Characterization and stability of light statistics distributions

Suggested by experiments concerning photon statistics, we study two kinds of problems: firstly, we give a complete characterization of the distributions for random sums of i.i.d. Bernoulli variables sharing a suitable stability property.Next, we study the distribution of a multivariate generalization of the univariate random sums and we obtain the explicit distribution of their L1L1 norm in terms of Bernstein polynomials.