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.

翻译：本文推导了$k$阶马尔可夫链的一些关键极值特征，这些特征可用于理解过程如何在极端状态与主体状态间转换。研究聚焦于阈值超出事件，且阈值趋于分布上端点时的链行为。与以往针对$k>1$的研究不同，我们考虑的是标准极限理论将每个极端事件描述为单一观测值、不包含任何关于分布状态间转换信息的过程。本研究采用不同的渐近理论，从而得到此类过程的非退化极限分布律。我们在弱假设下，针对涵盖渐近相依和渐近独立马尔可夫链的广义极值相依结构类别，研究了初始分布和马尔可夫链转移概率核的极值性质。对于$k>1$的链，当偏离超出状态时，其转移涉及$k$个先前状态的新颖函数，而非像$k=1$时仅依赖单一值。这增加了确定此类函数形式、性质及其在应用中推导方法的复杂度。我们发现，可以引入一种依赖于阈值超出量的仿射归一化，确保过程在所有滞后阶数下均具有非退化极限行为。这些归一化函数具有类似尤尔-沃克方程的优美结构。此外，极限过程始终关于创新项线性。我们通过基于广泛研究的连接函数相依结构、具有指数边缘分布的$k$阶平稳马尔可夫链案例，对结果进行了说明。