A central limit theorem with application to inference in α-stable regression models

It is well known that the α-stable distribution, while having no closed form density function in the general case, 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 straightforward Gaussian inference procedures. The PSR has to be truncated to a finite number of terms for practical purposes. The residual terms have been approximated in our previous work by a Gaussian distribution with fully characterised moments. In this paper we present a new Central Limit Theorem (CLT) for the residual terms which serves to justify our previous approximation of the residual as Gaussian. Furthermore, we provide an analysis of the asymptotic convergence rate expressed in the CLT.