The extremes of a univariate Markov chain with regulary varying stationary
marginal distribution and asymptotically linear behavior are known to exhibit a
multiplicative random walk structure called the tail chain. In this paper, we
extend this fact to Markov chains with multivariate regularly varying marginal
distribution in R^d. We analyze both the forward and the backward tail process
and show that they mutually determine each other through a kind of adjoint
relation. In a broader setting, it will be seen that even for non-Markovian
underlying processes a Markovian forward tail chain always implies that the
backward tail chain is Markovian as well. We analyze the resulting class of
limiting processes in detail. Applications of the theory yield the asymptotic
distribution of both the past and the future of univariate and multivariate
stochastic difference equations conditioned on an extreme event
