Geodesic
On Probabilities for Extreme Values of Sums of Random Variables Defined on a Homogeneous Markov Chain with a Finite Number of States
Teoriâ veroâtnostej i ee primeneniâ, Tome 5 (1960) no. 3, pp. 338-352

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This paper examines the probabilities for values of sums of random variables defined on a homogeneous Markov chain with a finite number of states. These values are such that their deviations from the smallest or largest possible value for each instant of time "$n$" are bounded in their sum. By separating traj ectorits in the random walk into classes defined by a proper method, regular components are picked out from the probabilities under consideration and exact and asymptotic formulas are found (for $n\to\infty$) for each of these components. 
@article{TVP_1960_5_3_a5,
     author = {I. S. Volkov},
     title = {On {Probabilities} for {Extreme} {Values} of {Sums} of {Random} {Variables} {Defined} on a {Homogeneous} {Markov} {Chain} with a {Finite} {Number} of {States}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {338--352},
     publisher = {mathdoc},
     volume = {5},
     number = {3},
     year = {1960},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1960_5_3_a5/}
}
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I. S. Volkov. On Probabilities for Extreme Values of Sums of Random Variables Defined on a Homogeneous Markov Chain with a Finite Number of States. Teoriâ veroâtnostej i ee primeneniâ, Tome 5 (1960) no. 3, pp. 338-352. http://geodesic.mathdoc.fr/item/TVP_1960_5_3_a5/