  The extremal behaviour of a Markov chain is typically characterized by its
tail chain. For asymptotically dependent Markov chains existing formulations
fail to capture the full evolution of the extreme event when the chain moves
out of the extreme tail region and for asymptotically independent chains recent
results fail to cover well-known asymptotically independent processes such as
Markov processes with a Gaussian copula between consecutive values. We use more
sophisticated limiting mechanisms that cover a broader class of asymptotically
independent processes than current methods, including an extension of the
canonical Heffernan-Tawn normalization scheme, and reveal features which
existing methods reduce to a degenerate form associated with non-extreme
states.
