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 \rightarrow \infty $) of this probability by assuming the behavior of the characteristic function of $X_{1}$ near zero.