Convergence of dependent walks in a random scenery to fBm-local time fractional stable motions

It is classical to approximate the distribution of fractional Brownian motion by a renormalized sum Sn of dependent Gaussian random variables. In this paper we consider such a walk Zn that collects random rewards ξj for j ∈ Z, when the ceiling of the walk Sn is located at j. The random reward (or scenery) ξj is independent of the walk and with heavy tail. We show the convergence of the sum of independent copies of Zn 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 Sn had independent increments limits.