The theory of sparse stochastic processes was recently introduced to model the statistical properties of natural images. In this theory, an image is described as a stochastic process. Based on the mathematical foundations that have been recently developed, research efforts are now focused on studying the regularity of sparse processes. We propose a project related to this question. To do so, a very promising method focuses on the control of the moments of infinitely divisible random variables. The project include a short review of the literature around the proposed method, as well as the formulation (and proof) of precise statements. This project is intended for students with a pronounced taste for abstract mathematics — mainly probability theory and functional analysis — and interested in applications to real-world signal analysis. Relevant documentation: http://www.sparseprocesses.org/sparseprocesses-chap7.pdf