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Interpretation of Continuous-Time Autoregressive Processes as Random Exponential Splines

J. Fageot, J.P. Ward, M. Unser

Proceedings of the Eleventh International Workshop on Sampling Theory and Applications (SampTA'15), Washington DC, USA, May 25-29, 2015, pp. 231–235.

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, Poisson-type 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. 

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