Extremal behavior of the autoregressive process with ARCH(1) errors

We investigate the extremal behavior of a special class of autoregressive processes with ARCH(1) errors given by the stochastic difference equationXn=αXn−1+β+λXn−12εn, n∈N,where (εn)n∈N are i.i.d. random variables. The extremes of such processes occur typically in clusters. We give an explicit formula for the extremal index and the probabilities for the length of a cluster.