One-dimensional Asymptotically Homogeneous Markov Chains: Cramér Transform and Large Deviation Probabilities

We consider a time-homogeneous ergodic Markov chain $\{X_n\}$ that takes values on the real line and has asymptotically homogeneous increments at infinity. We assume that the “limit jump” $\xi$ of $\{X_n\}$ has negative mean and satisfies the Cramér condition, i.e., the equation $\Bbb E\,e^{\beta\xi}=1$ has positive solution $\beta$. The asymptotic behavior of the probability $\mathbb P\{X_n>x\}$ is studied as $n\to\infty$ and $x\to\infty$. In particular, we distinguish the ranges of time $n$ where this probability is asymptotically equivalent to the tail of a stationary distribution.