Recent advances in the theory of Edgeworth expansion allow rigorous estimation of remainder terms, under explicit regularity conditions. However, even in the important case of Student's $t$ statistic, the moment conditions required are unrealistically stringent. In this note we relax those conditions to the minimum needed to define terms in the Edgeworth expansion.
"Edgeworth Expansion for Student's $t$ Statistic Under Minimal Moment Conditions." Ann. Probab. 15 (3) 920 - 931, July, 1987. https://doi.org/10.1214/aop/1176992073