Large-deviation probabilities for one-dimensional Markov chains. Part 1: Stationary distributions

In this paper, we consider time-homogeneous and asymptotically space-homogeneous Markov chains that take values on the real line and have an invariant measure. Such a measure always exists if the chain is ergodic. In this paper, we continue the study of the asymptotic properties of $\pi([x,\infty))$ as $x \rightarrow \infty$ for the invariant measure $\pi$, which was started in [A. A. Borovkov, Stochastic Processes in Queueing Theory, Springer-Verlag, New York, 1976], [A. A. Borovkov, Ergodicity and Stability of Stochastic Processes, TVP Science Publishers, Moscow, to appear], and [A. A. Brovkov and D. Korshunov, Ergodicity in a sense of weak convergence, equilibrium-type identities and large deviations for Markov chains, in Probability Theory and Mathematical Statistics, Coronet Books, Philadelphia, 1984, pp. 89–98]. In those papers, we studied basically situations that lead to a purely exponential decrease of $\pi([x,\infty))$. Now we consider two remaining alternative variants: the case of "power" decreasing of $\pi([x,\infty))$ and the "mixed" case when $\pi([x,\infty))$ is asymptotically $l(x)e^{-\beta x}$, where $l(x)$ is an integrable function regularly varying at infinity and $\beta>0$.