Limit Theorems for step reinforced random walks with regularly varying memory

TL;DR

We analyze the Regularly Varying Step Reinforced Random Walk (RVSRRW) with memory weights $\{\mu_n\}$ regularly varying of index $\gamma>-1$. The model couples recollection with probability $p$ to a memory kernel, yielding a phase diagram with a critical threshold $p_c=\frac{\gamma+1/2}{\gamma+1}$ that separates diffusive and superdiffusive behavior, and a rich critical regime whose limits depend on the boundedness of $\{v_n\}$. We establish a comprehensive set of results: a law of large numbers for linearly scaled location and process, almost sure limits when $\{v_n\}$ is bounded, and Gaussian weak limits when $\{v_n\}$ is unbounded, with a variety of scaling laws including a broad class $\sigma_n$ that extends beyond the classical $\sqrt{n\log n}$. The paper also provides detailed LLNs in the first, second, and higher moment settings, and a suite of illustrative examples for the critical regime, including linear, log-modulated, and nonlinear time scales, showing both natural and nonstandard convergence phenomena. Overall, the results yield a complete asymptotic portrait of RVSRRW across subcritical, critical, and supercritical regimes, along with practical guidance on time-space scalings in the critical regime.

Abstract

For a generalized step reinforced random walk, starting from the origin, the first step is taken according to the first element of an innovation sequence. Then in subsequent epochs, it recalls a past epoch with probability proportional to a regularly varying sequence $\{μ_n\}$ of index $γ>-1$; recalls and repeats the step taken with probability $p$, or with probability $1-p$ takes a fresh step from the innovation sequence. The innovation sequence is assumed to be i.i.d.\ with mean zero. We study the corresponding step reinforced random walk process with linearly scaled time as an r.c.l.l.\ function on $[0, \infty)$. We prove law of large numbers for the linearly scaled process almost surely and in $L^1$ for all possible values of $p$ and $γ$. Assuming finite second moments for the innovation sequence, we obtain interesting phase transitions based on the boundedness of a sequence associated with $\{μ_n\}$. The random walk suitably scaled converges almost surely to a process, which may not be Gaussian, when the sequence is bounded and the convergence is in distribution to a Gaussian process otherwise. This phase transition introduces the point of criticality at $p_c=\frac{γ+1/2}{γ+1}$ for $γ>-\frac12$. For the subcritical regime, the process is diffusive, while it is superdiffusive otherwise. However, for the critical regime, the scaled process can converge almost surely or in distribution depending on the choice of sequence $\{μ_n\}$. Almost sure convergence in the critical regime is new. In the critical regime, the scaling can include many more novel choices in addition to $\sqrt{n \log n}$. Further, we use linear time scale and time independent scales even for the critical regime. We argue the exponential time scale for the critical regime is not natural. All the convergences in all the regimes are obtained for the process as an r.c.l.l.\ function.

Theorems & Definitions (113)

• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Theorem 2.3
• proof
• Remark 2.4
• Proposition 2.5
• Theorem 2.6
• Theorem 2.7