  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.
