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

Johan Segers ()
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Johan Segers: Université catholique de Louvain, LIDAM/ISBA, Belgium

No 2020024, LIDAM Reprints ISBA from Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA)

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 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 to 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 independence; graphical model; H¨usler–Reiss distribution; max-linear model; Markov tree; multivariate Pareto distribution; Pickands dependence function; regular variation; root change formula; tail measure; tail tree; time change formula.

Keywords: Conditional independence; graphical model; Hüsler–Reiss distribution; max-linear model; Markov tree; multivariate Pareto distribution; Pickands dependence function; regular variation; root change formula; tail measure; tail tree; time change formula (search for similar items in EconPapers)
Date: 2020-09-24
Note: In: Advances in Applied Probability - Vol. 52, no.3, p. 855-878 (2020)
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Citations: View citations in EconPapers (4)

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Persistent link: https://EconPapers.repec.org/RePEc:aiz:louvar:2020024

DOI: 10.1017/apr.2020.22

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