  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.
