Hitting times for the perturbed reflecting random walk (Q1761487)

For the real valued process \((X_n)_{n\geq 0}\), denote by \(F_n^X\) the \(\sigma\)-algebra generated by \(X_0, X_1, \dots, X_n\) and set \(\overline{X}_n=\max\{X_0, X_1,\dots, X_n\}\). The perturbed reflecting random walk (PRRW) with reinforcement parameter \(r\in (-1,1)\) is a process \((X_n)_{n\geq 0}\) taking its values on \(\mathbb{Z}_+=\{0,1,2,\dots\}\) such that, for every \(n\geq 0\), \(X_{n+1}\in \{X_n-1, X_n+1\}\) and the transition probability \(\mathbb{P}(X_{n+1}=X_n+1 | F_n^X)\) is equal to \(\frac12\) if \(0<X_n<\overline{X}_n\); \(\frac{1-r}2\) if \(X_n={\overline{X}_n}\) and \(n\geq 1\); 1 if \(X_n=0\). Moreover \(X_0=0\). The goal of the paper is to study the PRRW via an excursion point of view. The law of the hitting times is computed. The invariance principles with explicit descriptions of the asymptotic laws are given. Some results on the almost sure asymptotic behavior are obtained.