Abstract
For stationary processes with infinite variance the notions of covariance and spectrum are not defined. We characterize regularity and minimality of such processes in the spirit of some classical results for second-order processes, namely values of the process forming conditional basis for their spans. Several open problems are discussed.
Similar content being viewed by others
REFERENCES
Cambanis, S., Hardin, C. D., and Weron, A. (1988). Innovations and Wold decompositions of stable processes. Prob. Th. Rel. Fields 79, 1–27.
Cambanis, S., and Soltani, A. R. (1984). Prediction of stable processes: Spectral and moving average representations. Z. Wahrsch. Verw. Geb. 66, 593–612.
Cheng, R., Miamee, A. G., and Pourahmadi, M. (1998). Some extremal problems in L p (w). Proc. of Amer. Math Soc. 126, 2333–2340.
Degerine, S. (1982). Partial autocorrelation function for a scalar stationary discrete-time process. Proc. Third Franco-Belgian Meeting of Statisticians, Bruxelles: Publications des Faculles Universitaires Sait-Louis, pp. 79–94.
Doob, J. L. (1953). Stochastic Processes, John Wiley, New York.
Kolmogorov, A. N. (1941). Stationary sequences in Hilbert space. Bull. Math. Univ. Moscow 2, 1–40.
Makagan, A., and Mandrekar, V. (1990). The spectral representation of stable processes. Harmonizability and regularity. Prob. Th. Rel. Fields 85, 1–11.
Miamee, A. G., and Pourahmadi, M. (1988a). Wold decomposition, prediction and parametrization of stationary processes with infinite variance. Prob. Th. Rel. Fields 79, 145–164.
Miamee, A. G., and Pourahmadi, M. (19881)). Best approximation in L p(dµ) and prediction problems of Szegö, Kolmogorov, Yaglom and Nakazi. J. London Math. Doc. 38, 133–145.
Pourahmadi, M. (1984). On minimality and interpolation of harmonizable stable processes. SIAM J. Appl. Math. 44, 1023–1030.
Rajput, B. S., and Sundberg, C. (1994). On some extermal problems in H p and the prediction of L p-harmonizable stochastic processes. Prob. Th. Rel. Fields 99, 197–210.
Rozanov, Yu. A. (1967). Stationary Random Processes, Holden-Day, San Francisco, California.
Samorodnitsky, G., and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance, New York, Chapman and Hall.
Singer, I. (1981). Bases in Banach Spaces II, Springer-Verlag, New York.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Cheng, R., Miamee, A.G. & Pourahmadi, M. Regularity and Minimality of Infinite Variance Processes. Journal of Theoretical Probability 13, 1115–1122 (2000). https://doi.org/10.1023/A:1007874226636
Issue Date:
DOI: https://doi.org/10.1023/A:1007874226636