Interpretation of continuous-time autoregressive processes as random exponential splines

Abstract

Abstract-We consider the class of continuous-time autoregressive (CAR) processes driven by (possibly non-Gaussian) Lévy white noises. When the excitation is an impulsive noise, also known as compound Poisson noise, the associated CAR process is a random non-uniform exponential spline. Therefore, Poissontype processes are relatively easy to understand in the sense that they have a finite rate of innovation. We show in this paper that any CAR process is the limit in distribution of a sequence of CAR processes driven by impulsive noises. Hence, we provide a new interpretation of general CAR processes as limits of random exponential splines. We illustrate our result with simulations.

Gaussian and sparse processes are limits of generalized Poisson processes

“…The present paper is an extension of our previous contribution [37] 1 . We believe that Theorem 1 is relevant for the conceptualization of random processes that are solution of linear SDE.…”

“…Here, we extend our preliminary result in several ways: The class of processes we study now is much more general since we consider arbitrary operators; moreover, we include multivariate random processes, often called random fields. Finally, our preliminary report contained a mere sketch of the proof of[37, Theorem 8], while the current work is complete in this respect.…”

Sampling and (sparse) stochastic processes: A tale of splines and innovation

“…where f (ω) is the Lévy exponent of the innovation w. The relevant theory is developed in [37] for the class of CARn processes associated with the generic operator (1). In particular, we note that λ e…”