Extremes and clustering of nonstationary max-AR(1) sequences

We consider general nonstationary max-autoregressive sequences Xi, i ⩾ 1, with Xi = Zimax(Xi − 1, Yi) where Yi, i ⩾ 1 is a sequence of i.i.d. random variables and Zi, i ⩾ 1 is a sequence of independent random variables (0 ⩽ Zi ⩽ 1), independent of Yi. We deal with the limit law of extreme values Mn = maxXi, i ⩽ n (as n → ∞) and evaluate the extremal index for the case where the marginal distribution of Yi is regularly varying at ∞. The limit of the point process of exceedances of a boundary un by Xi, i ⩽ n, is derived (as n → ∞) by analysing the convergence of the cluster distribution and of the intensity measure.