Assume we have a Markov process defined by $p(X_{k+1}|X_k)=N(f(X_k), \sigma^2)$, where the mapping f is nonlinear.
That is, $X_{k+1} = f(X_k) + e_k$ where the i.i.d additive noise $e_k\sim N(0, \sigma^2)$.
Is there any sufficient condition to guarantee that the asymptotic or the stationary distribution of $X_k$ is Gaussian, i.e. $X_k\xrightarrow{\text{dist}}N$? And is there similar result in multivariate cases when $e_k\sim N(0, \Sigma)$?
Any reference will be appreciated. Thank you!