Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

Linear time-invariant transformations of some nonstationary random processes


Authors: D. Tjøstheim and J. B. Thomas
Journal: Quart. Appl. Math. 34 (1976), 113-117
MSC: Primary 60G99
DOI: https://doi.org/10.1090/qam/448551
MathSciNet review: 448551
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Abstract: We consider the class of nonstationary processes $ Y\left( t \right)$ which can be represented as $ Y\left( t \right) = BX\left( t \right)$, where $ X\left( t \right)$ is wide sense stationary and $ B$ is a bounded self-adjoint operator with a bounded inverse. An equivalent characterization of this class of processes is given and is used to construct examples of nonstationary processes belonging to this class. A functional analytic treatment is given for describing the effects of linear time-invariant transformations.


References [Enhancements On Off] (What's this?)

  • [1] G. E. P. Box and G. M. Jenkins, Time series analysis forecasting and control, San Francisco, Holden Day, 1970 MR 0272138
  • [2] H. Cramér and M. R. Leadbetter, Stationary and related stochastic processes, New York, Wiley, 1967
  • [3] J. L. Doob, Stochastic processes, New York, Wiley, 1967 MR 0058896
  • [4] N. Dunford and J. T. Schwartz, Linear operators, Part III: spectral operators, New York, Wiley-Interscience, 1971 MR 1009164
  • [5] R. K. Getoor, The shift operator for nonstationary stochastic processes, Duke Math. J. 23, 175-187 (1956) MR 0074717
  • [6] V. Mandrekar, A characterization of oscillatory processes and their prediction, Proc. Am. Math. Soc. 32, 280-283 (1972) MR 0307310
  • [7] M. M. Martin, Utilisation des methodes de l'analyse spectrale á la prevision lineaire de certains processus non stationnaires, Rev. CETHEDEC 16, 137-148 (1968)
  • [8] M. B. Priestley, Evolutionary spectra and non-stationary processes, Roy. Statist. Soc. B. 27, 204-233 (1965) MR 0199886
  • [9] F. Riesz and B. de Sz.-Nagy, Functional analysis, New York, Ungar, 1955 (2nd ed.) MR 0071727
  • [10] B. de Sz.-Nagy, On uniformly bounded linear operators, Acta Sci. Math. Szeged. 11, 152-157 (1947)

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Additional Information

DOI: https://doi.org/10.1090/qam/448551
Article copyright: © Copyright 1976 American Mathematical Society

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