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July 1996 Generalized Ray-Knight theory and limit theorems for self-interacting random walks on $\mathbb{Z}^1$
Bálint Tóth
Ann. Probab. 24(3): 1324-1367 (July 1996). DOI: 10.1214/aop/1065725184


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.


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Bálint Tóth. "Generalized Ray-Knight theory and limit theorems for self-interacting random walks on $\mathbb{Z}^1$." Ann. Probab. 24 (3) 1324 - 1367, July 1996.


Published: July 1996
First available in Project Euclid: 9 October 2003

zbMATH: 0863.60020
MathSciNet: MR1411497
Digital Object Identifier: 10.1214/aop/1065725184

Primary: 60E99 , 60F05 , 60J15 , 60J55 , 82C41

Keywords: conjugate diffusions , limit theorems , Local time , Self-interacting random walks

Rights: Copyright © 1996 Institute of Mathematical Statistics


Vol.24 • No. 3 • July 1996
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