I am trying to find the stationary distribution of an AR(1) process, where the shock terms themselves are an AR(1) process. That is, the process moves subject to the following 2 equations: \begin{equation} z_{t+1} = \rho z_t + \sigma_t \eta_{z,t+1} \end{equation} \begin{equation} \sigma^2_{t+1} = \max\left\{\nu \sigma_t^2 + \sigma \eta_{\sigma,t+1},\: 0 \right\} \end{equation} where $\eta_{\sigma, t+1}, \eta_{z,t+1} \sim N(0,1)$ are IID.
I believe that the stationary distribution of $\sigma_t \eta_{z,t+1}$ is a linear combination of $\chi$-square distributions. (Based on answers here)
Does this mean that the stationary distribution of $z_{t+1}$ is also the same linear combination of $\chi$-square distributions?