Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-10T16:49:14.518Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotics of the sample mean and sample covariance of long-range-dependent series

Part of: Distribution theory

Published online by Cambridge University Press:  14 July 2016

Wen Dai
Wen Dai*
Affiliation:
School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia. Email address: wend@maths.usyd.edu.au
Get access Rights & Permissions [Opens in a new window]

Abstract

An asymptotic distribution is given for the partial sums of a stationary time-series with long-range dependence. The law of large numbers for the sample covariance of the series is also derived. The results differ from those given elsewhere in relaxing the assumption of the independence of the innovations of the series.

Keywords

Central limit theoremlaws of large numberslong-range dependencesample meansample covariance

MSC classification

Primary: 62E20: Asymptotic distribution theory
Type
Part 7. Time series analysis
Information
Journal of Applied Probability , Volume 41 , Issue A: Stochastic Methods and their Applications , 2004 , pp. 383 - 392
Copyright
Copyright © Applied Probability Trust 2004 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Anderson, T. W. (1958). The Statistical Analysis of Time Series. John Wiley, New York.Google Scholar
[2] Anh, V. V. and Heyde, C. C., (eds) (1999). (Special issue on long-range dependence.) J. Statist. Planning Infer. 80.Google Scholar
[3] Avram, F. and Taqqu, M. (1987). Noncentral limit theorems and Appell polynomials. Ann. Prob. 15, 767–775.Google Scholar
[4] Beran, J. (1994). Statistics for Long-Memory Processes. Chapman and Hall, New York.Google Scholar
[5] Davis, R. and Resnick, S. (1985). Limit theory for moving averages of random variables with regularly varying tail probabilities. Ann. Prob. 13, 179–195.Google Scholar
[6] Davis, R. and Resnick, S. (1985). More limit theory for the sample correlation function of moving averages. Stoch. Process. Appl. 20, 257–279.Google Scholar
[7] Davis, R. and Resnick, S. (1986). Limit theory for the sample covariance and correlation functions of moving averages. Ann. Statist. 14, 533–558.Google Scholar
[8] Davydov, Y. A. (1970). The invariance principle for stationary processes. Theory Prob. Appl. 15, 487–498.Google Scholar
[9] Gorodetskii, V. V. (1977). On convergence to semi-stable Gaussian processes. Theory Prob. Appl. 22, 498–508.Google Scholar
[10] Hall, R and Heyde, C. C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York.Google Scholar
[11] Hannan, E. J. (1970). Multiple Time Series. John Wiley, New York.Google Scholar
[12] Hannan, E. J. (1974). The uniform convergence of autocovariances. Ann. Statist. 2, 803–806.CrossRefGoogle Scholar
[13] Hannan, E. J. and Heyde, C. C. (1972). On limit theorems for quadratic functions of discrete time series. Ann. Math. Statist. 43, 2058–2066.Google Scholar
[14] Hosking, J. R. M. (1996). Asymptotic distributions of the sample mean, autocovariances, and autocorelations of long-memory time series. J. Econometrics 73, 261–284.Google Scholar
[15] Ibragimov, I. A. and Linnik, Y. V. (1971). Independent and Stationary Sequence of Random Variables. Wolters-Noorhoff, Groningen.Google Scholar
[16] Ivanov, A. V. and Leonenko, N. N. (1989). Statistical Analysis of Random Fields. Kluwer, Dordrecht.CrossRefGoogle Scholar
[17] Leonenko, N. N. (1999). Limit Theorems for Random Fields with Singular Spectrum (Math. Appl. 465). Kluwer, Dordrecht.Google Scholar
[18] Robinson, P. M. (1994). Time series with strong dependence. In Advances in Econometrics , Sixth World Congress (Econom. Soc. Monogr. 23), ed. Sims, C. A., Vol. 1, Cambridge University Press, pp. 47–95.CrossRefGoogle Scholar
[19] Robinson, P. M. (1996). Large-sample inference for nonparametric regression with dependent errors. Ann. Statist. 26, 2054–2083.Google Scholar
[20] Surgailis, D. (1981). Convergence of sums of nonlinear functions of moving averages to self-similar processes. Soviet Math. Dokl. 23, 247–250.Google Scholar
[21] Surgailis, D. (1982). Zone of attraction of self-similar multiple integrals. Lithuanian Math. J. 22, 327–340.Google Scholar