This paper is devoted to rejective sampling. We provide an expansion of joint inclusion probabilities of any order in terms of the inclusion probabilities of order one, extending previous results by Hájek (1964) and Hájek (1981) and making the remainder term more precise. Following Hájek (1981), the proof is based on Edgeworth expansions. The main result is applied to derive bounds on higher order correlations, which are needed for the consistency and asymptotic normality of several complex estimators.
"Approximation of rejective sampling inclusion probabilities and application to high order correlations." Electron. J. Statist. 6 1967 - 1983, 2012. https://doi.org/10.1214/12-EJS736