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

TL;DR

This paper demonstrates that sums of dependent walks collecting heavy-tailed rewards, when properly normalized, converge to a stable motion characterized by the local time of fractional Brownian motion, extending previous independent-increment results.

Contribution

It introduces a new convergence result for dependent walks in a random scenery with heavy tails to a stable motion involving fBm local time, broadening prior independent-increment findings.

Findings

Convergence of dependent walks to stable motions with fBm local time.

Extension of previous independent-increment results to dependent walks.

Heavy-tailed scenery influences the limiting stable process.

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.