Large-deviation probabilities for one-dimensional Markov chains. Part 2: Prestationary distributions in the exponential case

This paper continues investigations of [A. A. Borovkov and A. D. Korshunov, Theory Probab. Appl., 41 (1996), pp. 1–24]. We consider a time-homogeneous and asymptotically space-homogeneous Markov chain $\{X(n)\}$ that takes values on the real line and has increments possessing a finite exponential moment. The asymptotic behavior of the probability $\mathbf{P}\{X(n)\ge x\}$ is studied as $x\to\infty$ for fixed or growing values of time $n$. In particular, we extract the ranges of $n$ within which this probability is asymptotically equivalent to the tail of a stationary distribution $\pi(x)$ (the latter is studied in [A. A. Borovkov and A. D. Korshunov, Theory Probab. Appl., 41 (1996), pp. 1–24] and is detailed in section 27 of [A. A. Borovkov, Ergodicity and Stability of Stochastic Processes, Wiley, New York, 1998]).