Abstract
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.
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Doney, R.A. Local behaviour of first passage probabilities. Probab. Theory Relat. Fields 152, 559–588 (2012). https://doi.org/10.1007/s00440-010-0330-7
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DOI: https://doi.org/10.1007/s00440-010-0330-7