On the distribution of sample mean in finite populations. I

We obtain Bergström-type [2] asymptotic expansions for sample mean in finite population [8]. Analogues of the Cornish-Fisher transformation are obtained in the cases of limit distributions from class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{L}$$ \end{document} and distributions of sums of independent identically distributed random variables (in [2], the Bergström equality is used).