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Normal approximation in total variation for statistics in geometric probability
- Part of
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- Journal:
- Advances in Applied Probability / Volume 56 / Issue 1 / March 2024
- Published online by Cambridge University Press:
- 03 July 2023, pp. 106-155
- Print publication:
- March 2024
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- Article
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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 samples arising from random events driven by a marked Poisson point process on
$\mathbb{R}^d$. As in the study under the weaker Kolmogorov distance, the score functions are assumed to satisfy stabilisation and moment conditions. At the cost of an additional non-singularity condition, 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 tessellations, k-nearest-neighbours graphs, timber volume, and maximal layers.
