Extreme events of higher-order Markov chains: hidden tail chains and extremal Yule-Walker equations
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We derive some key extremal features for kth order Markov chains, which can be used to understand how the process moves to and fro between the body of the process and an extreme state. The chains are studied given that there is an exceedance of a threshold, as the threshold tends to the upper endpoint 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. We find that it is possible to find a simple 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 interesting structure that has a striking parallel 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 k+1 dimensional copula.
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