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

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.