Let S0 = 0, {Sn}n≥1 be a random walk generated by a sequence of i.i.d. random variables X1,X2, ... and let τ− := min {n ≥ 1 : Sn ≤ 0} and τ := min {n ≥ 1 : Sn > 0}. Assuming that the distribution of X1 belongs to the domain of attraction of an α-stable law, α 6= 1, we study the asymptotic behavior of P(τ± = n) as n → ∞.
