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On Wiener–Ito representation and the best linear predictors for bilinear time series

Published online by Cambridge University Press:  14 July 2016

Gy. Terdik  and
T. Subba Rao
Gy. Terdik*
Affiliation:
Kossuth Lajos University, Debrecen
T. Subba Rao*
Affiliation:
UMIST
*
∗ Postal address: Mathematical Institute, KLTM, Kossuth L. University, Pf 12 Debrecen, Hungary.
∗∗ Postal address: Department of Mathematics, UMIST, PO Box 88, Manchester M60 1QD, UK.
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Abstract

This paper gives the necessary and sufficient condition for the second-order stationarity of lower triangular bilinear models; this condition is in terms of norms of higher-order transfer functions. The spectral density function of the process is evaluated in terms of the transfer functions, and it is shown that the second-order structure is similar to a linear ARMA model with uncorrelated errors. The best linear predictor is obtained and it is shown that the variance of the prediction error is always greater than the optimal prediction error variance obtained from the bilinear model. An expression for the bispectral density function is also obtained.

Keywords

BILINEAR MODELSHIGHER-ORDER TRANSFER FUNCTIONSSPECTRALINEAR PREDICTORSSTATIONARITY
Type
Research Papers
Information
Journal of Applied Probability , Volume 26 , Issue 2 , June 1989 , pp. 274 - 286
Copyright
Copyright © Applied Probability Trust 1989 

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