Fractional Edgeworth expansions for one-dimensional heavy-tailed random variables and applications

TL;DR

This paper develops fractional Edgeworth expansions for heavy-tailed lattice random variables in the domain of attraction of stable laws, providing convergence rates, Green's function expansions, and applications to stochastic PDEs.

Contribution

It introduces a class of fractional Edgeworth expansions for heavy-tailed variables and analyzes their stability, convergence, and applications to stochastic PDEs.

Findings

Sharp convergence rates to stable laws in local CLT

Explicit Green's function expansions for heavy-tailed variables

Analysis of fluctuations in stochastic PDEs driven by heavy-tailed random walks

Abstract

In this article, we study a class of lattice random variables in the domain of attraction of an $\alpha$-stable random variable with index $\alpha \in (0,2)$ which satisfy a truncated fractional Edgeworth expansion. Our results include studying the class of such fractional Edgeworth expansions under simple operations, providing concrete examples; sharp rates of convergence to an $\alpha$-stable distribution in a local central limit theorem; Green's function expansions; and finally fluctuations of a class of discrete stochastic PDE's driven by the heavy-tailed random walks belonging to the class of fractional Edgeworth expansions.