  Accurate modelling of the joint extremal dependence structure within a
stationary time series is a challenging problem that is important in many
applications.\ Several previous approaches to this problem are only applicable
to certain types of extremal dependence in the time series such as asymptotic
dependence, or Markov time series of finite order.\ In this paper, we develop
statistical methodology for time series extremes based on recent probabilistic
results that allow us to flexibly model the decay of a stationary time series
after witnessing an extreme event.\ While Markov sequences of finite order are
naturally accommodated by our approach, we consider a broader setup, based on
the conditional extreme value model, which allows for a wide range of possible
dependence structures in the time series.\ We consider inference based on Monte
Carlo simulation and derive an upper bound for the variance of a commonly used
importance sampler.\ Our methodology is illustrated via estimation of cluster
functionals in simulated data and in a time series of daily maximum
temperatures from Orleans, France.
