Suppose $X_n$ is an ergodic Markov chain (i.e.,there exists an invariant probability measure $\pi$ s.t. $||P^n(x,\cdot)-\pi(\cdot)||_{TV}$) on a noncountable compact space. Define $\phi:\mathbb{R}\longrightarrow \{-1,1\}$, $\phi(X_n)=sgn(X_n)$. Denote $S_n=\frac{1}{\sqrt{n}}(\Sigma_{i=1}^n\phi(X_i)-\mathbb{E}[\Sigma_{i=1}^n\phi(X_i)])$.
Is $\frac{S_n}{\sqrt{n}}$ necessarily bounded in probability (i.e., $\underset{n}{sup}\ \mathbb{P}(\frac{S_n}{\sqrt{n}}>m)\overset{m\longrightarrow \infty}{\longrightarrow}0$)?
I would expect it to be true, since $\phi$ is bounded, but I really can't think of the right way to approach it.