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We use generalized Ray-Knight theorems introduced by B\'alint T\'oth in 1996together with techniques developed for excited random walks as main tools forestablishing positive and negative results concerning convergence of someclasses of diffusively scaled self-interacting random walks (SIRWs) to Brownianmotions perturbed at extrema (BMPE). T\'oth's work studied two classes ofSIRWs: asymptotically free and polynomially self-repelling walks. For bothclasses Toth has shown, in particular, that the distribution function of ascaled SIRW observed at independent geometric times converges to that of a BMPEindicated by the generalized Ray-Knight theorem for this SIRW. The question ofweak 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 alarge class of asymptotically free SIRWs which includes asymptotically freewalks considered in T\'oth's paper. On the other hand, we show that rescaledpolynomially self-repelling SIRWs do not converge to the BMPE predicted by thecorresponding generalized Ray-Knight theorems and, hence, do not converge toany BMPE.

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