We consider the probability that a two-dimensional random walk starting from the origin never returns to the half-line (−∞, 0] × {0} before time n. Let X = (X₁, X₂) be the increment of the two-dimensional random walk. For an aperiodic random walk with moment conditions E [X₂] = 0 and E [|X₁|^δ] < ∞, E [|X₂|^2+δ] < ∞ for some δ ∈ (0, 1), we obtain an asymptotic estimate (as n →∞) of this probability by assuming the behavior of the characteristic function of X near zero.