The range of once-reinforced random walk on the half-line
Zechun Hu, Ting Ma, Renming Song, Li Wang
TL;DR
This paper analyzes the range of the once-reinforced random walk (ORRW) on the half-line $\mathbb{Z}_{+}$ with reinforcement parameter $c>0$. By adapting the Pfaffelhuber–Stiefel method, it derives asymptotics for all moments of the range $R_n$, showing that $\mathbb{E}\left[(R_n/\sqrt{n})^{\ell}\right]$ converges to a constant multiple of a single integral $J_\ell(c)$, i.e. $\mathbb{E}\left[(R_n/\sqrt{n})^{\ell}\right] \sim \frac{1}{2^{(\ell-2)/2}\Gamma(\ell/2)} J_\ell(c)$ with $J_\ell(c)=2^{c}\int_{0}^{\infty} x^{\ell-1}\left(\frac{e^{x}}{e^{2x}+1}\right)^{c}dx$. Special cases recover known constants for $c=1$, and the results extend the full-line theory to the half-line, revealing how boundary effects modify the range’s moment growth while preserving the $\sqrt{n}$ scaling. The analysis hinges on a range-time decomposition, generating-function techniques, and Tauberian theorems to translate generating-function asymptotics into time-domain moment asymptotics. Overall, the paper provides a complete description of the asymptotic behavior of all moments of the range for ORRW on the half-line.
Abstract
In this paper, we consider a once-reinforced random walk on the half-line, and give the limiting behaviors of all the moments of its range.