I am new to Markov chain in continuous state space. It would be great if anyone can help me with this.
Consider a linear stochastic system: $x_{t+1}=Ax_t +w_t$, where $w_t$ i.i.d. Gaussian with zero mean, finite variance. Suppose $A$ is stable, i.e. the spectral norm is less than 1. Then, we know this Markov chain has time-invariant distribution $\pi$.
My question is, does central limit theorem (CLT) and strong law of large numbers (LLN) hold? In particular, CLT means, for any function $f(\cdot)$, there exists $\sigma>0$, such that $$\frac{1}{\sqrt T} \sum_{t=1}^T f(x_t) \to N(\mathbb E_{\pi} f(x), \sigma^2$$ where the convergence is in distribution; and LLN means $$\frac{1}{T} \sum_{t=1}^T f(x_t) \to \mathbb E_{\pi} f(x)$$ where the convergence is almost surely.
Thank you very much!