On the measure of the set of factorizable polynomial bispectra

In a recent work (1989), the authors have shown that factorization of bispectrum is not always possible. In the present work, they show that the subset of factorizable bispectra has Lebesgue measure zero in the set of polynomial bispectra, i.e. those that are obtained from finite-support bicumulants. Hence, a polynomial bispectrum cannot almost always be exactly realized as that of the output of a linear model driven by a third-order white input. This result can be generalized to multidimensional polynomial bispectra. Although it follows that a linear model driven by a third-order white input cannot almost always realize a given bispectrum or a bicumulant sequence, the use of a linear model as an approximation in certain applications can be justified if the computed bispectrum has an index value close to unity.<
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