Convergence of scaled asymptotically-free self-interacting random walks to Brownian motion perturbed at extrema

We consider a family of one-dimensional self-interacting walks whose dynamics characterized by a monotone weight function w on N∪{0}. The weight function takes the form w(n)=(1+2pBn−p+O(n−1−κ))−1, for some B∈R, κ>0 and p∈(0,1]. Our main model parameter is p, and for p∈(0,1∕2] we show the convergence of the SIRW to Brownian motion perturbed at extrema under the diffusive scaling. This completes the functional limit theorem in [8] for the asymptotically free case and extends the result to the full parameter range (0,1]. Our method depends on the generalized Ray-Knight theorems ([12], [8]) for the rescaled local times of this walk. The directed edge local times, described by the branching-like processes, are used to analyze the total drift experienced by the walker.