Geodesic
Limit behaviour of one-dimensional random walks in random environments
Teoriâ veroâtnostej i ee primeneniâ, Tome 27 (1982) no. 2, pp. 247-258

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We consider the simplest one-dimensional random walks with transitions $x\to x\pm 1$ having the probabilities $1/2\pm \xi(x)$ where $\xi(x)$ are independent random variables with zero mean and $|\xi(x)|\le c1/2$. Let $x(n)$ be the position of the moving particle after $n$ steps. We show that the limit distribution of $x(n)/\ln^2n$ is concentrated in a random point depending on a concrete realization of $\xi(\cdot)$. 
@article{TVP_1982_27_2_a3,
     author = {Ya. G. Sina{\^\i}},
     title = {Limit behaviour of one-dimensional random walks in random environments},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {247--258},
     publisher = {mathdoc},
     volume = {27},
     number = {2},
     year = {1982},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1982_27_2_a3/}
}
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Ya. G. Sinaî. Limit behaviour of one-dimensional random walks in random environments. Teoriâ veroâtnostej i ee primeneniâ, Tome 27 (1982) no. 2, pp. 247-258. http://geodesic.mathdoc.fr/item/TVP_1982_27_2_a3/