Some remarks on quadratic forms in stable random variables

• Autorzy: Mijnheer J.
• Języki publikacji: EN

Abstrakty

EN

We consider quadratic forms that appear in the least squares estimator for the unknown parameter in the AR(1) model with stable innovations. In three cases we obtain different limit distributions.

Słowa kluczowe

EN

quadratic forms; random variables

PL

zmienne losowe

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2001
• Tom: Vol. 34, nr 2
• Strony: 327--338
• Opis fizyczny: Bibliogr. 11 poz.

Twórcy

autor

Mijnheer J.

• University of Leiden Mathematical Institute P.O. Box 9512 2300 RA Leiden, The Netherlands

Bibliografia

• 1. Anderson, T.W. (1959). On asymptotic distributions of estimates of parameters of stochastic difference equations, Ann. Math. Statist. 30, 676-687.
• 2. Hu, Y. and W.A. Woyczynski (1995). Limit behavior of quadratic forms of moving averages and statistical solutions of the Burgers' Equation, J. Multiv. An. 52, 15-44.
• 3. Kwapień, S. and W. A. Woyczynski (1992). Random Series and Stochastic Integrals, Single and Multiple, Birkhauser, Boston.
• 4. Mijnheer, J. (1991). U-statistics and double stable integrals, Selected Proceedings of the Sheffield Symposium on Applied Probability, eds. I.V. Basawa & R.L. Taylor. IMS Lecture Notes 18, 256-268.
• 5. Mijnheer, J. (1995). On the influence of an extremal term in a (simple) U-statistic, Theory of Probability and Mathematical Statistics 50, 103-107.
• 6. Mijnheer, J. (1997). Asymptotic inference for AR(1) processes with (non-normal) stable errors III, Stable Ornstein-Uhlenbeck processes and integrals of stable O - U processes, Stochastic Models 13, 661-672.
• 7. Mijnheer, J. (1998a). Asymptotic inference for AR(1) processes with (non-normal) stable errors IV. A note on the case of a negative unit root, J. Math. Sciences 92, 4035-4038.
• 8. Mijnheer, J. (1998b). The asymptotic distribution of a sequential estimator for the parameter in an AR(1) model with stable errors. Proceedings of Conference „Asymptotic Methods in Probability and Statistics with Applications", St. Petersburg, August 1998. Publisher Birkhauser (in print).
• 9. Varberg, D.E. (1968). Almost sure convergence of quadratic forms in independent random variables, Ann. Math. Stat. 39, 1502-1506.
• 10. Williams, D. (1992). Probability with Martingales, Cambridge Math. Textbooks. Cambridge Univ. Press.
• 11. Zolotarev, V.M. (1986). One-dimensional stable distributions, vol. 65 of „Translations of Mathematical Monographs", Amer. Math. Soc. Translation from the original 1983 Russian edition.