Asymptotics of Markov Kernels and the Tail Chain

TL;DR

This paper develops a general framework for the tail chain model of Markov chains, connecting it with extreme value theory and demonstrating its implications for multivariate regular variation based on marginal tail behavior.

Contribution

It formalizes the distinction between extreme and non-extreme states and links the tail chain model to transition kernels and extreme value theory.

Findings

Establishes a connection between tail chains and multivariate regular variation.

Provides a generalized context for tail chain models in Markov chains.

Shows the applicability of the model under assumptions on marginal tails.

Abstract

An asymptotic model for extreme behavior of certain Markov chains is the "tail chain". Generally taking the form of a multiplicative random walk, it is useful in deriving extremal characteristics such as point process limits. We place this model in a more general context, formulated in terms of extreme value theory for transition kernels, and extend it by formalizing the distinction between extreme and non-extreme states. We make the link between the update function and transition kernel forms considered in previous work, and we show that the tail chain model leads to a multivariate regular variation property of the finite-dimensional distributions under assumptions on the marginal tails alone.