Convergence results for tractable inference in α-stable stochastic processes

The α-stable distribution is highly intractable for inference because of the lack of a closed form density function in the general case. However, it is well-established that the α-stable distribution admits a Poisson series representation (PSR) in which the terms of the series are a function of the arrival times of a unit rate Poisson process. In our previous work, we have shown how to carry out inference for regression models using this series representation, which leads to a very convenient conditionally Gaussian framework, amenable to tractable Gaussian inference procedures. The PSR has to be truncated to a finite number of terms for practical purposes. The residual error terms have been approximated in our previous work by a Gaussian distribution, and we have recently shown that this approximation can be justified through a Central Limit Theorem (CLT). In this paper we present a new and exact characterisation of the first and second moments of the residual series over finite time intervals for the unit rate Poisson process, correcting a previous version that was only true in the infinite time limit. This enables us to test through simulation the rapid convergence of the residual terms to a Gaussian distribution of the Poisson series residual. We test this convergence using both Q-Q plots and the classical Kolmogorov-Smirnov test of Gaussianity.