If we have a VAR process:
$\begin{align} \mathbf{y}_t = A_1\mathbf{y}_{t-1} + \dots + A_d\mathbf{y}_{t-d} + \boldsymbol{\epsilon}_t, \quad t \in \mathbb{Z} \end{align}$
With the stability condition met:
$\det(I - A_1z - \dots - A_d z^d) \neq 0$ for $|z| \leq 1$
Do we know that the process is then geometrically ergodic.
I know that from theorem 2.4(i) in this book, we know that an AR($d$) process is geometrically ergodic, but I do not know if this generalises to VAR($d$) processes.
The book takes that theorem from Theorem 3.1 of this paper where the conditions are listed as (3.1) and (3.2) above.
To summarise, we know that a stable AR model is geometrically ergodic, I want to know if this holds for a VAR process.
