Enhanced Poisson sum representation for alpha-stable processes

In this paper we present Poisson sum series representations for α-stable (αS) random variables and α-stable processes, in particular concentrating on continuous-time autoregressive (CAR) models driven by α-stable Levy processes. Our representations aim to provide a conditionally Gaussian framework, which will allow parameter estimation using Rao-Blackwellised versions of state of the art Bayesian computational methods such as particle filters and Markov chain Monte Carlo (MCMC). To overcome the issues due to truncation of the series, novel residual approximations are developed. Simulations demonstrate the potential of these Poisson sum representations for inference in otherwise intractable α-stable models.