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 P {X(n) greater than or equal to x} is studied as x --> infinity 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]).