Mean and Variance of Vacancy for Hard-Core Disc Processes and Applications

Hard-core Strauss disc processes with inhibition distance r and disc radius R are considered. The points in the Strauss point process are thought of as trees and the discs as crowns. Formulas for the mean and the variance of the vacancy (non-covered area) are derived. This is done both for the case of a fixed number of points and for the case of a random number of points. For tractability, the region is assumed to be a torus or, in one dimension, a circle in which case the discs are segments. In the one-dimensional case, the formulas are exact for all r. This case, although less important in practice than the two-dimensional case, has provided a lot of inspiration. In the two-dimensional case, the formulas are only approximate but rather accurate for r < R. Markov Chain Monte Carlo simulations confirm that they work well. For R ≤ r < 2R, no formulas are presented. A forestry estimation problem, which has motivated the research, is briefly considered as well as another application in spatial statistics.