Extremal behaviour of models with multivariate random recurrence representation

For the solution YY of a multivariate random recurrence model Yn=AnYn−1+ζnYn=AnYn−1+ζn in RqRq we investigate the extremal behaviour of the process yn=z∗′Yn, n∈Nn∈N, for z∗∈Rqz∗∈Rq with |z∗|=1|z∗|=1. This extends results for positive matrices AnAn. Moreover, we obtain explicit representations of the compound Poisson limit of point processes of exceedances over high thresholds in terms of its Poisson intensity and its jump distribution, which represents the cluster behaviour of such models on high levels. As a principal example we investigate a random coefficient autoregressive process.