A spectral-compensated scheme for space-parameter Poisson noise functionals: error bounds and complexity estimates

Abstract

This paper computes Poisson space noise functionals, $P^{\prime }(u)$, realised in a Gel'fand triple built from a Lévy measure $\hat{I}{»}_{\hat{I}}^2(u)du$. We isolate three discretisation parameters: a small-amplitude cut-off, a Donsker delta truncation M, and a chaos order N. For a stable-type intensity $\hat{I}{»}_{\hat{I}}^2(u)=c{u}^{-1-\hat{I}\pm }$ ($0<\hat{I}\pm <2$), replacing discarded small amplitudes with matched Gaussian space noise improves the Wasserstein-1 error from $O(\hat{I}{\mu }^{1-\hat{I}\pm /2})$ to $O(\hat{I}\mu )$. The residual is asymptotically normal at $O(\hat{I}{\mu }^{\hat{I}\pm /2})$. This compensation reduces computational complexity from $O(\ddot{I}{„}^{-2\hat{I}\pm /(2-\hat{I}\pm )})$ to $O(\ddot{I}{„}^{-\hat{I}\pm })$. We also evaluate the Gamma-type boundary ($\hat{I}\pm =0$) and exponential tempering. Truncations converge algebraically (M) and super-geometrically (N). All predicted rates are tightly confirmed by deterministic numerical experiments via Gil-Pelaez inversion, eliminating Monte Carlo noise.

24 pages, 7 figures. Submitted for publication

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