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

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 $E[|X_{1}|^δ]<\infty, E[|X_{2}|^{2+ δ}]< \infty$ for some $δ\in (0,1)$), 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.

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