  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 $(n^{-\frac 1\alpha}S_n)_{n\geq 1}$ converges in
distribution to a stable distribution of index $\alpha$, with $\alpha\in(1,2]$,
has been investigated. Here, we consider the cases where $(S_n)_{n\in\mathbb
N}$ is either: a) a transient random walk in $\mathbb{Z}^d$, b) a recurrent
random walk in $\mathbb{Z}^d$ such that $(n^{-\frac 1d}S_n)_{n\geq 1}$
converges in distribution to a stable distribution of index $d\in\{1,2\}$.
