Asymptotic properties with probability 1 for one-dimensional random walks in a random environment

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.