Abstract:
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.
Keywords:random walk in random environment, branching process in random environment with immigration, limit theorem.