Local Probabilities for Random Walks Conditioned to Stay Positive

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 we study the asymptotic behavior, as n → ∞, of the local probabilities P(τ = n) and the conditional local probabilities P(Sn ∈ [x, x+∆)|τ > n) for fixed ∆ and x = x(n) ∈ (0,∞) .