EXTREMES AND CLUSTERING OF NONSTATIONARY MAX-AR(1) SEQUENCES

We consider general nonstationary max-autoregressive sequences {X(i), i greater than or equal to 1}, with X(i) = Z(i)max(X(i-1), Y-i) where {Y-i, i greater than or equal to 1} is a sequence of i.i.d. random variables and {Z(i), i greater than or equal to 1} is a sequence of independent random variables (0 less than or equal to Z(i) less than or equal to 1), independent of {Y-i}. We deal with the limit law of extreme values M(n) = max{X(i), i less than or equal to n} (as n --> infinity) and evaluate the extremal index for the case where the marginal distribution of Y-i is regularly varying at infinity. The limit of the point process of exceedances of a boundary mu(n) by X(i), i less than or equal to n, is derived (as n --> infinity) by analysing the convergence of the cluster distribution and of the intensity measure.