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 $(W_n(s,t))_{s,tin[0,1]}$ with begin{equation*} W_n(s,t):=sum_{k=1}^{lfloor ntrfloor}big(1_{{xi_{S_k}leq s}}-sbig) end{equation*} where $(xi_x, xinmathbb{Z}^d)$ is a sequence of independent random variables uniformly distributed on $[0,1]$ and $(S_n)_{ninmathbb N}$ is a random walk evolving in $mathbb{Z}^d$, independent of the $xi$s. In Wendler (2016), the case where $(S_n)_{ninmathbb N}$ is a recurrent random walk in $mathbb{Z}$ such that $(n^{-frac 1alpha}S_n)_{ngeq 1}$ converges in distribution to a stable distribution of index $alpha$, with $alphain(1,2]$, has been investigated. Here, we consider the cases where $(S_n)_{ninmathbb N}$ is either: a) a transient random walk in $mathbb{Z}^d$, b) a recurrent random walk in $mathbb{Z}^d$ such that $(n^{-frac 1d}S_n)_{ngeq 1}$ converges in distribution to a stable distribution of index $din{1,2}$.