Large-Deviation Probabilities for One-Dimensional Markov Chains. Part 3: Prestationary Distributions in the Subexponential Case

Abstract

This paper continues investigations of A. A. Borovkov and D. A. Korshunov [Theory Probab. Appl., 41 (1996), pp. 1--24 and 45 (2000), pp. 379--405]. We consider a time-homogeneous Markov chain $\{X(n)\}$ that takes values on the real line and has increments which do not possess exponential moments. The asymptotic behavior of the probability ${\mathbf P}\{X(n)\ge x\}$ is studied as $x\to\infty$ for fixed values of time n and for unboundedly growing~n as well.