Adrian Casey | Searcharxiv

arXiv2023

Decomposable Tail Graphical Models

We develop an asymptotic theory for extremes in decomposable graphical models by presenting results applicable to a range of extremal dependence types. Specifically, we investigate the weak limit of the distribution of suitably normalised random vectors, conditioning on an extreme component, where the conditional independence relationships of the random vector are described by a chordal graph. Under mild assumptions, the random vector corresponding to the distribution in the weak limit, termed the tail graphical model, inherits the graphical structure of the original chordal graph. Our theory is applicable to a wide range of decomposable graphical models including asymptotically dependent and asymptotically independent graphical models. Additionally, we analyze combinations of copula classes with differing extremal dependence in cases where a normalization in terms of the conditioning variable is not guaranteed by our assumptions. We show that, in a block graph, the distribution of the random vector normalized in terms of the random variables associated with the separators converges weakly to a distribution we term tail noise. In particular, we investigate the limit of the normalized random vectors where the clique distributions belong to two widely used copula classes, the Gaussian copula and the max-stable copula.

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arXiv2019

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

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.

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