Asymptotics of Markov Kernels and the Tail Chain

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.

Insurance Applications of Some New Dependence Models Derived from Multivariate Collective Models

“…where the convergence above follows by Lemma 4.2 in Hashorva (2007) (see also Resnick and Zeber (2013) and Kulik and Soulier (2015) for more general results). Next, the convergence in (31) implies…”

Extreme events of Markov chains

“…It is covered in the literature usually on the Fréchet scale, cf. Perfekt (1994); Resnick and Zeber (2013); Kulik and Soulier (2015). The other two cases are concerned with asymptotically independent consecutive states of the original Markov chain.…”

“…In order to gain non-degenerate limits which allow for a refined analysis in such situations, we will introduce random change-points that can detect the misspecification of the norming and adapt the normings accordingly after change-points. The first of the change-points plays a similar role to the extremal boundary in Resnick and Zeber (2013). We also use this concept to resolve some of the subtleties arising from random negative dependence.…”

“…Remark 6. An alternative tail chain approach to Example 8 is to square the ARCH process, Y 2 t instead of Y t , which leads to a random walk tail chain as discussed in Resnick and Zeber (2013). An advantage of our approach is that we may condition on an upper (or by symmetry lower) extreme state whereas in the squared process this information is lost and one has to condition on its norm being large.…”

“…Extensions for asymptotically dependent processes to higher dimensions can be found in Perfekt (1997) and Janßen and Segers (2014) and to higher order Markov chains in Yun (1998) and multivariate Markov chains in Basrak and Segers (2009). Smith et al (1997), Segers (2007) and Janßen and Segers (2014) also study tail chains that go backwards in time and Perfekt (1994) and Resnick and Zeber (2013) include regularity conditions that prevent jumps from a nonextreme state back to an extreme state, and characterisations of the tail chain when the process can suddenly move to a non-extreme state. Almost all the above mentioned tail chains have been derived under regular variation assumptions on the marginal distribution, rescaling the Markov chain by the extreme observation resulting in the tail chain being a multiplicative random walk.…”