Poisson approximation of binomial point processes

Abstract

In the paper, we study properties of the vertex process from convex hulls generated by independent observations of a two-dimensional random vector, the distribution of which behaves like a regularly varying function near the boundary of the support of the disk. The problem of approximating the distributions of sums of random variables, despite its rich history, is still relevant today. The Poisson approximation, along with the normal approximation, remain intensively developing areas of modern probability theory. We note their importance in solving statistical problems, in which the presence of a simple asymptotic expansion makes it possible not only to obtain more accurate statistical estimates, but also to solve the problem of the error significance level. In this article, we use the Poisson approximation to study the limit distributions of functionals generated by aninhomogeneous binomial point process.