Poissonization Principle for a Class of Additive Statistics

In this paper, we consider a class of additive functionals of a finite or countable collection of the group frequencies of an empirical point process that corresponds to, at most, a countable partition of the sample space. Under broad conditions, it is shown that the asymptotic behavior of the distributions of such functionals is similar to the behavior of the distributions of the same functionals of the accompanying Poisson point process. However, the Poisson versions of the additive functionals under consideration, unlike the original ones, have the structure of sums (finite or infinite) of independent random variables that allows us to reduce the asymptotic analysis of the distributions of additive functionals of an empirical point process to classical problems of the theory of summation of independent random variables