We consider a random tree and introduce a metric in the space of trees to define the “mean tree” as the tree minimizing the average distance to the random tree. When the resulting metric space is compact we have laws of large numbers and central limit theorems for sequence of independent identically distributed random trees. As application we propose tests to check if two samples of random trees have...
Let N, N1 and N2 be point processes such that N1 is obtained from N by homogeneous independent thinning and N2=N−N1. We give a new elementary proof that N1 and N2 are independent if and only if N is a Poisson point process. We also present an application of this result to test if a homogeneous point process is a Poisson point process.
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