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 π([x,∞)) as x → ∞ for the invariant measure π, 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 π([x,∞)). Now we consider two remaining alternative variants: the case of “power” decreasing of π([x,∞)) and the “mixed” case when π([x,∞)) is asymptotically l(x)e−βx, where l(x) is an integrable function regularly varying at infinity and β > 0.