  We derive some key extremal features for $k$th order Markov chains that can
be used to understand how the process moves between an extreme state and the
body of the process. The chains are studied given that there is an exceedance
of a threshold, as the threshold tends to the upper endpoint of the
distribution. Unlike previous studies with $k>1$, we consider processes where
standard limit theory describes each extreme event as a single observation
without any information about the transition to and from the body of the
distribution. Our work uses different asymptotic theory which results in
non-degenerate limit laws for such processes. We study the extremal properties
of the initial distribution and the transition probability kernel of the Markov
chain under weak assumptions for broad classes of extremal dependence
structures that cover both asymptotically dependent and asymptotically
independent Markov chains. For chains with $k>1$, the transition of the chain
away from the exceedance involves novel functions of the $k$ previous states,
in comparison to just the single value, when $k=1$. This leads to an increase
in the complexity of determining the form of this class of functions, their
properties and the method of their derivation in applications. We find that it
is possible to derive an affine normalization, dependent on the threshold
excess, such that non-degenerate limiting behaviour of the process is assured
for all lags. These normalization functions have an attractive structure that
has parallels to the Yule-Walker equations. Furthermore, the limiting process
is always linear in the innovations. We illustrate the results with the study
of $k$th order stationary Markov chains with exponential margins based on
widely studied families of copula dependence structures.
