Geodesic
Large-deviation probabilities for one-dimensional Markov chains. Part 2: Prestationary distributions in the exponential case
Teoriâ veroâtnostej i ee primeneniâ, Tome 45 (2000) no. 3, pp. 437-468

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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]).
Mots-clés : Markov chain
Keywords: rough and exact asymptotic behavior of large-deviation probabilities, transition phenomena, invariant measure. 
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     author = {A. A. Borovkov and D. A. Korshunov},
     title = {Large-deviation probabilities for one-dimensional {Markov} chains. {Part} 2: {Prestationary} distributions in the exponential case},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {437--468},
     publisher = {mathdoc},
     volume = {45},
     number = {3},
     year = {2000},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2000_45_3_a1/}
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A. A. Borovkov; D. A. Korshunov. Large-deviation probabilities for one-dimensional Markov chains. Part 2: Prestationary distributions in the exponential case. Teoriâ veroâtnostej i ee primeneniâ, Tome 45 (2000) no. 3, pp. 437-468. http://geodesic.mathdoc.fr/item/TVP_2000_45_3_a1/