Geodesic
On the $\sigma$-algebra of symmetrical events for a countable Markov chain
Teoriâ veroâtnostej i ee primeneniâ, Tome 24 (1979) no. 1, pp. 198-204

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It is proved that the $\sigma$-algebra of symmetrical events for a countable Markov chain $\{x_n, n\ge 1\}$ coincides a. s. with $T(\sigma_1\bigcap\sigma_2)$, where $T$ is a left shift and $\sigma_i=\sigma(x_i)$, $i=1,2$. 
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     author = {L. A. Grigorenko},
     title = {On the $\sigma$-algebra of symmetrical events for a countable {Markov} chain},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
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     volume = {24},
     number = {1},
     year = {1979},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1979_24_1_a20/}
}
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L. A. Grigorenko. On the $\sigma$-algebra of symmetrical events for a countable Markov chain. Teoriâ veroâtnostej i ee primeneniâ, Tome 24 (1979) no. 1, pp. 198-204. http://geodesic.mathdoc.fr/item/TVP_1979_24_1_a20/