Empirical polygon simulation and central limit theorems for the homogeneous Poisson line process (Q2475268)

In the paper a new simulation of the Poisson line process in the plane is considered. This simulation based on the theoretical results of \textit{K. Paroux} [Adv. Appl. Probab. 30, No.~3, 640--656 (1998; Zbl 0914.60013)]. The first simulation of the Poisson line process which was performed by Dufour [cited in \textit{H. Solomon}, ``Geometric Probability'' (1978; Zbl 0382.60016)] was of 947 polygons generated by 65 lines. Grain and Miles (see [\textit{I. Grain} and \textit{R. E. Miles}, J. Stat. Comput. Simulation 4, 293--325 (1976; Zbl 0345.65001)]) performed 45 realizations of the line process in the disc of radius 1, thus generating a few thousands of polygons, and then they had to perform a non trivial statistical treatment to their values because of some dependence problems. Later George (see [\textit{E. I. George}, J. Appl. Probab. 24, 557--573 (1987; Zbl 0633.60020)] generated 2,500,000 individual Poisson polygons using the empirical law of the length of the sides of the polygons as well as the empirical law of the angles. The first procedure of the present paper follows the natural idea to perform the Poisson line process in a large domain (or equivalently in a fixed domain with high intensity), thus generating billions of polygons, and then to use some tools and remarks developed in [Zbl 0914.60013] to obtain efficiently some characteristics of classical empirical distributions. In a second part of the paper, the authors compute the laws of some characteristics of the polygon containing the origin, and relate these results to the ones on the empirical laws. This is done by generating the Poisson line process in a smaller domain near the origin and by a direct extraction of the connected component of the tessellation containing the point of coordinates (0,0). All the programs for the central limit theorems were written in C++ and ran relatively fast, the simulation of the polygons around the origin were written in C++ and in Matlab. Those simulations also gave a confirmation of the theorems proved in [Zbl 0914.60013].