XJ; Hu, SH, Complete moment convergence for weighted sums of weakly dependent random variables and its application in nonparametric regression model, Stat. Probab. Lett., 127, 56-66 (2017) · Zbl 1378.60063 · doi:10.1016/j.spl.2017.03.027 |
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