  The theory of sparse stochastic processes offers a broad class of statistical
models to study signals. In this framework, signals are represented as
realizations of random processes that are solution of linear stochastic
differential equations driven by white L\'evy 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 L\'evy 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.
