Central limit theorems for functionals of ergodic stationary Markov chains with general state space
Let \(P(s,A)\) be a transition probability on a general measurable space $(\mathcal{S},\Sigma)$ with invariant
probability $m$, and let $\Omega_1 =\mathcal{S}^{\mathbb N}$ be the space of trajectories with $\sigma$-field
$\mathcal{B}_1 = \Sigma^{\otimes \mathbb N}$, with coordinate projections $\{\xi_n\}$. Let $\mathbb{P}_s$ be the probability
on $\mathcal{B}:=\Sigma \times \mathcal{B}_1$ defined by the transition probability $P$ and initial distribution $\delta_s$.
The probability $\mathbb{P}_m:= \int_\mathcal{S} \mathbb{P}_s dm(s)$ is shift invariant on $\mathcal{S} \times \Omega_1$.
The Markov operator $Pf(s):=\int_S f(t)P(s,dt)$ is a contraction of all the $L_p(\mathcal{S},m)$ spaces
($1 \le p \le \infty$). We assume $P$ to be ergodic: $Pf=f \in L_\infty$ implies $f$ is constant a.e. This
implies ergodicity of the shift, and for any $f \in L_1(\mathcal{S},m)$ the ergodic theorem yields
the SLLN for the chain $\{\xi_n\}$:
$\frac1n \sum_{k=1}^n f(\xi_k) \to \int_\mathcal{S} f\,dm$ $\mathbb{P}_m$ a.e.
Given $f \in L_2(\mathcal{S},m)$ with zero integral, we look for conditions on $f$ for the CLT:
when does $\frac1{\sqrt{n}} \sum_{k=1}^n f(\xi_k)$ converge in distribution (in $(\Omega,\mathcal{B},\mathbb{P}_m)$ ) to
a normal random variable?