Summary
Let τ x denote the time at which a random walk with finite positive mean first passes into (x, ∞), wherex≧0. This paper establishes the asymptotic behaviour of Pr {τ x >n} asn→∞ for fixedx in two cases. In the first case the left hand tail of the step-distribution is regularly varying, and in the second the step-distribution satisfies a one-sided Cramér type condition. As a corollary, it follows that in the first case\(\mathop {\lim }\limits_{n \to \infty } \) Pr {τ x >n}/Pr{τ 0 >n} coincides with the limit of the same quantity for recurrent random walk satisfying Spitzer's condition, but in the second case the limit is more complicated.
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Doney, R.A. On the asymptotic behaviour of first passage times for transient random walk. Probab. Th. Rel. Fields 81, 239–246 (1989). https://doi.org/10.1007/BF00319553
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DOI: https://doi.org/10.1007/BF00319553