Trimmed stable AR(1) processes

In this paper we investigate the distribution of trimmed sums of dependent observations with heavy tails. We consider the case of autoregressive processes of order one with independent innovations in the domain of attraction of a stable law. We show if the $d$ largest (in magnitude) terms are removed from the sample, then the sum of the remaining elements satisfies a functional central limit theorem with random centering provided $d=d\left(n\right)\ge n^{\gamma }$ (for some $\gamma >0$) and $d\left(n\right)/n\to 0$. This result is used to get asymptotics for the widely used CUSUM process in case of dependent heavy tailed observations.