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

## TL;DR

This paper develops a comprehensive theory of multi-component regular variation for Markov trees, analyzing how tail behaviors change with different conditioning variables and establishing connections via a generalized time change formula.

## Contribution

It introduces a novel multi-component regular variation framework for Markov trees, extending tail analysis beyond single-component cases and linking tail trees through a generalized formula.

## Key findings

- Weak convergence to tail trees under tail assumptions
- Balance of marginal tails leads to a generalized time change formula
- Multi-component regular variation applies to broader models beyond Markov trees

## 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.