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

Abstract

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.

References in corpus (1)

• Discrete approximation of a stable self-similar stationary increments process

Cited by in corpus (1)

• A local limit theorem for random walks in random scenery and on randomly oriented lattices