On Limit Theorem for the Number of Vertices of the Convex Hulls in a Unit Disk

This paper is devoted to further investigation of the property of a number of vertices of convex hulls generated by independent observations of a two-dimensional random vector with regular distributions near the boundary of support when it is a unit disk. Following P. Groeneboom [4], the Binomial point process is approximated by the Poisson point process near the boundary of support and vertex processes of convex hulls are constructed. The properties of strong mixing and martingality of vertex processes are investigated. Using these properties, asymptotic expressions are obtained for the expectations and variance of the vertex processes that correspond to the results previously obtained by H. Carnal [2]. Further, using the properties of strong mixing of vertex processes, the central limit theorem for a number of vertices of a convex hull is proved

BiPAD: Binomial Point Process Based Energy-Aware Data Dissemination in Opportunistic D2D Networks

In opportunistic device-to-device (D2D) networks, the epidemic routing protocol can be used to optimize the message delivery ratio. However, it has the disadvantage that it causes excessive coverage overlaps and wastes energy in message transmissions because devices are more likely to receive duplicates from neighbors. We therefore propose an efficient data dissemination algorithm that can reduce undesired transmission overlap with little performance degradation in the message delivery ratio. The proposed algorithm allows devices further away than the k-th furthest distance from the source device to forward a message to their neighbors. These relay devices are determined by analysis based on a binomial point process (BPP). Using a set of intensive simulations, we present the resulting network performances with respect to the total number of received messages, the forwarding efficiency and the actual number of relays. In particular, we find the optimal number of relays to achieve almost the same message delivery ratio as the epidemic routing protocol for a given network deployment. Furthermore, the proposed algorithm can achieve almost the same message delivery ratio as the epidemic routing protocol while improving the forwarding efficiency by over 103% when k≥10.

Focusing of the scan statistic and geometric clique number

Given sets C and R in d-dimensional space, take a constant intensity Poisson point process on R; the associated scan statistic S is the maximum number of Poisson points in any translate of C. As R becomes large with C fixed, bounded and open but otherwise arbitrary, the distribution of S becomes concentrated on at most two adjacent integers. A similar result holds when the underlying Poisson process is replaced by a binomial point process, and these results can be extended to a large class of nonuniform distributions. Also, similar results hold for other finite-range scanning schemes such as the clique number of a geometric graph.

Focusing of the scan statistic and geometric clique number

Given sets C and R in d-dimensional space, take a constant intensity Poisson point process on R; the associated scan statistic S is the maximum number of Poisson points in any translate of C. As R becomes large with C fixed, bounded and open but otherwise arbitrary, the distribution of S becomes concentrated on at most two adjacent integers. A similar result holds when the underlying Poisson process is replaced by a binomial point process, and these results can be extended to a large class of nonuniform distributions. Also, similar results hold for other finite-range scanning schemes such as the clique number of a geometric graph.