Summary
A central limit theorem for quadratic forms in strongly dependent linear (or moving average) variables is proved, generalizing the results of Avram [1] and Fox and Taqqu [3] for Gaussian variables. The theorem is applied to prove asymptotical normality of Whittle's estimate of the parameter of strongly dependent linear sequences.
References
Avram, F.: On bilinear forms in Gaussian random variables and Toeplitz matrices. Probab. Th. Rel. Fields79, 37–45 (1988)
Dahlhaus, R.: Efficient parameter estimation for self-similar processes. Ann. Stat.17, 1749–1766 (1989)
Fox, R., Taqqu, M.S.: Central limit theorems for quadratic forms in random variables having long-range dependence. Probab. Th. Rel. Fields74, 213–240 (1987)
Fox, R., Taqqu, M.S.: Large sample properties of parameter estimates for strongly dependent stationary Gaussian time series. Ann. Statist.14, 517–532 (1986)
Grenander, V., Szegő, G.: Toeplitz forms and their applications. University of California Press 1958
Hannan, E.J.: The asymptotic theory of linear time series models. J. Appl. Probab.10, 130–145 (1973)
Ibragimov, I.A., Linnik, J.V.: Independent and stationary sequences of random variables. Gröningen: Walters-Noordhoff 1971
Natanson, I.P.: Theory of functions of a real variable, vol. I. New York: Ungar 1955
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Giraitis, L., Surgailis, D. A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotical normality of Whittle's estimate. Probab. Th. Rel. Fields 86, 87–104 (1990). https://doi.org/10.1007/BF01207515
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01207515
Keywords
- Stochastic Process
- Probability Theory
- Quadratic Form
- Dependent Linear
- Limit Theorem