A method is described for deriving the sampling moments of means of random vectors obtained by sampling without replacement from a finite $k$-variate population of $n$ vector members. A table of results is presented listing the moments of order less than or equal to six as a function of the population moments. These moments were originally derived, in a less general form, in the course of developing the Simplex-Sum Designs discussed in . Their possible wider applicability to sampling problems, however, motivated the extension of the work to the general formulas given here.
"Sampling Moments of Means from Finite Multivariate Populations." Ann. Math. Statist. 32 (2) 406 - 413, June, 1961. https://doi.org/10.1214/aoms/1177705049