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 $\mathbb{E}[f(\zeta_t)\otimes f(\zeta_t)]$ 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?
New edit. I'm going to post a few helpful references I discovered, for anyone who stumbles across this question:
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[2] Lei Guo, Lennart Ljung, and Guan-Jun Wang, "Necessary and Sufficient Conditions for Stability of LMS", IEEE Transactions On Automatic Control, Vol. 42, no. 6, June 1997, 761-770, MR1455707, Zbl 0886.93073.
[3] Lei Guo and Lennart Ljung, "Exponential Stability of General Tracking Algorithms", IEEE Transactions on Automatic Control, Vol. 40, no. 8, August 1995, 1376-1387, MR1343803, Zbl 0834.93059.
[4] Robert R. Bitmead and Brian D. O. Anderson, "Lyapunov Techniques for the Exponential Stability of Linear Difference Equations with Random Coefficients", IEEE Transactions on Automatic Control, Vol. 25, no. 4, 782-787 (1980), MR583456, Zbl 0451.93065.
[5] Yuzhen Qin, Ming Cao, and Brian D. O. Anderson, "Lyapunov Criterion for Stochastic Systems and Its Applications in Distributed Computation", IEEE Transactions on Automatic Control, Vol. 65, No. 2, 546-560 (2020), MR4060252, Zbl 7256184.
