Limit Theorems for the One-Dimensional Random Walk with Random Resetting to the Maximum

Abstract

The first part of this paper is devoted to study a model of one-dimensional random walk with memory to the maximum position described as follows. At each step the walker resets to the rightmost visited site with probability $r \in (0,1)$ and moves as the simple random walk with remaining probability. Using the approach of renewal theory, we prove the laws of large numbers and the central limit theorems for the random walk. These results reprove and significantly enhance the analysis of the mean value and variance of the process established in Majumdar et al. (Phys Rev E 92:052126, 2015). In the second part, we expand the analysis to the situation where the memory of the walker decreases over time by assuming that at the step n the resetting probability is $r_n = \min \{rn^{-a}, \tfrac{1}{2}\}$ with r, a positive parameters. For this model, we first establish the asymptotic behavior of the mean values of $X_n$-the current position and $M_n$-the maximum position of the random walk. As a consequence, we observe an interesting phase transition of the ratio ${{\mathbb {E}}}[X_n]/{{\mathbb {E}}}[M_n]$ when a varies. Precisely, it converges to 1 in the subcritical phase $a\in (0,1)$, to a constant $c\in (0,1)$ in the critical phase $a=1$, and to 0 in the supercritical phase $a>1$. Finally, when $a>1$, we show that the model behaves closely to the simple random walk in the sense that $\frac{X_n}{\sqrt{n}} \overset{(d)}{\longrightarrow } {\mathcal {N}}(0,1)$ and $\frac{M_n}{\sqrt{n}} \overset{(d)}{\longrightarrow } \max _{0 \le t \le 1} B_t$, where ${\mathcal {N}}(0,1)$ is the standard normal distribution and $(B_t)_{t\ge 0}$ is the standard Brownian motion.