Hitting time of a half-line by a two-dimensional nonsymmetric random walk

Abstract

We consider the probability that a two-dimensional random walk starting from the origin never returns to the half-line $(-\infty ,0]\times 0$ before time $n$. Let $X^{(1)}=(X_1,X_2)$ be the increment of the two-dimensional random walk. For an aperiodic random walk with moment conditions ($E[X_2]=0$ and $E[|X_1{|}^{\delta }]<\infty ,E[|X_2{|}^{2+\delta }]<\infty$ for some $\delta \in (0,1)$), we obtain an asymptotic estimate (as $n\to \infty$) of this probability by assuming the behavior of the characteristic function of $X_1$ near zero.