Geodesic
The Geometry of Finite Markov Chains
Canadian mathematical bulletin, Tome 8 (1965) no. 3, pp. 345-358

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The purpose of this paper is to present a geometric theorem which provides a proof of a fundamental theorem of finite Markov chains.The theorem, stated in matrix theoretic terms, concerns the asymptotic behaviour of the powers of an n by n stochastic matrix, that is, a matrix of non-negative entries each of whose row sums is 1. The matrix might arise from a repeated physical process which goes from one of n possible states to another at each iteration and whose probability of going to a state depends only on the state it is in at present and not on its more distant history. 
Pullman, N. The Geometry of Finite Markov Chains. Canadian mathematical bulletin, Tome 8 (1965) no. 3, pp. 345-358. doi: 10.4153/CMB-1965-025-3
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     title = {The {Geometry} of {Finite} {Markov} {Chains}},
     journal = {Canadian mathematical bulletin},
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     year = {1965},
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     number = {3},
     doi = {10.4153/CMB-1965-025-3},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1965-025-3/}
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