Inequalities for the total variation between the distributions of a sequence and its translate and applications

Let $\xi=(\xi_k)_{k\in\mathbf{N}^*}$ be a stationary homogeneous Markov chain and its translate $\xi+a=(\xi_k+a_k)_{k\in\mathbf{N}^*}$ be a real sequence. We prove an inequality for the total variation between the distributions of $\xi$ and $\xi+a$. This result allows us to give sufficient conditions for absolute continuity of these distributions. Next, we consider $\xi=(\xi_k)_{k\in\mathbf{N}^*}$ a sequence of independent and identically distributed random variables and another sequence of independent variables $\eta=(\eta_k)_{k\in\mathbf{N}^*}$, which is independent of $\xi$. We estimate the total variation between the distributions of $\xi$ and $\xi+\eta$ and apply the obtained results to the problem of absolute continuity.