Local behaviour of first passage probabilities

Suppose that S is an asymptotically stable random walk with norming sequence c n and that T x is the time that S first enters (x, ∞), where x ≥ 0. The asymptotic behaviour of P(T 0 = n) has been described in a recent paper of Vatutin and Wachtel (Probab. Theory Relat. Fields 143:177-217, 2009), and here we build on that result to give three estimates for P(T x = n), which hold uniformly as n → ∞ in the regions x = o(c n), x = O(c n), and x/c n → ∞, respectively. © 2010 Springer-Verlag.