When the bispectrum is real-valued

TL;DR

This paper explores conditions under which the bispectrum of a stationary time series is real-valued, revealing implications for reversibility and causality in linear processes with positive spectra.

Contribution

It broadens the scope of bispectrum analysis and establishes new links between bispectrum properties, reversibility, and causality in stationary time series.

Findings

Bispectrum is real-valued and nonzero implies reversibility for linear processes.

Linear processes with nonzero skewness are reversible if third order reversibility holds.

The paper extends bispectrum analysis beyond traditional definitions.

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.