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

Abstract

We use generalized Ray–Knight theorems, introduced by B. Tóth 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 (SIRW) to Brownian motions perturbed at extrema (BMPE). Tóth’s work studied two classes of SIRWs: asymptotically free and polynomially self-repelling walks. For both classes Tóth 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 the asymptotically free walks considered by Tóth. 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.