The $\sigma$-algebra of pasts of a random walk on the orbits of the Bernoulli action of the group $Z^d$

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 $h(\Xi)=\frac1{2d}\log(2d)$.