Convergence results for compound Poisson distributions and applications to the standard Luria-Delbruck distribution

We provide a scaling for compound Poisson distributions that leads (under certain conditions on the Fourier transform) to a weak convergence result as the parameter of the distribution tends to infinity. We show that the limiting probability measure belongs to the class of stable Cauchy laws with Fourier transfonn t -> exp(-c vertical bar t vertical bar - iat log vertical bar t vertical bar). We apply this convergence result to the standard discrete Luria-Delbruck distribution and derive an integral representation for the corresponding limiting density, as an alternative to that found in a closely related paper of Kepler and Oprea. Moreover, we verify local convergence and we derive an integral representation for the distribution function of the limiting continuous Luria-Delbruck distribution.