Extremal behaviour of models with multivariate random recurrence representation

For the solution Y of a multivariate random recurrence model Y n = A n Y n − 1 + ζ n in R q we investigate the extremal behaviour of the process y n = z ∗ ′ Y n , n ∈ N , for z ∗ ∈ R q with | z ∗ | = 1 . This extends results for positive matrices A n . 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.