On the existence of autoregressive models for third-order cumulant matching

Parametric modeling of third-order cumulant sequences has assumed importance recently because of the applications of the bispectrum. Approaches have been developed for autoregressive modeling of random processes using third-order cumulants, and these are based on solving linear equations which a re necessary but not sufficient conditions for matching samples of the cumulant sequence of the model to those of the process under consideration. This paper provides necessary and sufficient conditions for the third moment sequence of a white noise driven finite order AR model to match given samples of the third moment sequence of an arbitrary stationary process. These lead to a set of nonlinear equations to be solved for the model parameters. A method for finding the third moment sequence of a white noise driven AR model from its parameters is also provided. One of the key results of the paper is that as opposed to the extendability of a finite set of autocorrelation samples, a finite set of third moment sequence samples is not always linearly extendable to an infinite third moment sequence.