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This paper is devoted to rejective sampling. We provide an expansion of jointinclusion probabilities of any order in terms of the inclusion probabilities oforder one, extending previous results by H\'ajek (1964) and H\'ajek (1981) andmaking the remainder term more precise. Following H\'ajek (1981), the proof isbased on Edgeworth expansions. The main result is applied to derive bounds onhigher order correlations, which are needed for the consistency and asymptoticnormality of several complex estimators.

Approximation of rejective sampling inclusion probabilities and application to high order correlations