Appendix of Infinitely divisible cascades to model the statistics of natural images

Let us recall some basics on infinitely divisible probability distributions or laws. We denote the set of strictly positive integers by N∗. Definition A distribution G is called infinitely divisible if for all n ∈ N∗ there exists a distribution Gn such that G equals the n-fold convolution of Gn with itself, denoted as (Gn) . In other words, the distribution of a random variable S is infinitely divisible if and only if for all n ∈ N∗ the variable S can be written in law as the sum of n i.i.d. variables: