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Normal approximation in total variation for statistics in geometric probability (2011.07781v1)
Published 16 Nov 2020 in math.PR
Abstract: We use Stein's method to establish the rates of normal approximation in terms of the total variation distance for a large class of sums of score functions of marked Poisson point processes on $\mathbb{R}d$. As in the study under the weaker Kolmogorov distance, the score functions are assumed to satisfy stabilizing and moment conditions. At the cost of an additional non-singularity condition for score functions, we show that the rates are in line with those under the Kolmogorov distance. We demonstrate the use of the theorems in four applications: Voronoi tessellation, $k$-nearest neighbours, timber volume and maximal layers.