Convergence of randomly oscillating point patterns to the Poisson point process

summary:Oscillating point patterns are point processes derived from a locally finite set in a finite dimensional space by i.i.d. random oscillation of individual points. An upper and lower bound for the variation distance of the oscillating point pattern from the limit stationary Poisson process is established. As a consequence, the true order of the convergence rate in variation norm for the special case of isotropic Gaussian oscillations applied to the regular cubic net is found. To illustrate these theoretical results, simulated planar structures are compared with the Poisson point process by the quadrat count and distance methods