  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.
