On the first positive position of a random walker

2025/01/28 by Claude Godrèche, J. M. Luck, Godrèche, Claude +1 · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #Diffusion and Search Dynamics #FOS: Mathematics #FOS: Physical sciences #Probability (math.PR) #Statistical Mechanics (cond-mat.stat-mech) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2501.17268

openalex publication_date 2025/01/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Lévy flights.

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