Stochastic Properties of Minimal Arc Distance and Cosine Similarity between a Random Point and Prespecified Sites on Sphere

Abstract: In applications such as wireless communication, it is important to study the statistical properties of $L_{2}$, the minimal arc distance between a random point (e.g., a cellphone user) uniformly distributed on a sphere to a set of pre-defined seeds (e.g., wireless towers) on that sphere. In this study, we first derive the distribution (CDF) and density (PDF) functions of the arc distance between a selected vertex of a spherical triangle to a random point uniformly distributed within this triangle. Next, using computational techniques based on spherical Voronoi diagram and triangular partition of Voronoi cells, we derive moments of $L_{2}$ and $\cos L_{2}$. These results are verified by extensive Monte Carlo simulations.