Moderate deviation principle for ergodic Markov chain. Lipschitz summands
Résumé
For $\frac12<\alpha<1$, we propose the MDP analysis for family
\[
S_n^\alpha = \frac1{n^\alpha}\sum_{i=1}^n H(X_{i-1}),\ n\geq1,
\]
where $(X_n)_{n\geq0}$ 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’s equation and Stochastic Exponential; the first enables to replace the original family by $\frac1{n^\alpha}M_n$ with a martingale $M_n$ while the second to avoid the direct Laplace transform analysis.