  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.

Probability