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Non-singularity and asymptotic independence

Published online by Cambridge University Press:  14 July 2016

Peter Bloomfield
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Abstract

A stationary stochastic process must satisfy various requirements to make it a realistic model for a phenomenon in the real world. Some of these requirements are quantitative, such as agreement of distribution or moments. Other, more qualitative requirements deal with the general behavior of the process. Two such requirements are non-singularity and asymptotic independence. Each will be discussed from a variety of points of view, and given precise definition in a succession of progressively stronger forms.

Keywords

DETERMINISM AND NON-DETERMINISMPREDICTABILITYMINIMALITYCANONICAL AND ESSENTIAL CORRELATIONCOMPLETE AND ABSOLUTE REGULARITY

Information

Type
Part 1—Structure and General Methods for Time Series
Information
Journal of Applied Probability , Volume 23 , Issue A: Essays in Time Series and Allied Processes , 1986, pp. 9 - 21
Copyright
Copyright © 1986 Applied Probability Trust 

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References

Bloomfield, P., Jewell, N. P. and Hayashi, E. (1983) Characterization of completely non-deterministic stochastic process. Pacific J. Math. 197, 307–317.Google Scholar
Deleeuw, K. and Rudin, W. (1958) Extreme points and extremum problems in H1. Pacific J. Math. 8, 467–485.Google Scholar
Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
Gelfand, I. M. and Yaglom, A. M. (1959) Calculation of the amount of information about a random function contained in another such function. Amer. Math. Soc. Translations 12, 199–246.Google Scholar
Hannan, E. J. (1961) The general theory of canonical correlation and its relation to functional analysis. J. Austral. Math. Soc. 2, 229–242.CrossRefGoogle Scholar
Hannan, E. J. (1970) Multiple Time Series. Wiley, New York.Google Scholar
Helson, H. and Sarason, D. E. (1967) Past and future. Math. Scand. 21, 5–16.Google Scholar
Helson, H. and Szego, S. (1960) A problem in prediction theory. Ann. Math. Pura Appl. 51, 107–138.Google Scholar
Hunt, R. A., Muckenhoupt, B. and Wheeden, R. L. (1973) Weighted norm inequalities for the conjugate function and Hilbert transform. Trans. Amer. Math. Soc. 176, 227–251.Google Scholar
Ibragimov, I. A. and Rozanov, Y. A. (1978) Gaussian Random Processes. Springer-Verlag, New York.Google Scholar
Jewell, N. P. and Bloomfield, P. (1983) Canonical correlations of past and future for time series: definitions and theory. Ann. Statist. 11, 837–847.Google Scholar
Kolmogorov, A. N. (1941) Stationary sequences in Hilbert space. Bull. Moscow State Univ. 2, 1–40. Reprinted in translation in Linear Least Squares Estimation , ed. Kailath, T., Benchmark Papers in Electrical Engineering and Computer Science, 17, Dowden, Hutchinson & Ross, Stroudsburg, PA, 66-89.Google Scholar
Rosenblatt, M. (1956) A central limit theorem and a strong mixing condition. Proc. Nat. Acad. Sci. USA 42, 43–47.Google Scholar
Sarason, D. E. (1972) An addendum to “Past and future.” Math. Scand. 30, 62–64.Google Scholar
Zygmund, A. (1959) Trigonometric Series , Vol. I. Cambridge University Press, Cambridge.Google Scholar