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Linear processes and bispectra

Published online by Cambridge University Press:  14 July 2016

M. Rosenblatt*
University of California, San Diego
Postal address: Department of Mathematics, University of California, San Diego, La Jolla, CA 92093, U.S.A.


A linear process is generated by applying a linear filter to independent, identically distributed random variables. Only the modulus of the frequency response function can be estimated if only the linear process is observed and if the independent identically distributed random variables are Gaussian. In this case a number of distinct but related problems coalesce and the usual discussion of these problems assumes, for example, in the case of a moving average that the zeros of the polynomial given by the filter have modulus greater than one. However, if the independent identically distributed random variables are non-Gaussian, these problems become distinct and one can estimate the transfer function under appropriate conditions except for a possible linear phase shift by using higher-order spectral estimates.

Short Communications
Copyright © Applied Probability Trust 1980 

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Research partially supported by the Office of Naval Research Contract N00014–75–C–0428.


[1] Anderson, T. W. (1977) Estimation for autoregressive moving average models in the time and frequency domain, Ann. Statist. 5, 842865.CrossRefGoogle Scholar
[2] Åström, K. (1970) Introduction to Stochastic Control Theory. Academic Press, New York.Google Scholar
[3] Brillinger, D. R. and Rosenblatt, M. (1967) Asymptotic theory of estimates of k th order spectra. In Spectral Analysis of Time Series, ed. B. Harris, Wiley, New York, 153–188.Google Scholar
[4] Hannan, E. J. (1970) Multiple Time Series. Wiley, New York.CrossRefGoogle Scholar
[5] Whittle, P. (1963) Prediction and Regulation. English Universities Press, London.Google Scholar