Homework assignment 4

Consider a simple stationary point process \( \Phi \)  with intensity \( \lambda > 0 \) and the spherical contact distribution function \( F(r) \) and the nearest neighbour distance distribution function \( G(r) \). Denote the values corresponding to the homogeneous Poisson process with intensity \( \lambda \) as \( F_0(r), G_0(r) \). Assume \( R > 0 \) is a large value. Decide if the following properties correspond to clustering or repulsion between the points of the process:

a) \( F(r) > F_0(r) \) for \( r \in (0,R) \),

b) \( F(r) < F_0(r) \) for \( r \in (0,R) \),

c) \( G(r) > G_0(r) \) for \( r \in (0,R) \),

d) \( G(r) < G_0(r) \) for \( r \in (0,R) \).