Stable limit theorems for additive functionals of one-dimensional diffusion processes

TL;DR

This paper establishes stable central limit theorems for additive functionals of positive recurrent one-dimensional diffusion processes, identifying conditions under which their fluctuations resemble an alpha-stable process over large times.

Contribution

It provides explicit conditions on additive functionals ensuring their fluctuations converge to an alpha-stable process, extending classical CLT results for diffusions.

Findings

Conditions for stable convergence of additive functionals

Explicit characterization of alpha-stable fluctuation behavior

Extension of classical CLT to stable laws for diffusions

Abstract

We consider a positive recurrent one-dimensional diffusion process with continuous coefficients and we establish stable central limit theorems for a certain type of additive functionals of this diffusion. In other words we find some explicit conditions on the additive functional so that its fluctuations behave like some $\alpha$-stable process in large time for $\alpha\in(0,2]$.