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

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