The analog formulation of sparsity implies infinite divisibility and rules out Bernoulli-Gaussian priors

Motivated by the analog nature of real-world signals, we investigate continuous-time random processes. For this purpose, we consider the stochastic processes that can be whitened by linear transformations and we show that the distribution of their samples is necessarily infinitely divisible. As a consequence, such a modeling rules out the Bernoulli-Gaussian distribution since we are able to show in this paper that it is not infinitely divisible. In other words, while the Bernoulli-Gaussian distribution is among the most studied priors for modeling sparse signals, it cannot be associated with any continuous-time stochastic process. Instead, we propose to adapt the priors that correspond to the increments of compound Poisson processes, which are both sparse and infinitely divisible.