I am trying to derive the stationary distribution of the AR(1) process
$\mathbf{x}_t = A \mathbf{x}_{t-1} + \varepsilon_t$ with $\varepsilon_t \sim \mathcal{N}(0, Q)$ and $A = \begin{pmatrix}
1 & c \\
0 & 1 \\
\end{pmatrix}$ where $0 \le c \le 1$.
As the noise term is Gaussian, the stationary distribution will be Gaussian as well, i.e. $\lim_{t\rightarrow \infty} p(X_t) = \mathcal{N}(\mu,\Sigma)$. In this case $\mathbb{E}[x_t] = \mu = 0$. However, I run into a problem when trying to determine the covariance matrix $\Sigma$ of the stationary distribution:
\begin{align}
\Sigma &= A^2 Var[\mathbf{x}_{t-1}] + Var[\varepsilon_t] \\
\Leftrightarrow \Sigma &= (I – A^2)^{-1} Q
\end{align}
$I – A^2 = \begin{pmatrix}
0 & 2c \\
0 & 0 \\
\end{pmatrix}$ is not invertible. From a simulation I ran, I know that $\Sigma$ converges, so I assume the stationary distribution should exist. Is there a way I can derive it analytically?
Best Answer
I realized that the given AR(1) process is not wide-sense stationary as $|A| = 1$. As a result the variance of $\mathbf{x}_t$ depends on the time lag $dt$ and diverges as $dt$ goes to infinity.