Limit behaviour of one-dimensional random walks in random environments

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 $|\xi(x)|\le c<1/2$. 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 $\xi(\cdot)$.