A Riemann-Hilbert approach yields exact generating functions for first-passage statistics of 1D random walks, valid for continuous and discrete, symmetric and asymmetric jumps, with new exact asymptotics for Lévy flights.
On the first positive position of a random walker
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abstract
The distribution of the first positive position reached by a random walker starting from the origin is fundamental for understanding the statistics of extremes and records in one-dimensional random walks. We present a comprehensive study of this distribution, focusing particularly on its moments and asymptotic tail behaviour, in the case where the step distribution is continuous and symmetric, encompassing both diffusive random walks and L\'evy flights.
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First-passage statistics of random walks: a general approach via Riemann-Hilbert problems
A Riemann-Hilbert approach yields exact generating functions for first-passage statistics of 1D random walks, valid for continuous and discrete, symmetric and asymmetric jumps, with new exact asymptotics for Lévy flights.