Gaussian and Sparse Processes Are Limits of Generalized Poisson Processes

TL;DR

This paper shows that Gaussian and sparse stochastic processes can be viewed as limits of generalized Poisson processes, providing new insights and algorithms for modeling and generating such signals.

Contribution

It establishes that all generalized Le9vy processes, including Gaussian and sparse types, are limits of generalized Poisson processes, offering a novel conceptual framework.

Findings

Generalized Le9vy processes are limits of generalized Poisson processes.

Provides a new understanding of sparse processes as limits of simpler models.

Suggests algorithms for numerical generation of these processes.

Abstract

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.