Poisson convergence for point processes on the plane

On the classification problem for Poisson point processes

Convergence of Rare Events Point Processes to the Poisson for Billiards

We show that for planar dispersing billiards the return times distribution is in the limit Poisson for metric balls almost everywhere w.r.t. the SRB measure. Since the Poincaré return map is piecewise smooth but becomes singular at the boundaries of the partition elements, recent results on the limiting distribution of return times cannot be applied as they require the maps to have bounded seco...

Poisson convergence, in large deviations, for the superposition of independent point processes

For a small buffer queueing system fed by a large class of traffic processes we show that the single server queue and the associated sample paths behave as if fed by marked Poisson traffic; in a large deviations limit, for a system fed by many flows. The timescale upon which events occur in this setting tends to zero, hence we study a limit on the log moment generating function as time tends to...

Transforming spatial point processes into Poisson processes

In 1986, Merzbach and Nualart demonstrated a method of transforming a two-parameter point process into a planar Poisson process of unit rate, using random stopping sets. Merzbach and Nualart's theorem applies only to a special class of point processes, since it requires two restrictive conditions: the (F4) condition of conditional independence and the convexity of the 1-compensator. The (F4) co...

Poisson Cox Point Processes for Vehicular Networks

This paper analyzes statistical properties of the Poisson line Cox point process useful in the modeling of vehicular networks. The point process is created by a two-stage construction: a Poisson line process to model road infrastructure and independent Poisson point processes, conditionally on the Poisson lines, to model vehicles on the roads. We derive basic properties of the point process, in...