Semi-stable non-Gaussian limits arising in sampling of finite renewal processes

Bernoulli, 22(2) (2016), 1055–1092
https://projecteuclid.org/euclid.bj/1447077769

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.