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ASYMPTOTIC PROPERTIES OF SELF-NORMALIZED LINEAR PROCESSES WITH LONG MEMORY

Published online by Cambridge University Press:  25 November 2011

Magda Peligrad  and
Hailin Sang
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Abstract

In this paper we study the convergence to fractional Brownian motion for long memory time series having independent innovations with infinite second moment. For the sake of applications we derive the self-normalized version of this theorem. The study is motivated by models arising in economic applications where often the linear processes have long memory, and the innovations have heavy tails.

Type
Research Article
Information
Econometric Theory , Volume 28 , Issue 3 , June 2012 , pp. 548 - 569
Copyright
Copyright © Cambridge University Press 2011

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Footnotes

The authors are grateful to the referees for carefully reading the paper and for numerous suggestions that significantly improved the presentation of the paper. The first author’s research was supported in part by a Charles Phelps Taft Memorial Fund grant and NSA grants H9823009-1-0005 and H98230-11-1-0135. The second author worked on this paper during visits to the University of Cincinnati and the National Institute of Statistical Sciences.

References

REFERENCES

Billingsley, P. (1968) Convergence of Probability Measures. Wiley.Google Scholar
Chistyakov, G.P. & Götze, F. (2004) Limit distributions of Studentized means. Annals of Probability 32, 28–77.CrossRefGoogle Scholar
Csörgő, M., Szyszkowicz, B., & Wang, Q. (2003) Donsker’s theorem for self-normalized partial sums processes. Annals of Probability 31, 1228–1240.Google Scholar
Davydov, Y.A. (1970) The invariance principle for stationary processes. Theory of Probability and Its Applications 15, 487–498.CrossRefGoogle Scholar
Dedecker, J., Merlevède, F., & Peligrad, M. (2011) Invariance principles for linear processes: Application to isotonic regression. Bernoulli 17, 88–113.CrossRefGoogle Scholar
de la Peña, V. & Giné, E. (1999) Decoupling: From Dependence to Independence. Springer-Verlag.CrossRefGoogle Scholar
de la Peña, V., Lai, T.L., & Shao, Q. (2009) Self-Normalized Processes: Limit Theory and Statistical Applications. Springer-Verlag.CrossRefGoogle Scholar
Dickey, D.A. & Fuller, W.A. (1979) Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association 74, 427–431.Google Scholar
Dickey, D.A. & Fuller, W.A. (1981) Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica 49, 1057–1072.CrossRefGoogle Scholar
Feller, W. (1966) An Introduction to Probability Theory and Its Applications, vol. 2. Wiley.Google Scholar
Giné, E., Götze, F., & Mason, D.M. (1997) When is the Student t-statistic asymptotically standard normal? Annals of Probability 25, 1514–1531.CrossRefGoogle Scholar
Knight, K. (1991) Limit theory for M-estimates in an integrated infinite variance process. Econometric Theory 7, 200–212.CrossRefGoogle Scholar
Kuelbs, J. (1985) The LIL when X is in the domain of attraction of a Gaussian law. Annals of Probability 13, 825–859.CrossRefGoogle Scholar
Kulik, R. (2006) Limit theorems for self-normalized linear processes. Statistics and Probability Letters 76, 1947–1953.CrossRefGoogle Scholar
Lamperti, J. (1962) Semi-stable stochastic processes. Transactions of the American Mathematical Society 104, 62–78.CrossRefGoogle Scholar
Mikosch, T., Gadrich, T., Kliippelberg, C., & Adler, R.J. (1995) Parameter estimation for ARMA models with infinite variance innovations. Annals of Statistics 23, 305–326.CrossRefGoogle Scholar
Peligrad, M. & Sang, H. (2011) Central limit theorem for linear processes with infinite variance. Journal of Theoretical Probability; available at arXiv:1105.6129.Google Scholar
Phillips, P.C.B. (1987) Time series regression with a unit root. Econometrica 55, 277–301.CrossRefGoogle Scholar
PhillipsP,C.B. P,C.B. & Solo, V. (1992) Asymptotics for linear processes. Annals of Statistics 20, 971–1001.CrossRefGoogle Scholar
Seneta, E. (1976) Regularly Varying Functions. Lecture Notes in Mathematics 508. Springer-Verlag.CrossRefGoogle Scholar
Sowell, F. (1990) The fractional unit root distribution. Econometrica 58, 495–505.CrossRefGoogle Scholar
Taqqu, M.S. (1975) Weak convergence to fractional Brownian motion and to the Rosenblatt process. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 31, 287–302.CrossRefGoogle Scholar
Wang, Q., Lin, Y.X., & Gulati, C.M. (2003) Strong approximation for long memory processes with applications. Journal of Theoretical Probability 16, 377–389.CrossRefGoogle Scholar
Wu, W.B. (2003) Additive functionals of infinite-variance moving averages. Statistica Sinica 13, 1259–1267.Google Scholar
Wu, W.B. (2006) Unit root testing for functional of linear processes. Econometric Theory 22, 1–14.CrossRefGoogle Scholar
Wu, W.B. & Min, W. (2005) On linear processes with dependent innovations. Stochastic Processes and Their Applications 115, 939–958.Google Scholar