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.

中文翻译：

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泊松点过程产生的凸包顶点数和面积的联合概率分布

摘要 考虑由平面锥体中的齐次泊松点过程生成的凸包。本文证明了以半径为T的圆盘为界（圆盘的中心在圆锥的顶点）的圆锥中顶点数和凸包面积的联合概率分布的中心极限定理，对于 T → ∞ 。根据本文的结果，遵循 Groeneboom (1988) 和 Cabo and Groeneboom (1994) 的先前已知结果，其中中心极限定理证明了正方形中顶点数和凸包面积的中心极限定理为通过齐次泊松点过程逼近二项式点过程。