Random walks in doubly random scenery

TL;DR

This paper introduces a new class of stable self-similar processes with stationary increments, represented via random walks in doubly random scenery, expanding the known instances of these processes as limits.

Contribution

It constructs a model that generates a broad range of processes within this new class as limits, beyond the single instance previously identified.

Findings

A new representation of stable self-similar processes using random walks in doubly random scenery.

Demonstration of multiple processes in the class as limits, not just a single instance.

Extension of the functional limit theorem to encompass a wider set of processes.

Abstract

We provide a random walk in random scenery representation of a new class of stable self-similar processes with stationary increments introduced recently by Jung, Owada and Samorodnitsky. In the functional limit theorem they provided, only a single instance of this class arose as a limit. We construct a model in which a significant portion of processes in this new class is obtained as a limit.