Geodesic
The local limit theorem for Markov chains and regularity conditions
Teoriâ veroâtnostej i ee primeneniâ, Tome 13 (1968) no. 1, pp. 183-190

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The following stochastic process $x_t$ is considered. Given a set of segments $\{\Delta_i=[a_i,b_i],\ i=\overline{1,s}\}$, a point $x_t$ is moving uniformly along $\Delta_i$, having reached $b_i$ it jumps to $a_j$ with a probability $p_{ij}$ and then it goes on moving uniformly. In the present paper the necessary and sufficient conditions of regularity of $x_t$ are obtained. At the same time the connexion between these conditions and those of the local limit theorem for finite Markov chains is established. 
@article{TVP_1968_13_1_a18,
     author = {B. M. Gurevich},
     title = {The local limit theorem for {Markov} chains and regularity conditions},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {183--190},
     publisher = {mathdoc},
     volume = {13},
     number = {1},
     year = {1968},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1968_13_1_a18/}
}
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B. M. Gurevich. The local limit theorem for Markov chains and regularity conditions. Teoriâ veroâtnostej i ee primeneniâ, Tome 13 (1968) no. 1, pp. 183-190. http://geodesic.mathdoc.fr/item/TVP_1968_13_1_a18/