Asymptotic behavior of large deviation probabilities for a simple oscillating random walk

This paper considers simple oscillating random walks with $\tilde{S}_n=\sum\limits^n_{i=1} \tilde{X}_i$, under the assumption that $\mathbf P (\tilde{X}_{n+1}=1\mid \tilde{S}_n>0)=p>1/2$. We show that the asymptotic behavior of probability to reach high level for the oscillating random walk and a standard random walk are similar up to a constant multiplier. The asymptotics for the maximum of a random walk and for the moment of the first exit beyond the high level are obtained.