Orientation in Poisson Cluster Processes via Imaginary Bispectra

TL;DR

This paper investigates how to detect orientation in Poisson cluster processes after orientation information is removed, using spectral and bispectral methods to identify directional properties.

Contribution

It introduces new spectral and bispectral techniques to determine orientation in Poisson cluster processes, even when orientation information is erased.

Findings

Second-order structure alone cannot identify temporal direction.

A nonzero imaginary bispectrum certifies orientation in the $L^1$ third-cumulant regime.

Explicit null models and spectral matches are provided for various cases.

Abstract

We study what remains detectable about one-sided Poisson cluster processes after cluster orientation is erased. We construct matched reversible cluster nulls preserving intensity and the full Bartlett spectrum, showing that second-order structure alone need not identify temporal direction. For stationary Poisson branching clusters, we derive the Fourier--Stieltjes transform of the reduced third cumulant and show that, in the $L^1$ third-cumulant regime, a nonzero imaginary factorial bispectrum certifies orientation. We also give explicit orientation-erased nulls, reversible spectral matches for monotone Hawkes kernels, and finite-window third-order orientation contrasts.