When the bispectrum is real-valued

Abstract

Let {X(t)} be a stationary time series with a.e. positive spectrum. Two consequences of that the bispectrum of {X(t)} is real-valued but nonzero: 1) if {X(t)} is also linear, then it is reversible; 2) {X(t),} can not be causal linear. A corollary of the first statement: if {X(t)} is linear, and the skewness of X(0) is nonzero, then third order reversibility implies reversibility. In this paper the notion of bispectrum is of a broader scope.