Generalized Ray-Knight theory and limit theorems for self-interacting random walks on $\mathbb{Z}^1$

Abstract

We consider non-Markovian, self-interacting random walks (SIRW) on the one-dimensional integer lattice. The walk starts from the origin and at each step jumps to a neighboring site. The probability of jumping along a bond is proportional to w (number of previous jumps along that lattice bond), where $w: \mathbb{N} \to \math{R}_+$ is a monotone weight function. Exponential and subexponential weight functions were considered in earlier papers. In the present paper we consider weight functions w with polynomial asymptotics. These weight functions define variants of the "reinforced random walk." We prove functional limit theorems for the local time processes of these random walks and local limit theorems for the position of the random walker at late times. A generalization of the Ray-Knight theory of local time arises.