Decomposition of gamma-distributed domains constructed from poisson point processes

A known gamma-type result for the Poison process states that certain domains defined through configuration of the points (or 'particles') of the process have volumes which are gamma distributed. By proving the corresponding sequential gamma-type result, we show that in some cases such a domain allows for decomposition into subdomains each having independent exponentially distributed volumes. We consider other examples-based on the Voronoi and Delaunay tessellations-where a natural decomposition does not produce subdomains with exponentially distributed volumes. A simple algorithm for the construction of a typical Voronoi flower arises in this work. In our theoretical development, we generalize the classical theorem of Slivnyak, relating it to the strong Markov property of the Poison process and to a result of Mecke and Muche (1995). This new theorem has interest beyond the specific problems being considered here.