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Hidden tail chains and recurrence equations for dependence parameters associated with extremes of higher-order Markov chains

Abstract

We derive some key extremal features for kth order Markov chains, which 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. The extremal properties of the Markov chain at lags up to k are determined by the kernel of the chain, through a joint initialisation distribution, with the subsequent values determined by the conditional independence structure through a transition behaviour. We study the extremal properties of each of these elements under weak assumptions for broad classes of extremal dependence structures. For chains with k>1, these transitions involve 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 find 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 kth order stationary Markov chains based on widely studied families of copula dependence structures.

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