Limit Theorems for step reinforced random walks with regularly varying memory

Abstract

We study and prove limit theorems for a class of generalized step reinforced random walks. At every step, the walker chooses a step from the past with probability proportional to a given regularly varying sequence, called the memory sequence. Then it either repeats the chosen step with probability $p$ or uses an innovation with probability $1-p$. We provide functional law of large numbers for the linearly scaled process, viewed at a linearly scaled time. The convergence is almost sure and in $L^1$ under finite mean assumption of the innovation steps. A stronger finite variance assumption gives us $L^2$ convergence. Under finite variance assumption, the suitably scaled walk exhibits a novel phase transition based on the boundedness of a sequence related to the memory sequence. For the subcritical regime, the scaling is diffusive, while it is superdiffusive otherwise. The most interesting contribution of the paper is in the critical regime. We show that the process convergence of the scaled walk, viewed in the linear time scale, can be either in distribution or almost sure, depending on the choice of the memory sequence. We argue that the exponential time scale for the critical regime, traditionally used in the literature, is not natural and we obtain the asymptotic behavior under the linear time scale. In addition, we provide novel scalings other than $\sqrt{n \log n}$ in the critical regime. We also raise some open problems.