Properties of convex hull generated by inhomogeneous Poisson point process

Abstract

The paper is devoted to the limit distribution study of the exterior of a convex hull generated by independent observations of two-dimensional random points having Poisson distributions above the parabola. Following P. Groeneboom [1], we note that near the boundary of support, the Binomial point process is almost indistinguishable from the Poisson point process. Therefore, the approximation of a Binomial point process to a Poisson process is not considered here; it is believed that it is sufficient to study the functionals of the convex hull generated by the Poisson point process. Using the modified P. Groeneboom technique, the so-called strong mixing and martingale properties of the vertex Markovian jump stationary process, the asymptotic expressions are obtained for the expectation and variance of the external part of the area of the convex hull inside the parabola. This is a continuation of results by H. Carnal in [2], where an asymptotic expression was found only for mean values of basic functionals of a convex hull. The asymptotic expression for the variance of the area of a convex hull was later obtained by J. Pardon [3] as no regularity conditions were imposed on the boundary of the support of a uniform distribution. The asymptotic expressions obtained here are used in the proofs of the central limit theorem for the area of the convex hull. Similar results were established in the studies by A. J. Cabo and P. Groeneboom [4] for the case as the initial distribution in a convex polygon is uniform.