Two-boundary problem for a random walk in a random environment

Given a sequence of independent identically distributed pairs of random variables $(p_i,q_i)$, $i\in\mathbf{Z}$, with $p_0+q_0=1$, and $p_0>0$ a.s., $q_0>0$ a.s., one considers a random walk in the random environment $(p_i,q_i)$, $i\in\mathbf{Z}$. This means that, for a fixed random environment, a walking particle transits from the state $i$ either to the state $(i+1)$ with probability $p_i$ or to the state $(i-1)$ with probability $q_i$. It is assumed that $\mathbf{E}\ln (p_0/q_0)=0$, that is, the walk is oscillating. We are concerned with the exit problem of the walk under consideration from the interval $(-\lfloor an\rfloor,\lfloor bn\rfloor)$, where $a$, $b$ are arbitrary positive constants. We find the asymptotics of the exit probability of the walk from the above interval from the right (the left). A limit theorem for the exit time of the walk from this interval is obtained.