  A finite point process is characterized by the distribution of the number of
points (the size) of the process. In some applications, for example, in the
context of packet flows in modern communication networks, it is of interest to
infer this size distribution from the observed sizes of sampled point
processes, that is, processes obtained by sampling independently the points of
i.i.d. realizations of the original point process. A standard nonparametric
estimator of the size distribution has already been suggested in the
literature, and has been shown to be asymptotically normal under suitable but
restrictive assumptions. When these assumptions are not satisfied, it is shown
here that the estimator can be attracted to a semi-stable law. The assumptions
are discussed in the case of several concrete examples. A major theoretical
contribution of this work are new and quite general sufficient conditions for a
sequence of i.i.d. random variables to be attracted to a semi-stable law.
