Geodesic
Stochasticity of languages that can be recognized by two-sided finite probabilistic automata
Diskretnaya Matematika, Tome 1 (1989) no. 4, pp. 63-77 
Ya. Ya. Kanep. Stochasticity of languages that can be recognized by two-sided finite probabilistic automata. Diskretnaya Matematika, Tome 1 (1989) no. 4, pp. 63-77. http://geodesic.mathdoc.fr/item/DM_1989_1_4_a6/
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     author = {Ya. Ya. Kanep},
     title = {Stochasticity of languages that can be recognized by two-sided finite probabilistic automata},
     journal = {Diskretnaya Matematika},
     pages = {63--77},
     year = {1989},
     volume = {1},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_1989_1_4_a6/}
}
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We prove that with two-sided finite probabilistic automata only stochastic languages can be recognized, i.e., the capabilities of one-sided and two-sided finite probabilistic automata with a nonisolated point of intersection are identical with respect to recognition of languages. We give estimates for the increase in the number of states on passing from two-sided to one-sided automata that recognize the same language.