The theory of sparse stochastic processes offers a broad class of statistical models to study signals, far beyond the more classical class of Gaussian processes. In this framework, signals are represented as realizations of random processes that are solution of linear stochastic differential equations driven by Levy white noises. Among these processes, generalized Poisson processes based on compound-Poisson noises admit an interpretation as random L-splines with random knots and weights. We demonstrate that every generalized Levy process-from Gaussian to sparse-can be understood as the limit in law of a sequence of generalized Poisson processes. This enables a new conceptual understanding of sparse processes and suggests simple algorithms for the numerical generation of such objects. (C) 2018 Elsevier Inc. All rights reserved.