Linear prediction and filter stability
Ok, I found out from a book that linear prediction always results in minimum phase filters, which implies stability.

This can be shown using the mean-squared error (MSE) criterion which is to be minimized in the autocorrelation method. The proof is done by contradiction; if an all-pole filter had a single pole outside of the unit circle, the derivative of the MSE in regards to the pole radius is always greater than zero. In terms of the MSE, this implies the pole position is not optimal, so it cannot be a result from linear prediction (under ideal conditions, I assume).

The actual proof can be found here.