Convergence and non-convergence of scaled self-interacting random walks to Brownian motion perturbed at extrema

TL;DR

This paper investigates the convergence of scaled self-interacting random walks to Brownian motions perturbed at extrema, establishing a full limit theorem for some classes and showing non-convergence for others using advanced probabilistic techniques.

Contribution

It provides a comprehensive functional limit theorem for asymptotically free SIRWs and demonstrates non-convergence for polynomially self-repelling SIRWs, clarifying their limiting behaviors.

Findings

Full convergence for a broad class of asymptotically free SIRWs.

Non-convergence of polynomially self-repelling SIRWs to BMPE.

Application of generalized Ray-Knight theorems and excited random walk techniques.

Abstract

We use generalized Ray-Knight theorems introduced by B\'alint T\'oth in 1996 together with techniques developed for excited random walks as main tools for establishing positive and negative results concerning convergence of some classes of diffusively scaled self-interacting random walks (SIRWs) to Brownian motions perturbed at extrema (BMPE). T\'oth's work studied two classes of SIRWs: asymptotically free and polynomially self-repelling walks. For both classes Toth has shown, in particular, that the distribution function of a scaled SIRW observed at independent geometric times converges to that of a BMPE indicated by the generalized Ray-Knight theorem for this SIRW. The question of weak convergence of one-dimensional distributions of scaled SIRW remained open. In this paper, on the one hand, we prove a full functional limit theorem for a large class of asymptotically free SIRWs which includes asymptotically free walks considered in T\'oth's paper. On the other hand, we show that rescaled polynomially self-repelling SIRWs do not converge to the BMPE predicted by the corresponding generalized Ray-Knight theorems and, hence, do not converge to any BMPE.