Extremes for the inradius in the Poisson line tessellation. (English) Zbl 1342.60011

Chenavier, Nicolas; Hemsley, Ross

Adv. Appl. Probab. 48, No. 2, 544-573 (2016).

Summary: A Poisson line tessellation is observed in the window \(\boldsymbol{W}_\rho:=B(0,\pi^{-1/2}\rho^{1/2})\) for \(\rho>0\). With each cell of the tessellation, we associate the inradius, which is the radius of the largest ball contained in the cell. Using the Poisson approximation, we compute the limit distributions of the largest and smallest order statistics for the inradii of all cells whose nuclei are contained in \(\boldsymbol{W}_\rho\) as \(\rho\) goes to \(\infty\). We additionally prove that the limit shape of the cells minimising the inradius is a triangle.

MSC:

60D05	Geometric probability and stochastic geometry
60G70	Extreme value theory; extremal stochastic processes
60G55	Point processes (e.g., Poisson, Cox, Hawkes processes)
60F05	Central limit and other weak theorems
62G32	Statistics of extreme values; tail inference

Keywords:

line tessellation; Poisson point process; extreme value; order statistic