  We present a new approach to the statistical study and modelling of number
counts of faint point sources in astronomical images, i.e. counts of sources
whose flux falls below the detection limit of a survey. The approach is based
on the theory of alpha-stable distributions. We show that the non-Gaussian
distribution of the intensity fluctuations produced by a generic point source
population -- whose number counts follow a simple power law -- belongs to the
alpha-stable family of distributions. Even if source counts do not follow a
simple power law, we show that the alpha-stable model is still useful in many
astrophysical scenarios. With the alpha-stable model it is possible to totally
describe the non-Gaussian distribution with a few parameters which are closely
related to the parameters describing the source counts, instead of an infinite
number of moments. Using statistical tools available in the signal processing
literature, we show how to estimate these parameters in an easy and fast way.
We demonstrate that the model proves valid when applied to realistic point
source number counts at microwave frequencies. In the case of point
extragalactic sources observed at CMB frecuencies, our technique is able to
successfully fitting the P(D) distribution of deflections and to precisely
determining the main parameters which describe the number counts. In the case
of the Planck mission, the relative errors on these parameters are small either
at low and at high frequencies. We provide a way to deal with the presence of
Gaussian noise in the data using the empirical characteristic function of the
P(D). The formalism and methods here presented can be very useful also for
experiments in other frequency ranges, e.g. X-ray or radio Astronomy.
