Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-22T17:57:28.794Z Has data issue: false hasContentIssue false
  • English
  • Français

Covariances Estimation for Long-Memory Processes

Part of: Stochastic processes Limit theorems Inference from stochastic processes

Published online by Cambridge University Press:  01 July 2016

Wei Biao Wu ,
Yinxiao Huang  and
Wei Zheng
Wei Biao Wu*
Affiliation:
University of Chicago
Yinxiao Huang*
Affiliation:
University of Chicago
Wei Zheng*
Affiliation:
University of Illinois at Chicago
*
∗ Postal address: Department of Statistics, University of Chicago, Chicago, IL 60637, USA.
∗ Postal address: Department of Statistics, University of Chicago, Chicago, IL 60637, USA.
∗∗∗ Postal address: Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S. Morgan Street, Chicago, IL 60607-7045, USA.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For a time series, a plot of sample covariances is a popular way to assess its dependence properties. In this paper we give a systematic characterization of the asymptotic behavior of sample covariances of long-memory linear processes. Central and noncentral limit theorems are obtained for sample covariances with bounded as well as unbounded lags. It is shown that the limiting distribution depends in a very interesting way on the strength of dependence, the heavy-tailedness of the innovations, and the magnitude of the lags.

Keywords

Asymptotic normalitycovariancedichotomylinear processlong-range dependenceRosenblatt distribution

MSC classification

Primary: 60F05: Central limit and other weak theorems 62M10: Time series, auto-correlation, regression, etc.
Secondary: 60G10: Stationary processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 42 , Issue 1 , March 2010 , pp. 137 - 157
Copyright
Copyright © Applied Probability Trust 2010 

References

Anderson, T. W. (1971). The Statistical Analysis of Time Series. John Wiley, New York.Google Scholar
Anderson, T. W. and Walker, A. M. (1964). On the asymptotic distribution of the autocorrelations of a sample from a linear stochastic process. Ann. Math. Statist. 35, 1296–1303.CrossRefGoogle Scholar
Avram, F. and Taqqu, M. S. (1987). Noncentral limit theorems and Appell polynomials. Ann. Prob. 15, 767–775.Google Scholar
Bartlett, M. S. (1946). On the theoretical specification and sampling properties of autocorrelated time-series. J. R. Statist. Soc. 8, 27–41.Google Scholar
Berlinet, A. and Francq, C. (1999). Estimation of the asymptotic behavior of sample autocovariances and empirical autocorrelations of multivalued processes. Canad. J. Statist. 27, 525–546.Google Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation. Cambridge University Press.Google Scholar
Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods, 2nd edn. Springer, New York.Google Scholar
Chow, Y. S. and Teicher, H. (1988). Probability Theory, 2nd edn. Springer, New York.Google Scholar
Chung, C. F. (2002). Sample means, sample autocovariances, and linear regression of stationary multivariate long memory processes. Econometric Theory 18, 51–78.Google Scholar
Csörgő, S. and Mielniczuk, J. (2000). The smoothing dichotomy in random-design regression with long-memory errors based on moving averages. Statistica Sinica 10, 771–787.Google Scholar
Dai, W. (2004). Asymptotics of the sample mean and sample covariance of long-range-dependent series. In Stochastic Methods and Their Applications (J. Appl. Prob. Spec. Vol. 41A), eds Gani, J. and Seneta, E., Applied Probability Trust, Sheffield, pp. 383–392.Google Scholar
Dehling, H. and Taqqu, M. (1991). Bivariate symmetric statistics of long-range dependent observations. J. Statist. Planning Infer. 28, 153–165.Google Scholar
Giraitis, L. and Taqqu, M. S. (1999). Convergence of normalized quadratic forms. J. Statist. Planning Infer. 80, 15–35.Google Scholar
Hall, P. and Heyde, C. C. (1980). Martingale Limit Theorem and Its Application. Academic Press, New York.Google Scholar
Hannan, E. J. (1970). Multiple Time Series. John Wiley, New York.Google Scholar
Hannan, E. J. (1976). The asymptotic distribution of serial covariances. Ann. Statist. 4, 396–399.Google Scholar
Harris, D., McCabe, B. and Leybourne, S. (2003). Some limit theory for autocovariances whose order depends on sample size. Econometric Theory 19, 829–864.CrossRefGoogle Scholar
Ho, H.-C. and Hsing, T. (1997). Limit theorems for functionals of moving averages. Ann. Prob. 25, 1636–1669.Google Scholar
Horváth, L. and Kokoszka, P. (2008). Sample autocovariances of long-memory time series. Bernoulli 14, 405–418.Google Scholar
Hosking, J. R. M. (1996). Asymptotic distributions of the sample mean, autocovariances, and autocorrelations of long-memory time series. J. Econometrics 73, 261–284.CrossRefGoogle Scholar
Hsieh, M.-C., Hurvich, C. M. and Soulier, P. (2007). Asymptotics for duration-driven long range dependent processes. J. Econometrics 141, 913–949.Google Scholar
Keenan, D. M. (1997). A central limit theorem for m(n) autocovariances. J. Time Ser. Anal. 18, 61–78.Google Scholar
Major, P. (1981). Multiple Wiener-Itô Integrals (Lecture Notes Math. 849). Springer, New York.Google Scholar
Mikosch, T., Resnick, S., Rootzen, H. and Stegeman, A. (2002). Is network traffic approximated by stable Lévy motion or fractional Brownian motion? Ann. Appl. Prob. 12, 23–68.Google Scholar
Phillips, P. C. B. and Solo, V. (1992). Asymptotics for linear processes. Ann. Statist. 20, 971–1001.Google Scholar
Porat, B. (1987). Some asymptotic properties of the sample covariances of Gaussian autoregressive moving average processes. J. Time Ser. Anal. 8, 205–220.Google Scholar
Rosenblatt, M. (1979). Some limit theorems for partial sums of quadratic forms in stationary Gaussian variables. Z. Wahrscheinlichkeitsth. 49, 125–132.Google Scholar
Sly, A. and Heyde, C. (2008). Nonstandard limit theorem for infinite variance functionals. Ann. Prob. 36, 796–805.Google Scholar
Surgailis, D. (1982). Zones of attraction of self-similar multiple integrals. Lithuanian Math. J. 22, 327–340.Google Scholar
Surgailis, D. (2004). Stable limits of sums of bounded functions of long-memory moving averages with finite variance. Bernoulli 10, 327–355.Google Scholar
Taqqu, M. S. (1979). Convergence of integrated processes of arbitrary Hermite rank. Z. Wahrscheinlichkeitsth. 50, 53–83.Google Scholar
Wu, W. B. (2007). Strong invariance principles for dependent random variables. Ann. Prob. 35, 2294–2320.Google Scholar
Wu, W. B. (2009). An asymptotic theory for sample covariances of Bernoulli shifts. Stoch. Process. Appl. 119, 453–467.Google Scholar
Wu, W. B. and Min, W. (2005). On linear processes with dependent innovations. Stoch. Process. Appl. 115, 939–958.Google Scholar
Wu, W. B. and Woodroofe, M. (2004). Martingale approximations for sums of stationary processes. Ann. Prob. 32, 1674–1690 Google Scholar
Yajima, Y. (1993). Asymptotic properties of estimates in incorrect ARMA models for long-memory time series. In New Directions in Time Series Analysis. Part II (IMA Vol. Math. Appl. 46), eds Brillinger, D. et al., Springer, New York, pp. 375–382.Google Scholar