On small perturbations of stable Markov operators: unbounded case

We consider the problem of estimating the bounds for generic expressions of the type $\mathbb{E}[\varphi(\gamma,X_1)\cdots\varphi(\gamma,X_{n})]$, where $(X_i)$ is a not necessarily bounded Markov process, $\varphi$ is a smooth function, and $\gamma$ is a small parameter. We show that when the chain $(X_i)$ is exponentially ergodic, some tight bounds can be obtained by small perturbation of the transition operator of the chain. The result is then applied to prove exponential convergence of matrix products and exponential inequalities for Markov chains.