Nearest-Neighbor and Contact Distance Distributions for Thomas Cluster Process

We characterize the statistics of nearest-neighbor and contact distance distributions for Thomas cluster process (TCP), which is a special case of Poisson cluster process. In particular, we derive the cumulative distribution function of the distance to the nearest point of TCP from a reference point for three different cases: 1) reference point is not a part of the point process; 2) it is chosen uniformly at random from the TCP; and 3) it is a randomly chosen point from a cluster chosen uniformly at random from the TCP. While the first corresponds to the contact distance distribution, the other two provide two different viewpoints for the nearest-neighbor distance distribution. Closed-form bounds are also provided for the first two cases.