Nearest neighbor estimation of the intensity function of a cyclic Poisson process

We consider the problem of estimating the intensity function of a cyclic Poisson point process. We suppose that only a single realization of the cyclic Poisson point process is observed within a bounded 'window', and our aim is to estimate consistently the intensity function at a given point. A nearest neighbor estimator of the intensity function is proposed, and we show that our estimator is weakly and strongly consistent, as the window expands.