Abstract
In this article, we study a class of lattice random variables in the domain of attraction of an α-stable random variable with index 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 α-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.
Funding Statement
L. Chiarini was financially supported by CAPES and the NWO grant OCENW.KLEIN.083. M. Jara was funded by the ERC Horizon 2020 grant 715734, the CNPq grant 305075/2017-9 and the FAPERJ grant E-29/203.012/201. W. M. Ruszel is funded by OCENW.KLEIN.083 and the Vidi grant VI.Vidi.213.112 from the Dutch Research Council.
Acknowledgments
The authors would like to thank S. Frómeta for conversations about early versions of this article. We would also like to thank the anonymous referees, their suggestions helped to significantly improve this article.
Citation
Leandro Chiarini. Milton Jara. Wioletta M. Ruszel. "Fractional Edgeworth expansions for one-dimensional heavy-tailed random variables and applications." Electron. J. Probab. 28 1 - 42, 2023. https://doi.org/10.1214/23-EJP996
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