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.
Publié le : 1996-07-14
Classification:
Self-interacting random walks,
local time,
limit theorems,
conjugate diffusions,
60F05,
60J15,
60J55,
60E99,
82C41
@article{1065725184,
author = {T\'oth, B\'alint},
title = {Generalized Ray-Knight theory and limit theorems for
self-interacting random walks on $\mathbb{Z}^1$},
journal = {Ann. Probab.},
volume = {24},
number = {2},
year = {1996},
pages = { 1324-1367},
language = {en},
url = {http://dml.mathdoc.fr/item/1065725184}
}
Tóth, Bálint. Generalized Ray-Knight theory and limit theorems for
self-interacting random walks on $\mathbb{Z}^1$. Ann. Probab., Tome 24 (1996) no. 2, pp. 1324-1367. http://gdmltest.u-ga.fr/item/1065725184/