Limit behavior of linearly edge-reinforced random walks on the half-line

Abstract

Motivated by the article [M. Takei, Electron. J. Probab. 26 (2021), article no. 104], we study the limit behavior of linearly edge-reinforced random walks on the half-line ${\mathbb{Z}}_{+}$ with reinforcement parameter $\delta >0$, and each edge $\{x,x+1\}$ has the initial weight $x^{\alpha }{\ln }^{\beta }x$ for $x>1$ and $1$ for $x=0,1$. The aim of this paper is to study the almost sure limit behavior of the walk in the recurrent regime, and extend the results of Takei mentioned above.

22 pages