  An asymptotic model for extreme behavior of certain Markov chains is the
"tail chain". Generally taking the form of a multiplicative random walk, it is
useful in deriving extremal characteristics such as point process limits. We
place this model in a more general context, formulated in terms of extreme
value theory for transition kernels, and extend it by formalizing the
distinction between extreme and non-extreme states. We make the link between
the update function and transition kernel forms considered in previous work,
and we show that the tail chain model leads to a multivariate regular variation
property of the finite-dimensional distributions under assumptions on the
marginal tails alone.
