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Mathematics  2009 

Poisson--Voronoi approximation

DOI: 10.1214/08-AAP561

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Abstract:

Let $X$ be a Poisson point process and $K\subset\mathbb{R}^d$ a measurable set. Construct the Voronoi cells of all points $x\in X$ with respect to $X$, and denote by $v_X(K)$ the union of all Voronoi cells with nucleus in $K$. For $K$ a compact convex set the expectation of the volume difference $V(v_X(K))-V(K)$ and the symmetric difference $V(v_X(K)\triangle K)$ is computed. Precise estimates for the variance of both quantities are obtained which follow from a new jackknife inequality for the variance of functionals of a Poisson point process. Concentration inequalities for both quantities are proved using Azuma's inequality.

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