Quantitative CLTs on the Poisson space via Skorohod estimates and p-Poincaré inequalities

We establish new explicit bounds on the Gaussian approximation of Poisson functionals based on novel estimates of moments of Skorohod integrals. Combining these with the Malliavin–Stein method, we derive bounds in the Wasserstein and Kolmogorov distances whose application requires minimal moment assumptions on add-one cost operators—thereby extending the results from (Last, Peccati and Schulte, 2016). Our paper also removes a redundant term from the bound in Kolmogorov distance and replaces a complicated term by a simpler one, without requiring additional assumptions. Our applications include a central limit theorem (CLT) for the online nearest neighbour graph, whose validity was conjectured in (Wade, 2009; Penrose and Wade, 2009). We also apply our techniques to derive quantitative CLTs for edge functionals of the Gilbert graph, of the k-nearest neighbour graph and of the radial spanning tree. In most cases, even the qualitative CLTs are new.