Suppose that $\zeta_t$ is a univariate, stationary stochastic process ($t\in\mathbb{N}^+$).

Let $x_0\in\mathbb{R}^n$, and let $f:\mathbb{R}\rightarrow\mathbb{R}^{n\times n}$ be a continuously differentiable function.

Finally, suppose that for $t\in\mathbb{N}^+$, $x_t$ is defined by:
$$x_t=f(\zeta_t)x_{t-1}.$$

Question: Under what conditions does $x_t$ converge in probability to $0$ as $t\rightarrow\infty$?

Partial answer: If $\zeta_t$ is independent across time, and all of the eigenvalues of $\mathbb{E}f(\zeta_t)$ are in the unit circle, then the results of Conlisk (1974) imply that $x_t$ converges in probability to $0$ as $t\rightarrow\infty$. How does this generalize to the autocorrelated case?