GARCH Models, 2nd Edition

Appendix B Ergodicity, Martingales, Mixing

A.1. Ergodicity

A stationary sequence is said to be ergodic if it satisfies the strong law of large numbers.

Definition A.1 Ergodic stationary processes

A real valued strictly stationary process (Z _t ) _{t ∈ ℤ} , is said to be ergodic if and only if, for any Borel set B and any integer k ,

with probability 1. ¹

General transformations of ergodic sequences remain ergodic. The proof of the following result can be found, for instance, in Billingsley (1995, Theorem 36.4).

Theorem A.1

If (Z _t ) _{t ∈ ℤ} is an ergodic strictly stationary sequence and if (Y _t ) _{t ∈ ℤ} is defined by

where f is a measurable function from ℝ^∞ to ℝ, then (Y _t ) _{t ∈ ℤ} is also an ergodic strictly stationary sequence.

In particular, if (X _t ) _{t ∈ ℤ} is the non‐anticipative stationary solution of the AR(1) equation

A.1

then the theorem shows that (X _t ) _{t ∈ ℤ} , (X _t − 1 η _t ) _{t ∈ ℤ} and are also ergodic stationary sequences.

Theorem A.2 The ergodic theorem for stationary sequences

If (Z _t ) _{t ∈ ℤ} is strictly ...