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

الأصل · EN

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.