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.

本文的第一部分专门研究具有记忆到最大位置的一维随机游走模型，如下所述。在每个步骤中，步行者都以概率\（r \ in（0,1）\）重置到最右边的访问站点，并以剩余概率作为简单随机游走进行移动。使用更新理论的方法，我们证明了随机游动的大数定律和中心极限定理。这些结果证实并显着增强了对在Majumdar等人中建立的过程的均值和方差的分析。（Phys Rev E 92：052126，2015）。在第二部分中，我们通过假设在步骤n中重置概率为\（r_n = \分钟\ {RN ^ { - A}，\ tfrac {1} {2} \} \）与[R， 一个正的参数。对于此模型，我们首先建立\（X_n \） -当前位置的平均值和\（M_n \） -随机游走的最大位置的平均值的渐近行为。因此，我们观察到的比率的一个有趣的相转变\（{{\ mathbb {E}}} [X_n] / {{\ mathbb {E}}} [M_n] \）当一个变化。精确地，它在亚临界阶段\（a \ in（0,1）\）收敛为1，在临界阶段\（a = 1 \）收敛为常数\（c \ in（0,1）\），在超临界阶段\（a> 1 \）中为0 。最后，当\（a> 1 \），我们证明模型的行为与\（\ frac {X_n} {\ sqrt {n}} \ overset {（d）} {\ longrightarrow} { \ mathcal {N}}（0,1）\）和\（\ frac {M_n} {\ sqrt {n}} \ overset {（d）} {\ longrightarrow} \ max _ {0 \ le t \ le 1 } B_t \），其中\（{\ mathcal {N}}（0,1）\）是标准正态分布，\（（B_t）_ {t \ ge 0} \）是标准布朗运动。