Skip to main content
SpringerLink
Log in

Strong Convergence for Weighted Sums of Widely Orthant Dependent Random Variables and Applications

  • Published:
Methodology and Computing in Applied Probability Aims and scope Submit manuscript

Abstract

In this paper, the complete convergence and the Kolmogorov strong law of large numbers for weighted sums of widely orthant dependent random variables are presented. Some applications to simple linear errors-in-variables model, nonparametric regression model, and quasi-renewal counting process are provided. Simulation studies are also carried out to confirm the theoretical results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

Data Availability

NA.

References

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Chen PY, Sung SH (2018) On complete convergence and complete moment convergence for weighted sums of \(\rho ^*\)-mixing random variables. J Inequal Appl 2018:121, 16

  • Chen PY, Sung SH (2021) Complete convergence for weighted sums of widely orthant-dependent random variables. J Inequal Appl 2021:45, 16

  • Chen PY, Zhang T, Sung SH (2019) Strong laws for randomly weighted sums of random variables and applications in the bootstrap and random design regression. Stat Sin 29(4):1739–1749

    MathSciNet  MATH  Google Scholar 

  • Chen PY, Wen LL, Sung SH (2020) Strong and weak consistency of least squares estimators in simple linear EV regression models. J Stat Plan Inference 205:64–73

    Article  MathSciNet  MATH  Google Scholar 

  • Chen Y, Wang L, Wang YB (2013) Uniform asymptotics for the finite-time ruin probabilities of two kinds of nonstandard bidimensional risk models. J Math Anal Appl 401:114–129

    Article  MathSciNet  MATH  Google Scholar 

  • Chen YQ, Chen AY, Ng KW (2010) The strong law of large numbers for extended negatively dependent random variables. J Appl Probab 47(4):908–922

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Deaton A (1985) Panel data from time series of cross-sections. J Econom 30(1):109–126

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Fan GL, Liang HY, Wang JF, Xu HX (2010) Asymptotic properties for LS estimators in EV regression model with dependent errors. Adv Stat Anal 94:89–103

    Article  MathSciNet  MATH  Google Scholar 

  • Georgiev AA et al (1985) Local properties of function fitting estimates with applications to system identification. In: Grossmann W et al (eds) Mathematical Statistics and Applications, vol B. Bad Tatzmannsdorf, Austria, Reidel, Dordrecht, pp 141–151

    Chapter  Google Scholar 

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

  • Hu D, Chen PY, Sung SH (2017) Strong laws for weighted sums of \(\psi\)-mixing random variables and applications in errors-in-variables regression models. TEST 26:600–617

  • Hu TZ (2000) Negatively superadditive dependence of random variables with applications. Chinese J Appl Probab Statist 16:133–144

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Klüppelberg C, Mikosch T (1997) Large deviations of heavy-tailed random sums with applications in insurance and finance. J Appl Probab 34:293–308

    Article  MathSciNet  MATH  Google Scholar 

  • Kočetova J, Leipus R, Šiaulys J (2009) A property of the renewal counting process with application to the finite-time ruin probability. Lith Math J 49(1):55–61

  • Lehmann E (1966) Some concepts of dependence. Ann Math Stat 37:1137–1153

    Article  MathSciNet  MATH  Google Scholar 

  • Leipus R, Šiaulys J (2009) Asymptotic behaviour of the finite-time ruin probability in renewal risk models. Appl Stoch Models Bus Ind 25:309–321

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

    Article  MathSciNet  MATH  Google Scholar 

  • Liu L (2009) Precise large deviations for dependent random variables with heavy tails. Statist Probab Lett 79:1290–1298

    Article  MathSciNet  MATH  Google Scholar 

  • Liu JX, Chen XR (2005) Consistency of LS estimator in simple linear EV regression models. Acta Mathematica Scientia, Series B 25(1):50–58

    Article  MathSciNet  MATH  Google Scholar 

  • Miao Y, Wang K, Zhao FF (2011) Some limit behaviors for the LS estimator in simple linear EV regression models. Statist Probab Lett 81(1):92–102

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Shen AT (2013) Bernstein-type inequality for widely dependent sequence and its application to nonparametric regression models. Abstr Appl Anal 2013:862602, 9

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Stout WF (1974) Almost Sure Convergence. Academic Press, New York

    MATH  Google Scholar 

  • Sung SH (2010) Complete convergence for weighted sums of \(\rho ^*\)-mixing random variables. Discrete Dyn Nat Soc 2010:630608, 13

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

  • Thanh LV, Yin GG, Wang LY (2011) State observers with random sampling times and convergence analysis of double-indexed and randomly weighted sums of mixing processes. SIAM J Control Optim 49:106–124

    Article  MathSciNet  MATH  Google Scholar 

  • Thanh LV, Yin G (2015) Weighted sums of strongly mixing random variables with an application to nonparametric regression. Statist Probab Lett 105:195–202

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Wang KY, Wang YB, Gao QW (2013) Uniform asymptotics for the finite-time ruin probability of a new dependent risk model with a constant interest rate. Methodol Comput Appl Probab 15:109–124

    Article  MathSciNet  MATH  Google Scholar 

  • Wang XJ, Shen AT, Chen ZY, Hu SH (2015) Complete convergence for weighted sums of NSD random variables and its application in the EV regression model. TEST 24(1):166–184

    Article  MathSciNet  MATH  Google Scholar 

  • Wang XJ, Xu C, Hu T-C, Volodin A, Hu SH (2014) On complete convergence for widely orthant-dependent random variables and its applications in nonparametric regression models. TEST 23:607–629

    Article  MathSciNet  MATH  Google Scholar 

  • Wang XJ, Wu Y, Rosalsky A (2017) Complete convergence for arrays of rowwise widely orthant dependent random variables and its applications. Stoch - Int J Probab Stoch Process 89(8):1228–1252

    Article  MathSciNet  MATH  Google Scholar 

  • Wang YB, Cheng DY (2011) Basic renewal theorems for random walks with widely dependent increments. J Math Anal Appl 384:597–606

    Article  MathSciNet  MATH  Google Scholar 

  • Yang WZ, Liu TT, Wang XH, Hu SH (2014) On the Bahadur representation of sample quantiles for widely orthant dependent sequences. Filomat 28:1333–1343

    Article  MathSciNet  MATH  Google Scholar 

  • Yang Y, Leipus R, Šiaulys J, Cang YQ (2011) Uniform estimates for the finite-time ruin probability in the dependent renewal risk model. J Math Anal Appl 383:215–225

  • Zhang R, Wu Y, Xu WF, Wang XJ (2019) On complete consistency for the weighted estimator of nonparametric regression models. RACSAM 113(3):2319–2333

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

This work was supported by the National Natural Science Foundation of China (12201079, 11871072), the National Social Science Foundation of China (22BTJ059) and the Natural Science Foundation of Anhui Province (2108085MA06, 2108085QA15).

Author information

Authors and Affiliations

  1. School of Big Data and Statistics, Anhui University, Hefei, 230601, People’s Republic of China

    Yi Wu, Xuejun Wang & Aiting Shen

  2. School of Big Data and Artificial Intelligence, Chizhou University, Chizhou, 247000, People’s Republic of China

    Yi Wu

Authors
  1. Yi Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xuejun Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Aiting Shen
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Yi Wu, Xuejun Wang and Aiting Shen. The first draft of the manuscript was written by Yi Wu and Xuejun Wang. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Aiting Shen.

Ethics declarations

Competing Interests

All authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wu, Y., Wang, X. & Shen, A. Strong Convergence for Weighted Sums of Widely Orthant Dependent Random Variables and Applications. Methodol Comput Appl Probab 25, 15 (2023). https://doi.org/10.1007/s11009-023-09976-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11009-023-09976-3

Keywords

Mathematical Subject Classification

Search

Navigation