The Self-normalized Asymptotic Results for Linear Processes

  • Magda Peligrad
  • Hailin SangEmail author
Part of the Fields Institute Communications book series (FIC, volume 76)


The linear process is a tool for studying stationary time series. One can have a better understanding of many important time series by studying the corresponding linear processes. The strength of dependence and the tail properties of time series built upon linear processes can be expressed in terms of the linear process itself through the innovations and their weights. In this paper we survey recent developments on some asymptotics of linear processes. These asymptotics include central limit theorem, functional central limit theorem and their self-normalized forms.


Central Limit Theorem Linear Process Stationary Time Series White Noise Process Short Memory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We thank Rafal Kulik and the referees for helpful comments. Magda Peligrad was supported in part by a Charles Phelps Taft Memorial Fund grant, the NSA grant H98230-11-1-0135, and the NSF grant DMS-1208237.


  1. 1.
    Anderson, T.W.: The Statistical Analysis of Time Series. Wiley, New York (1971)zbMATHGoogle Scholar
  2. 2.
    Araujo, A., Giné, E.: The Central Limit Theorem for Real and Banach Valued Random Variables. Wiley Series in Probability and Mathematical Statistics. Wiley, New York/Chichester/Brisbane (1980)zbMATHGoogle Scholar
  3. 3.
    Baillie, R.T.: Long memory processes and fractional integration in econometrics. J. Econom. 73, 5–59 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Csörgő, M., Szyszkowicz, B., Wang, Q.: Donsker’s theorem for self-normalized partial sums processes. Ann. Probab. 31, 1228–1240 (2003)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Doukhan, P., Oppenheim, G., Taqqu, M.S. (eds.): Theory and Applications of Long-Range Dependence. Birkhäuser, Boston (2003)zbMATHGoogle Scholar
  6. 6.
    Fuller, W.A.: An Introduction to Probability Theory and Its Applications, vol. 2. Willey, New York (1966)Google Scholar
  7. 7.
    Fuller, W.A.: Introduction to Statistical Time Series. Wiley, New York (1976)zbMATHGoogle Scholar
  8. 8.
    Giné, E., Götze, F., Mason, D.: When is the student t-statistic asymptotically standard normal? Ann. Probab. 25, 1514–1531 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Giraitis, L., Surgailis, D.: A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotic normality of Whittle’s estimate. Probab. Theory Relat. Fields 86, 87–104 (1990)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Hannan, E.J.: Multiple Time Series. Wiley, New York (1970)zbMATHCrossRefGoogle Scholar
  11. 11.
    Ibragimov, I.A., Linnik, Y.V.: Independent and Stationary Sequences of Random Variables. Wolters, Groningen (1971)zbMATHGoogle Scholar
  12. 12.
    Juodis, M., Račkauskas, A.: A remark on self-normalization for dependent random variables. Lith. Math. J. 45, 142–151 (2005)zbMATHCrossRefGoogle Scholar
  13. 13.
    Juodis, M., Račkauskas, A.: A central limit theorem for self-normalized sums of linear process. Stat. Probab. Lett. 77, 1535–1541 (2007)zbMATHCrossRefGoogle Scholar
  14. 14.
    Kulik, R.: Limit theorems for self-normalized linear processes. Stat. Probab. Lett. 76, 1947–1953 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Peligrad, M., Sang, H.: Asymptotic properties of self-normalized linear processes with long memory. Econom. Theory 28, 548–569 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Peligrad, M., Sang, H.: Central limit theorem for linear processes with infinite variance. J. Theor. Probab. 26, 222–239 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Phillips, P.C.B., Solo, V.: Asymptotics for linear processes. Ann. Stat. 20, 971–1001 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Račkauskas, A., Suquet, C.: Functional central limit theorems for self-normalized partial sums of linear processes. Lith. Math. J. 51, 251–259 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Robinson, P.M. (ed.): Time Series with Long Memory. Oxford University Press, Oxford/New York (2003)zbMATHGoogle Scholar
  20. 20.
    Wu, W.B., Woodroofe, M.: Martingale approximations for sums of stationary processes. Ann. Probab. 32, 1674–1690 (2004)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of CincinnatiCincinnatiUSA
  2. 2.Department of MathematicsUniversity of MississippiUniversityUSA

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