Poissonian Tree Constructed from Independent Poisson Point Processes

In this work a connected graph without cycles and with a single infinite self-avoiding path, i.e., a tree with an end, is constructed. The vertices of the tree are point; of an infinite sequence of independent Poisson point processes defined on R-d, such that for every k >= 1, the rate of kth process X-k is lambda(k). This graph will be called a One-Ended Poissonian. Tree. The algorithm of construction of the Poissonian Tree is given, as well as the definition of its elements. This algorithm will be called algorithm A. We also give a sufficient condition for the generation of a unique tree. In the case where the sequence of rates is such that lim inf lambda(k) = 0, for processes defined on R, we prove that algorithm A generates a One-Ended Poissonian Tree.