Geodesic
The Perron--Frobenius theorem~-- a~proof with the use of Markov chains
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXI, Tome 359 (2008), pp. 5-16

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The Perron–Frobenius theorem for an irreducible nonnegative matrix is proved using the matrix graph and the ergodic theorem of the theory of Markov chains. Bibl. – 7 titles. 
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     author = {Yu. A. Al'pin and V. S. Al'pina},
     title = {The {Perron--Frobenius} theorem~-- a~proof with the use of {Markov} chains},
     journal = {Zapiski Nauchnykh Seminarov POMI},
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     volume = {359},
     year = {2008},
     language = {ru},
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Yu. A. Al'pin; V. S. Al'pina. The Perron--Frobenius theorem~-- a~proof with the use of Markov chains. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXI, Tome 359 (2008), pp. 5-16. http://geodesic.mathdoc.fr/item/ZNSL_2008_359_a0/