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.
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N. Dunford and J. T. Schwartz, Linear operators, Part III: spectral operators, New York, Wiley-Interscience, 1971
R. K. Getoor, The shift operator for nonstationary stochastic processes, Duke Math. J. 23, 175–187 (1956)
V. Mandrekar, A characterization of oscillatory processes and their prediction, Proc. Am. Math. Soc. 32, 280–283 (1972)
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)
M. B. Priestley, Evolutionary spectra and non-stationary processes, Roy. Statist. Soc. B. 27, 204–233 (1965)
F. Riesz and B. de Sz.-Nagy, Functional analysis, New York, Ungar, 1955 (2nd ed.)
B. de Sz.-Nagy, On uniformly bounded linear operators, Acta Sci. Math. Szeged. 11, 152–157 (1947)
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Article copyright:
© Copyright 1976
American Mathematical Society
