GARCH Models, 2nd Edition

3 Mixing*

It will be shown that, under mild conditions, GARCH processes are geometrically ergodic and β ‐mixing. These properties entail the existence of laws of large numbers and of central limit theorems (see Appendix A), and thus play an important role in the statistical analysis of GARCH processes. This chapter relies on the Markov chain techniques set out, for example, by Meyn and Tweedie (1996).

3.1 Markov Chains with Continuous State Space

Recall that for a Markov chain only the most recent past is of use in obtaining the conditional distribution. More precisely, (X _t ) is said to be a homogeneous Markov chain, evolving on a space E (called the state space) equipped with a σ‐field ℰ , if for all x ∈ E , and for all B ∈ ℰ ,

3.1

In this equation, P ^t (x, B) corresponds to the transition probability of moving from the state x to the set B in t steps. The Markov property refers to the fact that P ^t (x, B) does not depend on X _r ,  r < s . The fact that this probability does not depend on s is referred to as time homogeneity. For simplicity, we write P (x, B) = P ¹(x, B). The function P : E × ℰ → [0, 1] is called a transition kernel and satisfies:

1. ∀B ∈ ℰ , the function P(⋅, B) is measurable;
2. ∀x ∈ E , the function P(x, ⋅) is a probability measure on (E, ℰ).

The law of the process (X _t ) is characterised by an initial probability measure ...