A central limit theorem for Markov chains and applications to hypergroups

Let $(X_n)$ be a homogeneous Markov chain on an unbounded Borel subset of $\mathbb{R}$ with a drift function $d$ which tends to a limit $m_1$ at infinity. Under a very simple hypothesis on the chain we prove that $n^{-1/2} (X_n - \sum^ n_{k=1} d(X_{k-1}))$ converges in distribution to a normal law $N (0, \sigma^2)$ where the variance $\sigma^2$ depends on the asymptotic behaviour of $(X_n)$. When $d - m_1$ goes to zero quickly enough and $m_1 \neq 0$, the random centering may be replaced by $n m_1.$ These results are applied to the case of random walks on some hypergroups.