The first positive position of a lattice random walk

TL;DR

This paper provides a detailed analytical study of the distribution of the first positive position reached by symmetric finite-range lattice random walks, which is important for understanding extremes and records in such processes.

Contribution

It offers a novel, self-contained analytical characterization of the first positive position distribution for symmetric finite-range lattice walks.

Findings

Derived explicit formulas for the distribution

Analyzed asymptotic behavior of the distribution

Provided insights into extreme value statistics in lattice walks

Abstract

The distribution of the first positive position reached by a random walker starting at the origin is central to the analysis of extremes and records in one-dimensional random walks. In this work, we present a detailed and self-contained analytical study of this distribution for symmetric finite-range lattice walks, whose steps are drawn from a distribution supported on finitely many integers.