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：我们为美元1美元顺序的Markov 链条得出一些关键的极端极端特征,这些特征可用于理解这一过程如何在极端状态和进程主体之间移动。我们研究链条,因为阈值存在超值,因为阈值倾向于分布的上端点。与以前用美元1美元进行的研究不同,我们认为标准限值理论将每个极端事件描述为单一的观察,而没有任何关于向分配体过渡和从分配体向分配体过渡的信息。我们的工作使用了不同的零星理论,这导致这些过程的直线性差限制。我们研究了最初分布的极端性质和马尔科夫链的过渡概率内核,而最初分配的临界值与超值结构的过渡性骨质存在薄弱的假设,这些假设覆盖了极端依赖性的上端点,而仅包含对美元1美元的过渡,而从超值的链条条则包含新的固定功能,与单一值相比,当美元=1美元时。这导致马尔科夫链条链条的极值结构的极端性能和超值值结构的转变。我们研究了这一类的复杂性功能。