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One- versus multi-component regular variation and extremes of Markov trees

Published online by Cambridge University Press:  24 September 2020

Johan Segers Open the ORCID record for Johan Segers [Opens in a new window]
Johan Segers*
Affiliation:
Université catholique de Louvain
*
*Postal address: Université catholique de Louvain, LIDAM/ISBA, Voie du Roman Pays 20, B-1348 Louvain-la-Neuve, Belgium. Email: johan.segers@uclouvain.be
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Abstract

A Markov tree is a random vector indexed by the nodes of a tree whose distribution is determined by the distributions of pairs of neighbouring variables and a list of conditional independence relations. Upon an assumption on the tails of the Markov kernels associated to these pairs, the conditional distribution of the self-normalized random vector when the variable at the root of the tree tends to infinity converges weakly to a random vector of coupled random walks called a tail tree. If, in addition, the conditioning variable has a regularly varying tail, the Markov tree satisfies a form of one-component regular variation. Changing the location of the root, that is, changing the conditioning variable, yields a different tail tree. When the tails of the marginal distributions of the conditioning variables are balanced, these tail trees are connected by a formula that generalizes the time change formula for regularly varying stationary time series. The formula is most easily understood when the various one-component regular variation statements are tied up into a single multi-component statement. The theory of multi-component regular variation is worked out for general random vectors, not necessarily Markov trees, with an eye towards other models, graphical or otherwise.

Keywords

Conditional independencegraphical modelHüsler–Reiss distributionmax-linear modelMarkov treemultivariate Pareto distributionPickands dependence functionregular variationroot change formulatail measuretail treetime change formula
Type
Original Article
Information
Advances in Applied Probability , Volume 52 , Issue 3 , September 2020 , pp. 855 - 878
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Copyright
© Applied Probability Trust 2020

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