Moderate deviation principle for exponentially ergodic Markov chain

For $1\mathrm{/}2<\alpha <1$, we propose the MDP analysis for family $$S_n^{\alpha }=\frac{1}{n^{\alpha }}\sum _{i=1}^{n}H(X_{i-1}),n\ge 1,$$ 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.