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

• Published in: Advances in applied probability Vol. 52; no. 3; pp. 855 - 878
• Main Author: Segers, Johan
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.09.2020; Applied Probability Trust
• Subjects: Conditioning; Markov analysis; Original Article; Original Articles; Probability; Random variables; Random walk; River networks; Trees; Hüsler–Reiss distribution; Conditional independence; max-linear model; time change formula; graphical model; root change formula; tail measure; Markov tree; tail tree; Pickands dependence function; multivariate Pareto distribution; regular variation
• ISSN: 0001-8678; 1475-6064; 1475-6064
• DOI: 10.1017/apr.2020.22

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.