Statistics and Probability Letters 169 (2021) 108966 Contents lists available at ScienceDirect Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro On joint probability distribution of the number of vertices and area of the convex hulls generated by a Poisson point process Sh.K. Formanov, I.M. Khamdamov ∗ National University of Uzbekistan, 700174, Universitetskaya 4, Tashkent, Uzbekistan article info Article history: Received 17 February 2020 Received in revised form 25 September 2020 Accepted 1 October 2020 Available online 21 October 2020 MSC: primary 60F05 secondary 60D05 Keywords: Convex hull Random point Poisson point process abstract Consider a convex hull generated by a homogeneous Poisson point process in a cone in the plane. In the present paper the central limit theorem is proved for the joint probability distribution of the number of vertices and the area of a convex hull in a cone bounded by the disk of radius T (the center of the disk is at the cone vertex), for T →∞. From the results of the present paper the previously known results of Groeneboom (1988) and Cabo and Groeneboom (1994) are followed, in which the central limit theorem was proved for the number of vertices and the area of the convex hull in a square by approximating the binomial point process by a homogeneous Poisson point process. © 2020 Published by Elsevier B.V. 1. Introduction The joint probability distribution of the number of vertices and area of the convex hulls generated by realization of a Poisson point process in a cone in R 2 are investigated in the paper. The studies of this kind are usually attributed to the field of stochastic geometry. Convex hulls are analytically complex objects. Therefore, for a long time, research in this area was limited only to studying the average value of the main functionals such as, the number of vertices, perimeter, and area of the convex hull. The most complete bibliography on this issue is given in the studies of Efron (1965), Carnal (1970), Renyi and Sulanke (1963), Hueter (1994), Buchta (2005, 2013), Schneider (1987), Nagaev and Khamdamov (1991), Nagaev (1995), Groeneboom (2012) and Pardon (2011). It should be noted that in this list there are the studies in which a wider class of supports of the initial uniform distribution is considered. For the first time, a significant progress in this area was made by P. Groeneboom. In Groeneboom (1988) the limit distribution for the number of vertices of a convex hull was obtained in the case where the support of uniform distribution is either a convex polygon or a unit disk. The research technique in Groeneboom (1988) was based on the original idea of using the Poisson approximation of a binomial point process near the border of support of the initial distribution. There (in Groeneboom, 1988), powerful methods were used, such as strongly mixed stationary processes and martingales. Then, in Cabo and Groeneboom (1994) applied the method developed by Groeneboom (1988) to prove central limit theorems for the area and perimeter of a convex hull in a polygon. Hsing (1994), modifying the Groeneboom method, proved the central limit theorem for the area outside a convex hull in a circle. Later, Pardon (2011, 2012), without imposing any regularity conditions on the border of support, continued to study the limit distribution for the area and number of vertices of the convex hull in the case when the primary distribution is a uniform one. In the present work, the ∗ Corresponding author. E-mail addresses: shakirformanov@yandex.ru (Sh.K. Formanov), khamdamov.isakjan@gmail.com (I.M. Khamdamov). https://doi.org/10.1016/j.spl.2020.108966 0167-7152/© 2020 Published by Elsevier B.V.