Geodesic
Nonnegative chainable matrices and Kolmogorov's condition
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXIV, Tome 504 (2021), pp. 5-20

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The paper proves that an indecomposable nonnegative matrix is chainable if and only if it satisfies Kolmogorov's condition, under which the Markov chain determined by a stochastic matrix obeys the multidimensional local limit theorem. 
@article{ZNSL_2021_504_a0,
     author = {Yu. A. Al'pin and I. V. Bashkin},
     title = {Nonnegative chainable matrices and {Kolmogorov's} condition},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {5--20},
     publisher = {mathdoc},
     volume = {504},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2021_504_a0/}
}
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Yu. A. Al'pin; I. V. Bashkin. Nonnegative chainable matrices and Kolmogorov's condition. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXIV, Tome 504 (2021), pp. 5-20. http://geodesic.mathdoc.fr/item/ZNSL_2021_504_a0/