Tempered stable laws as random walk limits

TL;DR

This paper introduces random walk models that converge to tempered stable laws, which modify stable laws by reducing large jumps, providing a physical basis for phenomena in statistical physics.

Contribution

It develops new random walk models that converge to tempered stable laws under a triangular array scheme, linking probabilistic models to physical applications.

Findings

Models converge to tempered stable laws

Retain power law behavior at infinity

Applicable to statistical physics phenomena

Abstract

Stable laws can be tempered by modifying the L\'evy measure to cool the probability of large jumps. Tempered stable laws retain their signature power law behavior at infinity, and infinite divisibility. This paper develops random walk models that converge to a tempered stable law under a triangular array scheme. Since tempered stable laws and processes are useful in statistical physics, these random walk models can provide a basic physical model for the underlying physical phenomena.