A unified formulation of Gaussian vs. sparse stochastic processes - Part I: Continuous-domain theory

Abstract

We introduce a general distributional framework that results in a unifying description and characterization of a rich variety of continuous-time stochastic processes. The cornerstone of our approach is an innovation model that is driven by some generalized white noise process, which may be Gaussian or not (e.g., Laplace, impulsive Poisson or alpha stable). This allows for a conceptual decoupling between the correlation properties of the process, which are imposed by the whitening operator L, and its sparsity pattern which is determined by the type of noise excitation. The latter is fully specified by a Levy measure. We show that the range of admissible innovation behavior varies between the purely Gaussian and super-sparse extremes. We prove that the corresponding generalized stochastic processes are well-defined mathematically provided that the (adjoint) inverse of the whitening operator satisfies some Lp bound for p>=1. We present a novel operator-based method that yields an explicit characterization of all Levy-driven processes that are solutions of constant-coefficient stochastic differential equations. When the underlying system is stable, we recover the family of stationary CARMA processes, including the Gaussian ones. The approach remains valid when the system is unstable and leads to the identification of potentially useful generalizations of the Levy processes, which are sparse and non-stationary. Finally, we show how we can apply finite difference operators to obtain a stationary characterization of these processes that is maximally decoupled and stable, irrespective of the location of the poles in the complex plane.