On the range of self-interacting random walks on an integer interval

TL;DR

This paper studies the behavior of self-interacting random walks on an integer interval, analyzing their scaled range and its distributional limits, which satisfy specific functional equations and regularity properties.

Contribution

It introduces a framework for understanding the scaled range of self-interacting walks and characterizes the distributional limits via de Rham's functional equations.

Findings

Weak convergence of scaled range established

Distribution functions satisfy de Rham's functional equations

Limits exhibit specific regularity properties

Abstract

We consider the range of a one-parameter family of self-interacting walks on the integers up to the time of exit from an interval. We derive the weak convergence of an appropriately scaled range. We show that the distribution functions of the limits of the scaled range satisfy a certain class of de Rham's functional equations. We examine the regularity of the limits.