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

This paper computes Poisson space noise functionals, $P^{\prime}(u)$, realised in a Gel'fand triple built from a L\'evy measure $\lambda_{\beta}(u)du$. 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 $\lambda_{\beta}(u)=cu^{-1-\alpha}$ ($0<\alpha<2$), replacing discarded small amplitudes with matched Gaussian space noise improves the Wasserstein-1 error from $O(\epsilon^{1-\alpha/2})$ to $O(\epsilon)$. The residual is asymptotically normal at $O(\epsilon^{\alpha/2})$. This compensation reduces computational complexity from $O(\tau^{-2\alpha/(2-\alpha)})$ to $O(\tau^{-\alpha})$. We also evaluate the Gamma-type boundary ($\alpha=0$) 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.