# 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.

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

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

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

Get *GARCH Models, 2nd Edition* now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.