On the first positive position of a random walker

TL;DR

This paper thoroughly analyzes the distribution of the first positive position in symmetric, continuous step random walks, exploring its moments and tail behavior for both diffusive and Lévy flight models.

Contribution

It provides a comprehensive analysis of the distribution, moments, and asymptotic tail behavior of the first positive position in symmetric, continuous step random walks.

Findings

Derived explicit formulas for moments of the distribution.

Characterized the asymptotic tail behavior for different types of walks.

Unified treatment of diffusive and Lévy flight cases.

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.