Markov Chain Models for Threshold Exceedances

In recent research on extreme value statistics, there has been an extensive development of threshold methods, first in the univariate case but subsequently in the multivariate case as well. In this paper, we develop an alternative methodology for extreme values of univariate time series, by assuming that the time series is Markovian and using bivariate extreme value theory to suggest appropriate models for the transition distributions. We develop an alternative form of the likelihood representation for threshold methods, and then show how this can be applied to a Markovian time series. A major motivation for developing this kind of theory, in comparison with existing methods based on cluster maxima, is the possibility of calculating probability distributions for extremal functionals more complicated than the maxima or extreme quantiles of the series. In the latter part of the paper, we develop this theme, showing how a theory of compound Poisson limits for additive functionals can be combined with simulation to obtain numerical solutions for problems of practical interest.