Orientation in Poisson Cluster Processes via Imaginary Bispectra

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.