Strong convergence properties for weighted sums of m-asymptotic negatively associated random variables and statistical applications

Abstract

In this paper, we establish a general result on complete moment convergence and the Marcinkiewicz–Zygmund-type strong law of large numbers for weighted sums of m-asymptotic negatively associated random variables, which improve and extend some existing ones. As applications of our main results, we present a result on complete consistency for the weighted estimator in a nonparametric regression model and a result on strong consistency for conditional Value-at-risk estimator based on m-asymptotic negatively associated errors. We also carry out some numerical simulations to confirm the theoretical results.

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References

  1. Artzner P, Delbaen F, Eber JM, Heath D (1999) Coherent measures of risk. Math Financ 9(3):203–228

    MathSciNet  MATH  Article  Google Scholar 

  2. Bai ZD, Cheng PE (2000) Marcinkiewicz strong laws for linear statistics. Stat Probab Lett 46(2):105–112

    MathSciNet  MATH  Article  Google Scholar 

  3. Bradley RC (1992) On the spectral density and asymptotic normality of weakly dependent random fields. J Theor Probab 5:355–373

    MathSciNet  MATH  Article  Google Scholar 

  4. Bradley RC (1997) Every “lower psi-mixing” Markov chain is “interlaced rho-mixing”. Stoch Process Appl 72:221–239

    MathSciNet  MATH  Article  Google Scholar 

  5. Chen PY (2005) Limiting behavior of weighted sums of negatively associated random variables. Acta Mathematica Scientia 25A(4):489–495

    MathSciNet  MATH  Google Scholar 

  6. Chen PY, Sung SH (2018) On complete convergence and complete moment convergence for weighted sums of \(\rho ^*\)-mixing random variables. J Inequal Appl 2018:16

    MathSciNet  Article  Google Scholar 

  7. Chen PY, Bai P, Sung SH (2014) The von Bahr-Esseen moment inequality for pairwise independent random variables and applications. J Math Anal Appl 419(2):1290–1302

    MathSciNet  MATH  Article  Google Scholar 

  8. Chow YS (1966) Some convergence theorems for independent random variables. Ann Math Stat 37(6):1482–1493

    MathSciNet  MATH  Article  Google Scholar 

  9. Chow YS (1988) On the rate of moment convergence of sample sums and extremes. Bull Inst Math 16(3):177–201

    MathSciNet  MATH  Google Scholar 

  10. Cuzick J (1995) A strong law for weighted sums of iid random variables. J Theor Probab 8(3):625–641

    MathSciNet  MATH  Article  Google Scholar 

  11. Fan Y (1990) Consistent nonparametric multiple regression for dependent heterogeneous processes: the fixed design case. J Multivar Anal 33:72–88

    MathSciNet  MATH  Article  Google Scholar 

  12. Georgiev AA (1985) Local properties of function fitting estimates with applications to system identification. In: Grossmann W et al (eds) Mathematical Statistics and Applications, vol B. Proceedings 4th pannonian symposium on mathematical statistics, 4–10, September 1983. Bad Tatzmannsdorf, Austria, Reidel, Dordrecht, pp 141–151

  13. Hsu PL, Robbins H (1947) Complete convergence and the law of large numbers. Proc Nat Acad Sci USA 33:25–31

    MathSciNet  MATH  Article  Google Scholar 

  14. Hu TC, Chiang CY, Taylor RL (2009) On complete convergence for arrays of rowwise \(m\)-negatively associated random variables. Nonlinear Anal 71(12):1075–1081

    MathSciNet  MATH  Article  Google Scholar 

  15. Joag-Dev K, Proschan F (1983) Negative association of random variables with applications. Ann Stat 11(1):286–295

    MathSciNet  MATH  Article  Google Scholar 

  16. Liang HY, Jing BY (2005) Asymptotic properties for estimates of nonparametric regression models based on negatively associated sequences. J Multivar Anal 95:227–245

    MathSciNet  MATH  Article  Google Scholar 

  17. Liu XD, Liu JX (2009) Moments of the maximum of normed partial sums of \(\rho ^-\)-mixing random variables. Applied Mathematics-A Journal of China Universities, Series B 24(3):355–360

    MathSciNet  MATH  Article  Google Scholar 

  18. Luo ZD (2020) Nonparametric kernel estimation of CVaR under \(\alpha \)-mixing sequences. Stat Pap 61:615–643

    MathSciNet  MATH  Article  Google Scholar 

  19. Luo ZD, Yang SC (2013) The asymptotic properties of CVaR estimator under \(\rho \)-mixing sequences. Acta Mathematica Sinica 56(6):851–870

    MathSciNet  MATH  Google Scholar 

  20. Mausser H, Rosen D (1998) Beyond VaR: from measuring risk to managing risk. Algo Res Q 1(2):5–20

    Google Scholar 

  21. Mckay R, Keefer TE (1996) VaR is a dangerous technique. Corporate Finance, Searching for Systems Integration Supplement, September

  22. Peligrad M, Gut A (1999) Almost-sure results for a class of dependent random variables. J Theor Probab 12:87–104

    MathSciNet  MATH  Article  Google Scholar 

  23. Pflug GC (2000) Some remarks on the value-at-risk and the conditional value-at-risk, Probabilistic constrained optimization. Springer, Boston, MA, pp 272–281

    Google Scholar 

  24. Rochafellar RT, Uryasev S (2000) Optimization of conditional value-at-risk. J Risk 2(3):21–41

  25. Roussas GG (1989) Consistent regression estimation with fixed design points under dependence conditions. Stat Probab Lett 8:41–50

    MathSciNet  MATH  Article  Google Scholar 

  26. Roussas GG, Tran LT, Ioannides DA (1992) Fixed design regression for time series: asymptotic normality. J Multivar Anal 40:262–291

    MathSciNet  MATH  Article  Google Scholar 

  27. Shen AT (2016) Complete convergence for weighted sums of END random variables and its application to nonparametric regression models. J Nonparametr Stat 28(4):702–715

    MathSciNet  MATH  Article  Google Scholar 

  28. Shen AT, Zhang Y, Volodin A (2015) Applications of the Rosenthal-type inequality for negatively superadditive dependent random variables. Metrika 78:295–311

    MathSciNet  MATH  Article  Google Scholar 

  29. Shen AT, Xue MX, Volodin A (2016) Complete moment convergence for arrays of rowwise NSD random variables. Stochastics 88(4):606–621

    MathSciNet  MATH  Article  Google Scholar 

  30. Stone CJ (1977) Consistent nonparametric regression. Ann Stat 5:595–645

    MathSciNet  MATH  Article  Google Scholar 

  31. Sung SH (2010) Complete convergence for weighted sums of \(\rho ^*\)-mixing random variables. Discret Dyn Nat Soc 2010:13

    MathSciNet  Article  Google Scholar 

  32. Sung SH (2013) On the strong convergence for weighted sums of \(\rho ^{*}\)-mixing random variables. Stat Pap 54:773–781

    MathSciNet  MATH  Article  Google Scholar 

  33. Tran L, Roussas G, Yakowitz S, Van Truong B (1996) Fixed-design regression for linear time series. Ann Stat 24:975–991

    MathSciNet  MATH  Article  Google Scholar 

  34. Trindade AA, Uryasev S, Shapiro A, Zrazhevsky G (2007) Financial prediction with constrained tail risk. J Bank Financ 31(11):3524–3538

    Article  Google Scholar 

  35. Utev S, Peligrad M (2003) Maximal inequalities and an invariance principle for a class of weakly dependent random variables. J Theor Probab 16:101–115

    MathSciNet  MATH  Article  Google Scholar 

  36. Wang XJ, Hu SH (2014) Complete convergence and complete moment convergence for martingale difference sequence. Acta Mathematica Sinica, English Series 30:119–132

    MathSciNet  MATH  Article  Google Scholar 

  37. Wang JF, Lu FB (2006) Inequalities of maximum partial sums and weak convergence for a class of weak dependent random variables. Acta Mathematica Sinica, English Series 22(3):693–700

    MathSciNet  MATH  Article  Google Scholar 

  38. Wang JF, Zhang LX (2007) A Berry–Esseen theorem and a law of the iterated logarithm for asymptotically negatively associated sequences. Acta Mathematica Sinica 23(1):127–136

    MathSciNet  MATH  Article  Google Scholar 

  39. Wang XJ, Zheng LL, Xu C, Hu SH (2015) Complete consistency for the estimator of nonparametric regression models based on extended negatively dependent errors. Statistics 49(2):396–407

    MathSciNet  MATH  Article  Google Scholar 

  40. Wang XJ, Hu SH, Volodin AI (2018) Moment inequalities for \(m\)-NOD random variables and their applications. Theor Probab Appl 62(3):471–490

    MathSciNet  MATH  Article  Google Scholar 

  41. Wu QY, Jiang YY (2008) Some strong limit theorems for \(\tilde{\rho }\)-mixing sequences of random variables. Statistics & Probability Letters 78:1017–1023

    MathSciNet  MATH  Article  Google Scholar 

  42. Wu YF, Cabrea MO, Volodin A (2014a) Complete convergence and complete moment convergence for arrays of rowwise END random variables. Glasnik Matematički 49(69):449–468

    MathSciNet  Google Scholar 

  43. Wu YF, Sung SH, Volodin A (2014b) A note on the rates of convergence for weighted sums of \(\rho ^{*}\)-mixing random variables. Lith Math J 54:220–228

    MathSciNet  MATH  Article  Google Scholar 

  44. Wu Y, Wang XJ, Hu SH (2017) Complete moment convergence for weighted sums of weakly dependent random variables and its application in nonparametric regression model. Stat Probab Lett 127:56–66

    MathSciNet  MATH  Article  Google Scholar 

  45. Xing GD, Yang SC, Li YM (2014) Strong consistency of conditional value-at-risk estimate for \(\varphi \)-mixing samples. Commun Stat Theory Methods 43:5105–5113

    MathSciNet  MATH  Article  Google Scholar 

  46. Zhang LX (2000a) A functional central limit theorem for asymptotically negatively dependent random fields. Acta Mathematica Hungarica 86(3):237–259

    MathSciNet  MATH  Article  Google Scholar 

  47. Zhang LX (2000b) Central limit theorems for asymptotically negatively associated random fields. Acta Mathematica Sinica, English Series 16(4):691–710

    MathSciNet  MATH  Article  Google Scholar 

  48. Zhang LX, Wang XY (1999) Convergence rates in the strong laws of asymptotically negatively associated random fields. Appl Math A J China Univ Ser B 14(4):406–416

    MathSciNet  MATH  Article  Google Scholar 

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Acknowledgements

The authors are most grateful to the Editor and anonymous referee for carefully reading the manuscript and valuable suggestions which helped in improving an earlier version of this paper.

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Correspondence to Aiting Shen.

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Supported by the National Natural Science Foundation of China (Grant Nos. 11671012, 11871072), the Natural Science Foundation of Anhui Province (Grant No. 1908085QA01), the Provincial Natural Science Research Project of Anhui Colleges (Grant No. KJ2019A0003) and the Project on Reserve Candidates for Academic and Technical Leaders of Anhui Province (Grant No. 2017H123)

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Cite this article

Wu, Y., Wang, X. & Shen, A. Strong convergence properties for weighted sums of m-asymptotic negatively associated random variables and statistical applications. Stat Papers (2020). https://doi.org/10.1007/s00362-020-01179-z

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Keywords

  • m-Asymptotic negatively associated random variables
  • Complete moment convergence
  • Complete convergence
  • Strong law of large numbers
  • Nonparametric regression model
  • Conditional value-at-risk
  • Complete consistency
  • Strong consistency

Mathematics Subject Classification

  • 60F15
  • 62G05