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The theory of sparse stochastic processes offers a broad class of statisticalmodels to study signals. In this framework, signals are represented asrealizations of random processes that are solution of linear stochasticdifferential equations driven by white L\'evy noises. Among these processes,generalized Poisson processes based on compound-Poisson noises admit aninterpretation as random L-splines with random knots and weights. Wedemonstrate that every generalized L\'evy process-from Gaussian to sparse-canbe understood as the limit in law of a sequence of generalized Poissonprocesses. This enables a new conceptual understanding of sparse processes andsuggests simple algorithms for the numerical generation of such objects.

Gaussian and Sparse Processes Are Limits of Generalized Poisson Processes