Asymptotic Behavior of Large Deviation Probabilities for a Simple Oscillating Random Walk

Abstract

This paper considers simple oscillating random walks with \( {\overset{\sim }{S}}_n=\sum_{i=1}^n{\overset{\sim }{X}}_t \), under the assumption that \( \mathbf{P}\left({\overset{\sim }{X}}_{n+1}=1\left|{\overset{\sim }{S}}_n\right.>0\right)=p>1/2 \). 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.