An asymptotic expansion is obtained for the distribution function of the standardized mean of a sample of $s$ observations taken randomly without replacement from a finite population of $n$ numbers. The expansion is given to order $1/n$ and agrees with the formal Edgeworth expansion. The proof of the result is obtained using an approximation to the characteristic function of the standardized sum.
"An Asymptotic Expansion for Samples from a Finite Population." Ann. Statist. 6 (5) 1005 - 1011, September, 1978. https://doi.org/10.1214/aos/1176344306