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Research Article

Some strong convergence properties for randomly weighted maximum partial sums of END random variables with statistical applications

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Received 19 Dec 2022, Accepted 15 Jan 2024, Published online: 04 Feb 2024

References

  • Hsu PL, Robbins H. Complete convergence and the law of large numbers. Proc Natl Acad Sci. 1947;33(2):25–31. doi: 10.1073/pnas.33.2.25
  • Erdös P. On a theorem of Hsu and Robbins. Ann Math Statist. 1949;20(2):286–291. doi: 10.1214/aoms/1177730037
  • Erdös P. Remark on my paper ‘On a theorem of Hsu and Robbins’. Ann Math Statist. 1950;21(1):138–138. doi: 10.1214/aoms/1177729897
  • Katz ML. The probability in the tail of a distribution. Ann Math Statist. 1963;34(1):312–318. doi: 10.1214/aoms/1177704268
  • Baum LE, Katz M. Convergence rates in the law of large numbers. Trans Am Math Soc. 1965;120(1):108–123. doi: 10.1090/tran/1965-120-01
  • Chow YS. Delayed sums and borel summability of independent, identically distributed random variables. Bull Inst Math Acad Sinica. 1973;1(2):207–220.
  • Gut A. Marcinkiewicz laws and convergence rates in the law of large numbers for random variables with multidimensional indices. Ann Probab. 1978;6(3):469–482. doi: 10.1214/aop/1176995531
  • Gut A. Complete convergence for arrays. Period Math Hung. 1992;25(1):51–75. doi: 10.1007/BF02454383
  • Gut A. Complete convergence and Cesaro summation for i.i.d. random variables. Probab Theory Related Fields. 1993;97:169–178. doi: 10.1007/BF01199318
  • Marcinkiewicz J, Zygmund A. Sur les fonctions indépendantes. Fundam Math. 1937;29:60–90. doi: 10.4064/fm-29-1-60-90
  • Chow YS. Some convergence theorems for independent random variables. Ann Math Statist. 1966;37(6):1482–1493. doi: 10.1214/aoms/1177699140
  • Franck WE, Hanson DL. Some results giving rates of convergence in the law of large numbers for weighted sums of independent random variables. Bull Am Math Soc. 1966;72(2):266–268. doi: 10.1090/S0002-9904-1966-11488-X
  • Pruitt WE. Summability of independent random variables. J Math Mech. 1966;15(5):769–776.
  • Stout WF. Some results on the complete and almost sure convergence of linear combinations of independent random variables and martingale differences. Ann Math Statist. 1968;39(5):1549–1562. doi: 10.1214/aoms/1177698136
  • Chow YS. On the rate of moment convergence of sample sums and extremes. Bull Inst Math Acad Sinica. 1988;16(3):177–201.
  • Guo ML, Zhu DJ. Equivalent conditions of complete moment convergence of weighted sums for ρ∗-mixing sequence of random variables. Statist Probab Lett. 2013;83(1):13–20. doi: 10.1016/j.spl.2012.08.015
  • Hosseini SM, Nezakati A. Complete moment convergence for the dependent linear processes with random coefficients. Acta Math Sin Engl Ser. 2019;35(8):1321–1333. doi: 10.1007/s10114-019-8205-z
  • Ko MH. A note on complete moment convergence for coordinatewise negatively associated random vectors in Hilbert spaces. Commun Statist-Theory Methods. 2020;49(7):1780–1791. doi: 10.1080/03610926.2019.1565833
  • Shen AT, Xue MX, Volodin A. Complete moment convergence for arrays of rowwise NSD random variables. Stochastics. 2016;88(4):606–621. doi: 10.1080/17442508.2015.1110153
  • Wang XJ, Hu SH. Complete convergence and complete moment convergence for martingale difference sequence. Acta Math Sin Engl Ser. 2014;30(1):119–132. doi: 10.1007/s10114-013-2243-8
  • Wang XJ, Wu Y, Hu SH, et al. Complete moment convergence for negatively orthant dependent random variables and its applications in statistical models. Statist Papers. 2020;61:1147–1180. doi: 10.1007/s00362-018-0983-3
  • Wu YF, Cabrera MO, Volodin A. Complete convergence and complete moment convergence for arrays of rowwise END random variables. Glasnik Matematički. 2014;49(69):447–466. doi: 10.3336/gm
  • Wu Y, Wang XJ, Hu TC, et al. Complete f-moment convergence for extended negatively dependent random variables. RACSAM. 2019;113:333–351. doi: 10.1007/s13398-017-0480-x
  • Lu C, Chen Z, Wang XJ. Complete f-moment convergence for widely orthant dependent random variables and its application in nonparametric models. Acta Math Sin Engl Ser. 2019;35(12):1917–1936. doi: 10.1007/s10114-019-8315-7
  • Wang Y, Wang XJ. Complete f-moment convergence for Sung's type weighted sums and its application to the EV regression models. Statist Papers. 2021;62:769–793. doi: 10.1007/s00362-019-01112-z
  • Cheng N, Lu C, Qi JB, et al. Complete moment convergence for randomly weighted sums of extended negatively dependent random variables with application to semiparametric regression models. Statist Papers. 2022;63:397–419. doi: 10.1007/s00362-021-01244-1
  • Liu L. Precise large deviations for dependent random variables with heavy tails. Statistics and Probability Letters. 2009;79(9):1290–1298. doi: 10.1016/j.spl.2009.02.001
  • Joag-Dev K, Proschan F. Negative association of random variables with applications. Ann Statist. 1983;11(1):286–295. doi: 10.1214/aos/1176346079
  • Chen YQ, Chen AY, Ng KW. The strong law of large numbers for extended negatively dependent random variables. J Appl Probab. 2010;47(4):908–922. doi: 10.1239/jap/1294170508
  • Wu YF, Guan M. Convergence properties of the partial sums for sequences of END random variables. J Korean Math Soc. 2012;49(6):1097–1110. doi: 10.4134/JKMS.2012.49.6.1097
  • Wang SJ, Wang XJ. Precise large deviations for random sums of END real-valued random variables with consistent variation. J Math Anal Appl. 2013;402(2):660–667. doi: 10.1016/j.jmaa.2013.02.002
  • Yang WZ, Xu HY, Chen L, et al. Complete consistency of estimators for regression models based on extended negatively dependent errors. Statist Papers. 2018;59:449–465. doi: 10.1007/s00362-016-0771-x
  • Xi MM, Wu Y, Wang XJ. Complete convergence for arrays of rowwise END random variables and its statistical applications under sub-linear expectations. J Korean Stat Soc. 2019;48(3):412–425. doi: 10.1016/j.jkss.2018.12.002
  • Ding LW, Chen P, Li YM. Consistency for wavelet estimator in nonparametric regression model with extended negatively dependent samples. Statist Papers. 2020;61:2331–2349. doi: 10.1007/s00362-018-1050-9
  • Xu JP, Zhang LX. The law of logarithm for arrays of random variables under sub-linear expectations. Acta Math Appl Sinica, English Ser. 2020;36(3):670–688. doi: 10.1007/s10255-020-0958-8
  • Xu X, Yan JG. Complete moment convergence for randomly weighted sums of END sequences and its applications. Commun Statist-Theory Methods. 2021;50(12):2877–2899. doi: 10.1080/03610926.2019.1678637
  • Liu L. Necessary and sufficient conditions for moderate deviations of dependent random variables with heavy tails. Science China Mathematics. 2010;53(6):1421–1434. doi: 10.1007/s11425-010-4012-9
  • Li PH, Li XQ, Wu KH. Complete convergence of randomly weighted END sequences and its application. J Inequal Appl. 2017;2017:182. doi: 10.1186/s13660-017-1457-1
  • Wu Y, Wang XJ, Sung SH. Complete moment convergence for arrays of rowwise negatively associated random variables and its application in non-parametric regression model. Probab Eng Inf Sci. 2018;32(1):37–57. doi: 10.1017/S026996481600053X
  • Zhang GH. Complete convergence for Sung's type weighted sums of END random variables. J Inequal Appl. 2014;2014:353. doi: 10.1186/1029-242X-2014-353
  • Adler A, Rosalsky A. Some general strong laws for weighted sums of stochastically dominated random variables. Stoch Anal Appl. 1987;5(1):1–16. doi: 10.1080/07362998708809104
  • 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. 1989;12(3):507–530. doi: 10.1155/S0161171289000657
  • Ding Y, Tang XF, Wang H, et al. Complete moment convergence of moving-average process generated by a class of random variables. Commun Statist-Theory Methods. 2017;46(22):10903–10913. doi: 10.1080/03610926.2016.1252401
  • Shen AT, Yao M, Xiao BQ. Equivalent conditions of complete convergence and complete moment convergence for END random variables. Chin Ann Math Ser B. 2018;39(1):83–96. doi: 10.1007/s11401-018-1053-9
  • Georgiev AA. Local properties of function fitting estimates with application to system identification. In: Proceedings 4th Pannonian Symposium Mathematical Statistics, Mathematical Statistics and Applications; September 4-10, 1983; Bad Tatzmannsdorf, Austria, Reidel, Dordrecht, 1985, p. 141–151.
  • Chen X. Asymptotic properties for estimates of nonparametric regression model with martingale difference errors. Statistics. 2012;46(5):687–696. doi: 10.1080/02331888.2011.555546
  • Chen ZY, Wang HB, Wang XJ. The consistency for the estimator of nonparametric regression model based on martingale difference errors. Statist Papers. 2016;57:451–469. doi: 10.1007/s00362-015-0662-6
  • Roussas GG, Tran LT, Ioannides DA. Fixed design regression for time series: asymptotic normality. J Multivar Anal. 1992;40(2):262–291. doi: 10.1016/0047-259X(92)90026-C
  • Zhang SL, Hou TT, Qu C. Complete consistency for the estimator of nonparametric regression model based on martingale difference errors. Commun Statist-Theory Methods. 2021;50(2):358–370. doi: 10.1080/03610926.2019.1635160
  • Priestley MB, Chao MT. Non-parametric function fitting. J R Statist Soc Ser B: Statistical Methodology. 1972;34(3):385–392.
  • Yang SC, Wang YB. Strong consistency of regression function estimator for negatively associated samples. Acta Math Appl Sinica. 1999;22(4):522–530.
  • Wang KY, Wang YB, Gao QW. Uniform asymptotics for the finite-time ruin probability of a dependent risk model with a constant interest rate. Methodol Comput Appl Probab. 2013;15(1):109–124. doi: 10.1007/s11009-011-9226-y
  • Xiao YT, Li FX. Estimation in partially linear varying-coefficient errors-in-variables models with missing response variables. Comput Stat. 2020;35:1637–1658. doi: 10.1007/s00180-020-00967-3
  • Deng X, Wang XJ, Hu SH, et al. A general result on complete convergence for weighted sums of linear processes and its statistical applications. Statistics. 2019;53(4):903–920. doi: 10.1080/02331888.2019.1615912

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