Properties of the vertex of a convex hull generated by a Poisson point process inside a parabola

A convex hull generated by the implementation of a Poisson point process inside a parabola is considered in the article. At that, the measure of intensity of the Poisson law is related to regularly varying functions near the boundary of the support. It is proven that the domain bounded by the perimeters of the convex hull and the boundary of the support - a parabola, can be represented as a sum of independent identically distributed random variables. Moreover, this value does not depend on the vertices of the convex hull itself. It is worth noting that having approximated the binomial point process by a Poisson one, P.Groeneboom [6], A.J.Cabo and P.Groeneboom [3], I.Hueter [9], T.Hsing [8] and others, using the martingale properties of stationary vertex processes, proved various options of the central limit theorem for functionals of a random convex hull in the case when the original distribution is uniformly concentrated in a convex polygon or ellipse. In this paper, the exact distribution and conditional distribution of vertex processes are found when the convex hull is generated by a inhomogeneous Poisson point process inside a parabola. In some special cases, it is shown that the area between the perimeter of the convex hull and the support of the distribution is expressed by the sum of independent random variables.

Ҳали таржима қилинмаган