Gamma approximation and Poisson--Gaussian invariance principle on Poisson chaos

What the math gives to ML

The paper identifies the fourth add-one energy of a Poisson functional as an intrinsic Lindeberg quantity: when this energy vanishes after variance normalization, a fixed-order Poisson chaos and the Gaussian chaos built from the same kernel become indistinguishable in distribution. This is stronger and more structurally informative than matching a few ordinary moments, because rare jumps can preserve low-order moments while still preventing Gaussian behavior. A transferable use is to build sparse Poisson random-feature or noise layers whose kernels are explicitly regularized toward low add-one energy, obtaining Gaussian-like representations while retaining discrete or sparse sampling. The criterion can also diagnose when Gaussian initialization or Gaussian dropout is a faithful replacement for a sparse Poisson mechanism.