Discrete approximation of a stable self-similar stationary increments process

TL;DR

This paper develops new discrete approximation and convergence results for a class of stable self-similar stationary increments processes, extending known results beyond stable Lévy motions and fractional Brownian motion.

Contribution

It provides the first convergence theorems for the random rewards schema in a general setting, linking to random walk in random scenery models.

Findings

Proves convergence of the random rewards schema for stable self-similar processes.

Establishes strong connections with random walk in random scenery models.

Analyzes path properties of the limit process.

Abstract

The aim of this paper is to present a result of discrete approximation of some class of stable self-similar stationary increments processes. The properties of such processes were intensively investigated, but little is known on the context in which such processes can arise. To our knowledge, discretisation and convergence theorems are available only in the case of stable L\'evy motions and fractional Brownian motions. This paper yields new results in this direction. Our main result is the convergence of the random rewards schema, which was firstly introduced by Cohen and Samorodnitsky, and that we consider in a more general setting. Strong relationships with Kesten and Spitzer's random walk in random sceneries are evidenced. Finally, we study some path properties of the limit process.