On the classification problem for Poisson point processes

For Poisson processes taking values in a general metric space, we tackle the problem of supervised classification in two different ways: via the classical k-nearest neighbor rule, by introducing suitable distances between patterns of points; and via the Bayes rule, by nonparametrically estimating the intensity function of the process. In the first approach we prove that under the separability of the space, the rule turns out to be consistent. In the second case, we prove the consistency of the rule by proving the consistency of the estimated intensities. Both classifiers are shown to behave well under departures from the Poisson distribution.