A Note on Empirical Process Methods in the Theory of Poisson Point Processes

For a sample taken from an i.i.d. sequence of Poisson point processes with not necessarily finite unknown intensity measure the arithmetic mean is shown to be an estimator which is consistent uniformly on certain classes of functions. The method is a reduction to the case of finite intensity measure, which in turn can be dealt with using empirical process methods. A functional central limit theorem is also established in this context.