## The Annals of Probability

- Ann. Probab.
- Volume 24, Number 3 (1996), 1324-1367.

### 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.

#### Article information

**Source**

Ann. Probab., Volume 24, Number 3 (1996), 1324-1367.

**Dates**

First available in Project Euclid: 9 October 2003

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1065725184

**Digital Object Identifier**

doi:10.1214/aop/1065725184

**Mathematical Reviews number (MathSciNet)**

MR1411497

**Zentralblatt MATH identifier**

0863.60020

**Subjects**

Primary: 60F05: Central limit and other weak theorems 60J15 60J55: Local time and additive functionals 60E99: None of the above, but in this section 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]

**Keywords**

Self-interacting random walks local time limit theorems conjugate diffusions

#### Citation

Tóth, Bálint. Generalized Ray-Knight theory and limit theorems for self-interacting random walks on $\mathbb{Z}^1$. Ann. Probab. 24 (1996), no. 3, 1324--1367. doi:10.1214/aop/1065725184. https://projecteuclid.org/euclid.aop/1065725184