Information geometry of tempered stable processes

TL;DR

This paper explores the geometric structure of tempered stable processes, deriving divergence measures, Fisher information, and connections, and discusses applications with various process examples.

Contribution

It introduces the information geometry framework for tempered stable processes, including divergence, Fisher information, and connections, with practical examples.

Findings

Derived the $oldsymbol{ extit{ ext{alpha}}}$-divergence between tempered stable processes.

Computed Fisher information matrices and $oldsymbol{ extit{ ext{alpha}}}$-connections.

Presented applications and examples of different tempered stable processes.

Abstract

We find the information geometry of tempered stable processes. Beginning with the derivation of $\alpha$-divergence between two tempered stable processes, we obtain the corresponding Fisher information matrices and the $\alpha$-connections on their statistical manifolds. Furthermore, we explore statistical applications of this geometric framework. Various tempered stable processes such as generalized tempered stable processes, classical tempered stable processes, and rapidly-decreasing tempered stable processes are presented as illustrative examples.