Geodesic
The $\sigma$-algebra of pasts of a~random walk on the orbits of the Bernoulli action of the group~$Z^d$
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XII, Tome 325 (2005), pp. 103-112

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In the present paper, we study the $\sigma$-algebra of pasts $\Xi=\{\xi_n\}_n$ of a random walk $\mathcal T$ on the orbits of the Bernoulli action of the group $Z^d$. The proper scaling and the scaling entropy of this sequence of partitions is calculated. We show that the proper scaling entropy of the $\sigma$-algebra of pasts is $h(\Xi)=\frac1{2d}\log(2d)$. 
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     author = {A. D. Gorbul'skii},
     title = {The $\sigma$-algebra of pasts of a~random walk on the orbits of the {Bernoulli} action of the group~$Z^d$},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {103--112},
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A. D. Gorbul'skii. The $\sigma$-algebra of pasts of a~random walk on the orbits of the Bernoulli action of the group~$Z^d$. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XII, Tome 325 (2005), pp. 103-112. http://geodesic.mathdoc.fr/item/ZNSL_2005_325_a5/