Moderate deviation principle for exponentially ergodic Markov chain

For ${1/2}<\alpha<1$, we propose the MDP analysis for family $$S^\alpha_n=\frac{1}{n^\alpha}\sum_{i=1}^nH(X_{i-1}), n\ge 1,$$ where $(X_n)_{n\ge 0}$ be a homogeneous ergodic Markov chain, $X_n\in \mathbb{R}^d$, when the spectrum of operator $P_x$ is continuous. The vector-valued function $H$ is not assumed to be bounded but the Lipschitz continuity of $H$ is required. The main helpful tools in our approach are Poisson equation and Stochastic Exponential; the first enables to replace the original family by $\frac{1}{n^\alpha}M_n$ with a martingale $M_n$ while the second to avoid the direct Laplace transform analysis.