Markov chains

Abstract

Many time series models are either Markov chains or functions of a Markov chain. We will use the powerful theory of irreducible Markov chains to check the conditions needed to apply the results of Part II. We start with some well-known definitions and elementary properties in order to fix the notation.

14.7 Bibliographical notes

For a general introduction to Markov chains, see [MT09] or [DMPS18]. Extreme value theory for Markov chains has a very long history. See, for instance, the references in [Roo88, Per94] and [JS14]. The earlier references were mainly concerned with the extremal index. In this chapter, our main concern is the regular variation of Markov chains (as processes) and their tail processes, rather than the regular variation of the invariant distribution.

The main reference for existence and tails of solutions of stochastic recurrence equations is [Kes73]. Example 14.1.2 is taken from [Gol91] as well as Problems 14.16 and 14.17. The literature on this topic is extremely wide and still growing. References for regular variation of some specific Markovian-type models, such as AR with random coefficients, AR with GARCH errors, threshold AR, and GARCH, include: [dHRRdV89, BP92, BK01, KP04, Cli07]. See also [JOC12, Als16] for recent references. More reference and multivariate extensions are given in [BDM16].

The results of Section 14.2 are based on [JS14]. See also [RZ13a, RZ13b] for related results.

Section 14.3 builds on [KSW18] which relies on the application of the geometric drift condition initiated by [MW13]. The link between geometric ergodicity and a positive extremal index was first made by [RRSS06]. The asymptotic negligibility condition $\mathrm{{ANSJB}}(r_{n}, c_{n})$ is proved for all $\alpha >0$ in [MW13, Theorem 4.1] for atomic chains by regeneration techniques which can be extended to general geometrically ergodic chains.

Section 14.4 is essentially inspired from [Roo88]. A reference for regenerative processes and (14.4.1) is [Asm03, Chapter VI]. Example 14.4.5 is due to [Smi88].

For regenerative Markov chains special estimation techniques were developed in [BC06, BCT09, BCT13].

Section 14.5 follows [KS15, Section 3]. See also [RZ14].

The stochastic unit-root model of Problem 14.12 is inspired from [GR06] and [Rob07].

The Ryll-Nardzewski-Slivnyak formula (14.4.8) can be found in [BB03, Eq. (1.2.25)]