The concept of sparsity plays a central role in modern signal processing: (many) real-world signals are inherently sparse (in some adapted representation domain). The theory of sparse stochastic processes provides continuous-domain statistical models that are in adequacy with this paradigm, but there are still a number of issues that need to be clarified. The primary goal of this project is to investigate the compressibility of sparse processes in an adapted wavelet basis; that is, to characterize the decay of the approximation error as a function of the number of retained expansion coefficients. We are mainly interested in obtaining lower bounds on the compressibility, i.e. showing that the approximation rates for a given class of processes cannot be better than those that are currently predicted by the theory.