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

We consider general nonstationary max-autoregressive sequences Xi, i [greater-or-equal, slanted] 1, with Xi = Zimax(Xi - 1, Yi) where Yi, i [greater-or-equal, slanted] 1 is a sequence of i.i.d. random variables and Zi, i [greater-or-equal, slanted] 1 is a sequence of independent random variables (0 [less-than-or-equals, slant] Zi [less-than-or-equals, slant] 1), independent of Yi. We deal with the limit law of extreme values Mn = maxXi, i [less-than-or-equals, slant] n (as n --> [infinity]) and evaluate the extremal index for the case where the marginal distribution of Yi is regularly varying at [infinity]. The limit of the point process of exceedances of a boundary un by Xi, i [less-than-or-equals, slant] n, is derived (as n --> [infinity]) by analysing the convergence of the cluster distribution and of the intensity measure.