  It is classical to approximate the distribution of fractional Brownian motion
by a renormalized sum $ S_n $ of dependent Gaussian random variables. In this
paper we consider such a walk $ Z_n $ that collects random rewards $ \xi_j $
for $ j \in \mathbb Z,$ when the ceiling of the walk $ S_n $ is located at $
j.$ The random reward (or scenery) $ \xi_j $ is independent of the walk and
with heavy tail. We show the convergence of the sum of independent copies of $
Z_n$ suitably renormalized to a stable motion with integral representation,
whose kernel is the local time of a fractional Brownian motion (fBm). This work
extends a previous work where the random walk $ S_n$ had independent increments
limits.
