Empirical processes for recurrent and transient random walks in random scenery

In this paper, we are interested in the asymptotic behaviour of the sequence of processes $(W_n(s,t){)}_{s,t\in [0,1]}$ with \begin{equation*} W_n(s,t):=\sum_{k=1}^{\lfloor nt\rfloor}\big(1_{{\xi_{S_k}\leq s}}-s\big) \end{equation*} where $({\xi }_x,x\in {\mathbb{Z}}^d)$ is a sequence of independent random variables uniformly distributed on $[0,1]$ and $(S_n{)}_{n\in \mathbb{N}}$ is a random walk evolving in ${\mathbb{Z}}^d$, independent of the $\xi$’s. In Wendler (2016), the case where $(S_n{)}_{n\in \mathbb{N}}$ is a recurrent random walk in $\mathbb{Z}$ such that $(n^{-\frac{1}{\alpha }}{S}_n{)}_{n\ge 1}$ converges in distribution to a stable distribution of index $\alpha$, with $\alpha \in (1,2]$, has been investigated. Here, we consider the cases where $(S_n)_{n\in\mathbb N}$iseither:a)atransientrandomwalkin$$\mathbb{Z}$^d$, b) a recurrent random walk in ${\mathbb{Z}}^d$ such that $(n^{-\frac{1}{d}}{S}_n{)}_{n\ge 1}$ converges in distribution to a stable distribution of index $d\in \{1,2\}$.