When the bispectrum is real-valued

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