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 Lévy 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 Lévy 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.

稀疏随机过程的理论为研究信号提供了广泛的统计模型，远远超出了经典的高斯过程。在此框架中，信号表示为随机过程的实现，这些过程是由Lévy白噪声驱动的线性随机微分方程的解。在这些过程中，基于复合泊松噪声的广义泊松过程允许将其解释为具有随机结和权重的随机L样条。我们证明，从高斯到稀疏的每个广义Lévy过程都可以理解为一系列广义Poisson过程的法律极限。这使人们对稀疏过程有了新的概念性理解，并提出了用于此类对象数值生成的简单算法。