The first positive position of a lattice random walk

TL;DR

This work provides a complete, self-contained analytic treatment of the distribution of the first positive position $H$ for symmetric finite-range lattice random walks. Using a Wiener--Hopf factorization tailored to finite-range lattice steps, the authors express observables in terms of the complex zeros $z_a$ of the characteristic polynomial, yielding exact formulas for the distribution $f_k$ of $H$, its moments, and the related renewal process quantities. Key results include $f_k=S_{k-1}-S_k$ with $S_k$ the elementary symmetric polynomials of the zeros, the mean $\langle H\rangle={\cal E}\sqrt{D}$, higher moments in terms of factorial cumulants $c_m$, and explicit equilibrium backward/forward length distributions, all tied together by discrete Pollaczek--Spitzer-type representations. The paper also analyzes discrete uniform step distributions, deriving large-$K$ scaling to the continuous-limit problem and providing detailed asymptotics for moments and variances, illustrating the robustness and limits of the approach. Overall, the work furnishes a rigorous framework for extremes and records in one-dimensional lattice walks and connects discrete renewal structures to continuous analogues through precise scaling laws.

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.

Figures (3)

• Figure 1: Rescaled probabilities $K f_k$ against $x=(k-1/2)/K$ for discrete uniform step distributions with $K=4$, 7 and 10 (see legend). Continuous curve: density $f_c(x)$ entering the scaling behaviour (\ref{['fsca']}), obtained by Wiener-Hopf techniques (see I).
• Figure 2: Symbols: plots of ${\langle H\rangle}/K$ (red), of ${\langle H^2\rangle}/K^2$ (blue) and of $V$ (green) against $1/K$ for discrete uniform step distributions with $K$ ranging from 4 to 60. Straight lines with corresponding colours: expressions (\ref{['momasy']}) including linear corrections in $1/K$.
• Figure 3: The $K-1$ complex zeros $z_a$ (red symbols) and their reciprocals $1/z_a$ (blue symbols) in the complex $z$-plane, for discrete uniform step distributions with range $K$. Left panel: $K=11$ is odd: there are 10 complex zeros. Right panel: $K=22$ is even: there are 21 zeros, including a real positive one. Black: unit circle. Green square: double diffusive zero at $z=-1$.