Abstract
The problem of finding analytic conditions for subordination of harmonizable and periodically correlated sequences is studied. Sufficient conditions for subordination of harmonizable sequences (in the spirit of Kolmogorov’s work) and a simple counter-example showing that these conditions are not necessary are given. In the case of periodically correlated sequences, which is a subclass of harmonizable sequences, necessary and sufficient conditions for subordination, mutual subordination and necessary conditions for strong subordination of such processes in terms of their associated multivariate stationary sequences are derived. The problem of finding spectral conditions such that it is possible to find a mean-convergent series for a harmonizable sequence, when it is a linear transformation of another such sequence is studied. This idea is used to find an algorithm for the linear predictor and interpolator of a periodically correlated process in the time domain.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
L. Abreu, A note on harmonizable and stationary sequences, Bol. Soc. Mat. Mexicana 15 (1970), 48–51.
J.L. Abreu, Stationary prediction of non-stationary sequences in Hilbert space, Prediction theory and harmonic analysis (V. Mandrekar and H. Salehi editors) North-Holland, Amsterdam (1983), 1–12.
S. Cambanis and B. Liu, On harmonizable stochastic processes, Information and Control 17 (1970), 183–202.
S. Cambanis and S.T. Huang, Stochastic and multiple wiener integrals for Gaussian Processes, Ann. Probab. (1978), 585–614.
H. Cramér, A contribution to the theory of stochastic processes, 2nd Berkeley Symp. Math. Stat. Prob. (1951), 169–179.
E.G. GlĂdysev, On periodically correlated random sequences, Soviet Math. Dokl. 2 (1961), 385–388.
F. Graef, Optimal filtering of infinite-dimensional stationary signals, Lecture Notes in Math. 695, Springer-Verlag (1977) 63–75.
A.N. Kolmogorov, Stationary sequences in Hilbert space, Bull. Math. Univ. Moscow 2 (1941), 1–40.
V. Mandrekar and H. Salehi, Subordination of infinite-dimensional stationary stochastic processes, Ann. Inst. H. Poincare, Sec. B, 6 (1970), 115–130.
P. Masani, Recent trends in Multivariate prediction theory, Multivariate Analysis (P.R. Krishnaiah, editor), Academic Press, New York (1966), 351–382.
A.G. Miamee and H. Salehi, On the prediction of periodically correlated stochastic processes, Multivariate Analysis 5 (P.R. Krishnaiah, editor), North-Holland, Amsterdam (1980), 167–179.
H. Niemi, Diagonal measure of a positive definite bimeasure, Lecture Notes in Math. 945, Springer-Verlay (1982) 237–246.
M. Pourahmadi, On subordination, sampling theorem and "Past and Future" on some classes of second-order processes, Ph.D. thesis, Department of Statistics and Probability, Michigan State University (1980).
M. Pourahmadi, On complete regularity of vector-valued stationary processes (unpublished), (1981).
M. Pourahmadi, A matricial extension of the Helson-Szegö theorem and its application in multivariate prediction. J. Multivariate Analysis (to appear).
M. Pourahmadi, On the mean convergence of the best linear interpolator of multivariate stationary stochastic processes. Annals Probab. (to appear).
Yu. A. Rozanov, Stationary random processes, Holden-Day, San Francisco, 1967.
M. Rosenberg, The spectral analysis of multivariate weakly stationary stochastic processes, Ph.D. Thesis, Indiana Univ. 1964.
M. Rosenberg, Mutual subordination of multivariate stationary processes over any locally compact Abelian group, Z.W. verv. Gebiete 12 (1969), 333–343.
T.N. Siraya, On subordinate processes, Theory Probab. Appl. 22 (1977), 129–133.
T.N. Siraya, The projection of processes with orthogonal increments, and subordinate processes (Russian) Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 72 (1977), 132–139. Math. Revs. 57 #17803.
A Weron, Harmonizable stable processes on groups: spectral, ergodic and interpolation properties (manuscript) (1983).
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1984 Springer-Verlag
About this paper
Cite this paper
Pourahmadi, M., Salehi, H. (1984). On subordination and linear transformation of harmonizable and periodically correlated processes. In: Szynal, D., Weron, A. (eds) Probability Theory on Vector Spaces III. Lecture Notes in Mathematics, vol 1080. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0099797
Download citation
DOI: https://doi.org/10.1007/BFb0099797
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-13388-9
Online ISBN: 978-3-540-38939-2
eBook Packages: Springer Book Archive