Geodesic
Asymptotic properties with probability~1 for one-dimensional random walks in a~random environment
Sbornik. Mathematics, Tome 74 (1993) no. 2, pp. 455-473

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Random walks in a random environment are considered on the set $\mathbf Z$ of integers when the moving particle can go at most $R$ steps to the right and at most $L$ steps to the left in a unit of time. The transition probabilities for such a random walk from a point $x\in\mathbf Z$ are determined by the vector $\mathbf p(x)\in\mathbf R^{R+L+1}$. It is assumed that the sequence $\{\mathbf p(x),\,x\in\mathbf Z\}$ is a sequence of independent identically distributed random vectors. Asymptotic properties with probability 1 are investigated for such a random process. An invariance principle and the law of the iterated logarithm for a product of independent random matrices are proved as auxiliary results. 
@article{SM_1993_74_2_a9,
     author = {A. V. Letchikov},
     title = {Asymptotic properties with probability~1 for one-dimensional random walks in a~random environment},
     journal = {Sbornik. Mathematics},
     pages = {455--473},
     publisher = {mathdoc},
     volume = {74},
     number = {2},
     year = {1993},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1993_74_2_a9/}
}
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A. V. Letchikov. Asymptotic properties with probability~1 for one-dimensional random walks in a~random environment. Sbornik. Mathematics, Tome 74 (1993) no. 2, pp. 455-473. http://geodesic.mathdoc.fr/item/SM_1993_74_2_a9/